# questions

tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
the-marks-scored-by-two-students-a-and-b-in-six-subjects-are-given-below-a-b-71-55-56-74-45-83-89-54-54-38-44-52-which-one-of-the-following-statements

slug:
the-marks-scored-by-two-students-a-and-b-in-six-subjects-are-given-below-a-b-71-55-56-74-45-83-89-54-54-38-44-52-which-one-of-the-following-statements-67379

question:
<div class="question">The marks scored by two students A and B in six subjects are given below:<br/>$\begin{array}{ccccccc}\text { A } &amp; 71 &amp; 56 &amp; 45 &amp; 89 &amp; 54 &amp; 44 \\ \text { B } &amp; 55 &amp; 74 &amp; 83 &amp; 54 &amp; 38 &amp; 52\end{array}$<br/>Which one of the following statements is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The average scores of $\mathrm{A}$ and $\mathrm{B}$ are same but $\mathrm{A}$ is consistent</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The average scores of $A$ and $B$ are not same but $A$ is consistent</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The average scores of $\mathrm{A}$ and $\mathrm{B}$ are same but $\mathrm{B}$ is consistent</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">The average scores of $\mathrm{A}$ and $\mathrm{B}$ are not same but $\mathrm{B}$ is consistent $\quad$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The average scores of $\mathrm{A}$ and $\mathrm{B}$ are not same but $\mathrm{B}$ is consistent $\quad$</span> </div>

answer:
<div class="solution">Average score of $\mathrm{A}=\frac{71+56+45+89+54+44}{6}$ $=\frac{359}{6}=59.83$<br/>and average score of $\mathrm{B}=\frac{55+74+83+54+38+52}{6}=\frac{356}{6}=59.33$<br/>Variation is lesser in case of B than A. So, the average scores of A and B are not same but B is consistent.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
the-marks-scored-by-two-students-a-and-b-in-six-subjects-are-given-below-beginarraylllllll-hline-a-71-56-55-75-54-49-hline-b-55-74-83-54-38-52-hline-e

slug:
the-marks-scored-by-two-students-a-and-b-in-six-subjects-are-given-below-beginarraylllllll-hline-a-71-56-55-75-54-49-hline-b-55-74-83-54-38-52-hline-e-64428

question:
<div class="question">The marks scored by two students $A$ and $B$ in six subjects are given below \begin{array}{lllllll}<br/>\hline A &amp; 71 &amp; 56 &amp; 55 &amp; 75 &amp; 54 &amp; 49 \\<br/>\hline B &amp; 55 &amp; 74 &amp; 83 &amp; 54 &amp; 38 &amp; 52 \\<br/>\hline<br/>\end{array} Which one of the following statements is most appropriate'</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Theaverage scores of $A$ and $B$ are same but $A$ is consistent</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The average scores of $A$ and $B$ are not same but $A$ is consistent</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The average scores of $A$ and $B$ are same but $B$ is consistent</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The average scores of $A$ and $B$ are not same but $B$ is consistent</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The average scores of $A$ and $B$ are not same but $A$ is consistent</span> </div>

answer:
<div class="solution">Average of marks of A $=\frac{71+56+55+75+54+49}{6}=\frac{360}{6}=60$<br/>and SD<br/>$=\sqrt{\frac{121+16+25+225+36+121}{6}}=\sqrt{\frac{544}{6}}=9.52$<br/>Average of marks of B<br/>$=\frac{55+74+83+54+38+52}{6}=\frac{356}{6}=59.3$<br/>Thus, the average scores of $A$ and $B$ are not same but $A$ is consistent</div>


title: the-matrix-0-4-i-4-i-0-is
slug: the-matrix-0-4-i-4-i-0-is-56253
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The matrix $\left[\begin{array}{cc}0 &amp; -4+i \\ 4+i &amp; 0\end{array}\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">symmetric</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">skew-symmetric</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Hermitian</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">skew-Hermitian</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">skew-Hermitian</span> </div>

answer:
<div class="solution">The given matrix $\mathrm{A}=\left[\begin{array}{cc}0 &amp; -4+\mathrm{i} \\ 4+\mathrm{i} &amp; 0\end{array}\right]$<br/>Now, from options:<br/>From option (a): For Symmetric matrix<br/>$\mathrm{A}^{\mathrm{T}}=\mathrm{A}$<br/>Now, $\mathrm{A}^{\mathrm{T}}=\left[\begin{array}{cc}0 &amp; 4+\mathrm{i} \\ -4+\mathrm{i} &amp; 0\end{array}\right] \neq \mathrm{A}$<br/>$\therefore$ The given matrix is not symmetric<br/>$\therefore \quad$ option (a) is wrong. From option (b): For Skew-symmetric matrix $\mathrm{A}^{\mathrm{T}}=-\mathrm{A}$<br/>$=\left[\begin{array}{cc}0 &amp; 4+i \\ -4+i &amp; 0\end{array}\right] \neq-\mathrm{A}$<br/>$\therefore \quad$ Given matrix is not skew-symmetric<br/>$\therefore \quad$ option (b) is wrong. From option (c): For Hermitian matrix<br/>$\mathrm{A}^{\mathrm{T}}=\overline{\mathrm{A}}$, where $\overline{\mathrm{A}}$ is conjugate of matrix $\mathrm{A}$<br/>$\overline{\mathrm{A}}=\left[\begin{array}{cc}0 &amp; -4-\mathrm{i} \\ 4-\mathrm{i} &amp; 0\end{array}\right] \neq \mathrm{A}^{\mathrm{T}}$<br/>$\therefore \quad$ option (c) is wrong. From option (d): For Skew-Hermitian matrix The diagonal element of a skew-hermitian matrix are pure imaginary or zero.<br/>$\mathrm{A}=\left[\begin{array}{cc}0 &amp; -4+\mathrm{i} \\ 4+\mathrm{i} &amp; 0\end{array}\right]$<br/>Here, diagonal element indicates that the given matrix is skew-hermitian matrix.<br/>$\therefore$ option (d) is correct.</div>


title: the-matrix-a-1-1-2-3-x-1-7-2-1-x-3-will-have-inverse-for-every-real-number-x-except-for
slug: the-matrix-a-1-1-2-3-x-1-7-2-1-x-3-will-have-inverse-for-every-real-number-x-except-for-29383
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The matrix $\mathrm{A}=\left[\begin{array}{ccc}1 &amp; 3 &amp; 2 \\ 1 &amp; x-1 &amp; 1 \\ 2 &amp; 7 &amp; x-3\end{array}\right]$ will have inverse for every real number $x$ except for</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x=\frac{11 \pm \sqrt{5}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=\frac{9 \pm \sqrt{5}}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=\frac{11 \pm \sqrt{3}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=\frac{9 \pm \sqrt{3}}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=\frac{11 \pm \sqrt{5}}{2}$</span> </div>

answer:
<div class="solution">$A=\left[\begin{array}{ccc}1 &amp; 3 &amp; 2 \\ 1 &amp; x-1 &amp; 1 \\ 2 &amp; 7 &amp; x-3\end{array}\right]$<br/>$|\mathrm{A}|=1[(\mathrm{x}-1)(\mathrm{x}-3)-7]-3[(\mathrm{x}-3)-2]+2[7-2(\mathrm{x}-1)]$<br/>$=x^{2}-11 x+29$<br/>If inverse will not exist then $|\mathrm{A}|=0$ $x^{2}-11 x+29=0$<br/>$\mathrm{x}=\frac{11 \pm \sqrt{5}}{2}$</div>


title: the-matrix-a-1-2-2-2-satisfies-which-one-of-the-following-polynomial-equations
slug: the-matrix-a-1-2-2-2-satisfies-which-one-of-the-following-polynomial-equations-54943
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The matrix $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 2\end{array}\right]$ satisfies which one of the following polynomial equations?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$A^{2}+3 A+2 I=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A^{2}+3 A-2 I=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$A^{2}-3 A-2 I=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A^{2}-3 A+2 I=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A^{2}-3 A-2 I=0$</span> </div>

answer:
<div class="solution">Given that, $A=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 2\end{array}\right]$<br/>$\therefore \quad A^{2}=\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 2\end{array}\right]\left[\begin{array}{ll}1 &amp; 2 \\ 2 &amp; 2\end{array}\right]=\left[\begin{array}{cc}1+4 &amp; 2+4 \\ 2+4 &amp; 4+4\end{array}\right]$<br/>$=\left[\begin{array}{ll}5 &amp; 6 \\ 6 &amp; 8\end{array}\right]$<br/>Let $\mathrm{A}^{2}+\mathrm{xA}+\mathrm{yI}=0$ where $\mathrm{x}$ and $\mathrm{y}$ are constant.<br/>$\Rightarrow\left[\begin{array}{ll}5 &amp; 6 \\ 6 &amp; 8\end{array}\right]+\left[\begin{array}{cc}x &amp; 2 x \\ 2 x &amp; 2 x\end{array}\right]+\left[\begin{array}{ll}y &amp; 0 \\ 0 &amp; y\end{array}\right]=\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0\end{array}\right]$<br/>$\Rightarrow\left[\begin{array}{cc}5+x+y &amp; 6+2 x \\ 6+2 x &amp; 8+2 x+y\end{array}\right]=\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0\end{array}\right]$<br/>So, $\begin{aligned} 6+2 \mathrm{x}=0 &amp; \Rightarrow \mathrm{x}=-3 \\ 5+\mathrm{x}+\mathrm{y}=0 &amp; \Rightarrow \mathrm{y}=-5-\mathrm{x}=-2 \\ &amp; \Rightarrow \mathrm{A}^{2}-3 \mathrm{~A}-2 \mathrm{I}=0 \end{aligned}$</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
the-matrix-a-has-x-rows-and-x-5-columns-the-matrix-b-has-y-rows-and-11-y-columns-both-ab-and-ba-exist-what-are-the-values-of-x-and-y-respectively

slug:
the-matrix-a-has-x-rows-and-x-5-columns-the-matrix-b-has-y-rows-and-11-y-columns-both-ab-and-ba-exist-what-are-the-values-of-x-and-y-respectively-62411

question:
<div class="question">The matrix A has $x$ rows and $x+5$ columns. The matrix $B$ has y rows and $11-y$ columns. Both $\mathrm{AB}$ and BA exist. What are the values of $\mathrm{x}$ and $\mathrm{y}$ respectively?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8 and 3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3 and 4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3 and 8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8 and 8</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">3 and 8</span> </div>

answer:
<div class="solution">Given, Matrix A has $x$ rows and $x+5$ columns Matrix $\mathrm{B}$ has $\mathrm{y}$ rows and $11-\mathrm{y}$ columns. Also given $\mathrm{AB}$ and $\mathrm{BA}$ exists. If AB exists, then the number of rows in A most be equal to number of columns in $\mathrm{B}$.<br/>i.e., $x=11-y$ $\ldots .(1)$<br/>If BA exists, then the number of rows in $\mathrm{B}$ must be equal to number of rows in $\mathrm{A}$.<br/>i.e., $x+5=y$ $\Rightarrow 11-y+5=y($ from $(1))$<br/>$\Rightarrow 2 y=16$<br/>$\Rightarrow y=8$<br/>$(1) \Rightarrow x=11-8=3$<br/>So, $x=3, y=8$.</div>


title: the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals-1
tags: ['Mathematics', 'Matrices', 'JEE Main']
source_db: MarksBatch2_P3.db
slug:
the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals-1-34705

question:
<div class="question">The matrix \(A^2+4 A-5 I\), where \(I\) is identity matrix and \(A=\left[\begin{array}{cc}1 &amp; 2 \\ 4 &amp; -3\end{array}\right]\), equals :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4\left[\begin{array}{cc}0 &amp; -1 \\ 2 &amp; 2\end{array}\right]\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(32\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(32\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 0\end{array}\right]\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; A^2+4 A-5 I=A \times A+4 A-5 I \\<br/>= &amp; {\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right] \times\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]+4\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]-5\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9 &amp; -4 \\<br/>-8 &amp; 17<br/>\end{array}\right]+\left[\begin{array}{cc}<br/>4 &amp; 8 \\<br/>16 &amp; -12<br/>\end{array}\right]-\left[\begin{array}{cc}<br/>5 &amp; 0 \\<br/>0 &amp; 5<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9+4-5 &amp; -4+8-0 \\<br/>-8+16-0 &amp; 17-12-5<br/>\end{array}\right]=\left[\begin{array}{cc}<br/>8 &amp; 4 \\<br/>8 &amp; 0<br/>\end{array}\right] } \\<br/>= &amp; 4\left[\begin{array}{cc}<br/>2 &amp; 1 \\<br/>2 &amp; 0<br/>\end{array}\right]<br/>\end{aligned}\)</div>


title: the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
slug:
the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals-43455

question:
<div class="question">The matrix \(A^2+4 A-5 I\), where \(I\) is identity matrix and \(A=\left[\begin{array}{cc}1 &amp; 2 \\ 4 &amp; -3\end{array}\right]\), equals :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4\left[\begin{array}{cc}0 &amp; -1 \\ 2 &amp; 2\end{array}\right]\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(32\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(32\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 0\end{array}\right]\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; A^2+4 A-5 I=A \times A+4 A-5 I \\<br/>= &amp; {\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right] \times\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]+4\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]-5\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9 &amp; -4 \\<br/>-8 &amp; 17<br/>\end{array}\right]+\left[\begin{array}{cc}<br/>4 &amp; 8 \\<br/>16 &amp; -12<br/>\end{array}\right]-\left[\begin{array}{cc}<br/>5 &amp; 0 \\<br/>0 &amp; 5<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9+4-5 &amp; -4+8-0 \\<br/>-8+16-0 &amp; 17-12-5<br/>\end{array}\right]=\left[\begin{array}{cc}<br/>8 &amp; 4 \\<br/>8 &amp; 0<br/>\end{array}\right] } \\<br/>= &amp; 4\left[\begin{array}{cc}<br/>2 &amp; 1 \\<br/>2 &amp; 0<br/>\end{array}\right]<br/>\end{aligned}\)</div>


title: the-matrix-aleftbeginarrayccfrac1sqrt2-frac1sqrt2-frac1sqrt2-frac1sqrt2endarrayright-is
slug: the-matrix-aleftbeginarrayccfrac1sqrt2-frac1sqrt2-frac1sqrt2-frac1sqrt2endarrayright-is-30417
tags: ['Mathematics', 'Matrices', 'AP EAMCET', 'AP EAMCET 2020 (17 Sep Shift 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The matrix \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} &amp; \frac{-1}{\sqrt{2}}\end{array}\right]\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Unitary</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Orthogonal</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Nilpotent</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Involutory</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Nilpotent</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>A &amp; =\left[\begin{array}{cc}<br/>\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}} \\<br/>-\frac{1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}}<br/>\end{array}\right] \\<br/>A^2 &amp; =\left[\begin{array}{cc}<br/>\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}} \\<br/>-\frac{1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}}<br/>\end{array}\right]\left[\begin{array}{cc}<br/>\frac{1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}} \\<br/>-\frac{1}{\sqrt{2}} &amp; -\frac{1}{\sqrt{2}}<br/>\end{array}\right] \\<br/>\Rightarrow \quad A^2 &amp; =\left[\begin{array}{cc}<br/>\frac{1}{2}-\frac{1}{2} &amp; \frac{1}{2}-\frac{1}{2} \\<br/>-\frac{1}{2}+\frac{1}{2} &amp; -\frac{1}{2}+\frac{1}{2}<br/>\end{array}\right] \\<br/>A^2 &amp; =\left[\begin{array}{ll}<br/>0 &amp; 0 \\<br/>0 &amp; 0<br/>\end{array}\right] \text { null matrix }<br/>\end{aligned}\)<br/>So, \(A\) is nilpotent matrix</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
the-maximum-three-digit-integer-in-the-decimal-system-will-be-represented-in-the-binary-system-by-which-one-of-the-following

slug:
the-maximum-three-digit-integer-in-the-decimal-system-will-be-represented-in-the-binary-system-by-which-one-of-the-following-39290

question:
<div class="question">The maximum three digit integer in the decimal system will be represented in the binary system by which one of the following?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1111110001</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">111111110</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1111100111</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1111000111</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1111100111</span> </div>

answer:
<div class="solution">The maximum three digit integer in decimal system $=999$. We go on dividing till we get a dividend $   &lt; 2$ and write remainders from last to firstas shown below:<br/>\begin{array}{l|l|l}<br/>\hline 2 &amp; 999 &amp; \\<br/>\hline 2 &amp; 499 &amp; 1 \\<br/>\hline 2 &amp; 249 &amp; 1 \\<br/>\hline 2 &amp; 124 &amp; 1 \\<br/>\hline 2 &amp; 62 &amp; 0 \\<br/>\hline 2 &amp; 31 &amp; 0 \\<br/>\hline 2 &amp; 15 &amp; 1 \\<br/>\hline 2 &amp; 7 &amp; 1 \\<br/>\hline 2 &amp; 3 &amp; 1 \\<br/>\hline &amp; 1 &amp; 1 \\<br/>\hline<br/>\end{array}<br/>Hence, $(999)_{10}=(1111100111)_{2}$</div>


title: the-maximum-value-of-sin-x-5-cos-x-5-where-x-0-2-is-attained-at
slug: the-maximum-value-of-sin-x-5-cos-x-5-where-x-0-2-is-attained-at-99020
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of $\quad$ $\sin \left(x+\frac{\pi}{5}\right)+\cos \left(x+\frac{\pi}{5}\right)$, where $x \in\left(0, \frac{\pi}{2}\right)$, is attained at</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{20}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{15}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{10}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{20}$</span> </div>

answer:
<div class="solution">Let $\mathrm{y}=\sin \left(\mathrm{x}+\frac{\pi}{5}\right)+\cos \left(\mathrm{x}+\frac{\pi}{5}\right)$<br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\cos \left(\mathrm{x}+\frac{\pi}{5}\right)-\sin \left(\mathrm{x}+\frac{\pi}{5}\right)$<br/>for maximum value, $\frac{\mathrm{dy}}{\mathrm{dx}}=0$. $\therefore \cos \left(x+\frac{\pi}{5}\right)-\sin \left(x+\frac{\pi}{5}\right)=0$<br/>$\Rightarrow \frac{\cos \left(x+\frac{\pi}{5}\right)}{\sin \left(x+\frac{\pi}{5}\right)}=1$<br/>$\Rightarrow \cot \left(x+\frac{\pi}{5}\right)=\cot \frac{\pi}{4}$<br/>$\Rightarrow x+\frac{\pi}{5}=\frac{\pi}{4}$<br/>$\Rightarrow x=\frac{\pi}{4}-\frac{\pi}{5}=\frac{5 \pi-4 \pi}{20}=\frac{\pi}{20}$</div>


title: the-maximum-value-of-sin-x-6-cos-x-6-in-the-interval-0-2-is-attained-at
slug: the-maximum-value-of-sin-x-6-cos-x-6-in-the-interval-0-2-is-attained-at-73983
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of $\sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)$ in the<br/>interval $\left(0, \frac{\pi}{2}\right)$ is attained at</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{12}$</span> </div>

answer:
<div class="solution">$\quad \sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)$<br/>$=\sqrt{2}\left[\frac{1}{\sqrt{2}} \sin \left(x+\frac{\pi}{6}\right)+\frac{1}{\sqrt{2}} \cos \left(x+\frac{\pi}{6}\right)\right]$<br/>$=\sqrt{2}\left[\sin \left(x+\frac{\pi}{6}\right) \cos \frac{\pi}{4}+\cos \left(x+\frac{\pi}{6}\right) \sin \frac{\pi}{4}\right]$<br/>$=\sqrt{2}\left[\sin \left(x+\frac{\pi}{6}+\frac{\pi}{4}\right)\right]$<br/>$[\because \sin (A+B)=\sin A \cos B+\cos A \operatorname{Sin} B]$<br/>$=\sqrt{2}\left[\sin \left(x+\frac{5 \pi}{12}\right)\right]$<br/>Given interval is $\left(0, \frac{\pi}{2}\right)$ For, maximum value $x+\frac{5 \pi}{12}=\frac{\pi}{2}$ $\Rightarrow x=\frac{\pi}{2}-\frac{5 \pi}{12}=\frac{6 \pi-5 \pi}{12}=\frac{\pi}{12}$</div>


title: the-maximum-value-of-the-function-f-x-x-3-2-x-2-4-x-6-exists-at
slug: the-maximum-value-of-the-function-f-x-x-3-2-x-2-4-x-6-exists-at-66027
question: <div class="question">The maximum value of the function $f(x)=x^{3}+2 x^{2}-4 x+6$ exists at</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad x=-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=-1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad x=-2$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+2 \mathrm{x}^{2}-4 \mathrm{x}+6$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=3 \mathrm{x}^{2}+4 \mathrm{x}-4$<br/>$\mathrm{f}^{\prime \prime}(\mathrm{x})=6 \mathrm{x}+4$<br/>Put $\mathrm{f}^{\prime}(\mathrm{x})=0$<br/>$3 \mathrm{x}^{2}+4 \mathrm{x}-4=0$<br/>$\mathrm{x}=-2, \frac{2}{3}$<br/>$\mathrm{f}^{\prime \prime}(-2)=6 \mathrm{x}-2+4=-8   &lt; 0$<br/>$\mathrm{f}^{\prime \prime}\left(\frac{2}{3}\right)=6 \times \frac{2}{3}+4=8&gt;0$<br/>Value is maximumat $\mathrm{x}=-2$</div>


title: the-maximum-value-of-x-in-x-is
slug: the-maximum-value-of-x-in-x-is-94303
question: <div class="question">The maximum value of $\frac{\operatorname{In} x}{x}$ is</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">e</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\mathrm{e}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{\mathrm{e}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\mathrm{e}}$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{\ln \mathrm{x}}{\mathrm{x}}$<br/>$f^{\prime}(x)=\frac{\left(\frac{1}{x}\right) x-\ell n x}{x^{2}}=\frac{1-\ell n x}{x^{2}}$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=0 \Rightarrow \frac{1-\ln \mathrm{x}}{\mathrm{x}^{2}}=0 \Rightarrow 1-\ell \mathrm{nx}=0$<br/>$\Rightarrow \ell \mathrm{nx}=1 \quad \Rightarrow \mathrm{x}=\mathrm{e}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'KVPY', 'KVPY 2017 (19 Nov SB/SX)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">257</span> </div>
source_db: MarksBatch2_P3.db
title:
the-maximum-work-in-kj-mol-1-that-can-be-derived-from-complete-combustion-of-1-mol-of-co-at-298-k-and-1-atm-is-standard-enthalpy-of-combustion-of-co-2

slug:
the-maximum-work-in-kj-mol-1-that-can-be-derived-from-complete-combustion-of-1-mol-of-co-at-298-k-and-1-atm-is-standard-enthalpy-of-combustion-of-co-2-37580

question:
<div class="question">The maximum work (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) that can be derived from complete combustion of $1 \mathrm{~mol}$ of $\mathrm{CO}$ at $298 \mathrm{~K}$ and $1 \mathrm{~atm}$ is<br/>[Standard enthalpy of combustion of $\mathrm{CO}=-283.0 \mathrm{~kJ} \mathrm{~mol}^{-1} ;$ standard molar entropies at $298 \mathrm{~K} ; \mathrm{S}_{\mathrm{O} 2}=205.1 \mathrm{~J} \mathrm{~mol}^{-1}$ $\left.\mathrm{S}_{\mathrm{CO}}=197.7 \mathrm{~J} \mathrm{~mol}^{-1}, \mathrm{~S}_{\mathrm{CO} 2}=213.7 \mathrm{~J} \mathrm{~mol}^{-1}\right]$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">257</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">227</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">57</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">127</span> </li> </ul>

answer:
<div class="solution">$\mathrm{Co}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g}) \Delta \mathrm{N}=-283.5$<br/>$\Delta \mathrm{S}=213.7-197.7-\frac{20501}{2}=-86.5$<br/>$\Delta \mathrm{G}=-283-\frac{298 \times(-86.5)}{1000}=-2570 \mathrm{~kJ}$<br/>$\mathrm{~W}_{\max }=-\Delta \mathrm{G}=-(-257 \mathrm{~kJ})=257 \mathrm{~kJ}$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2023 (15 Apr Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-10-observations-are-20-and-8-respectively-later-on-it-was-observed-that-one-observation-was-recorded-as-50-instead-

slug:
the-mean-and-standard-deviation-of-10-observations-are-20-and-8-respectively-later-on-it-was-observed-that-one-observation-was-recorded-as-50-instead-18153

question:
<div class="question">The mean and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> observations are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> respectively. Later on, it was observed that one observation was recorded as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math>. Then the correct variance is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>Incorrect mean is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>x</mi><mi>i</mi></msub></mstyle><mn>10</mn></mfrac><mo>=</mo><mn>20</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>200</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>So, corrected mean is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>50</mn><mo>+</mo><mn>40</mn></mrow><mn>10</mn></mfrac><mo>=</mo><mfrac><mrow><mn>200</mn><mo>-</mo><mn>10</mn></mrow><mn>10</mn></mfrac><mo>=</mo><mn>19</mn></math></p><p>Incorrect variance is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><mn>400</mn><mo>=</mo><mn>64</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>4640</mn></math></p><p>Now, corrected variance is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msup><mn>40</mn><mn>2</mn></msup><mo>-</mo><msup><mn>50</mn><mn>2</mn></msup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mn>19</mn></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4640</mn><mo>-</mo><mn>900</mn></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mn>19</mn></mfenced><mn>2</mn></msup><mspace linebreak="newline"></mspace><mo>=</mo><mn>374</mn><mo>-</mo><mn>361</mn><mspace linebreak="newline"></mspace><mo>=</mo><mn>13</mn></math></p></blockquote></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-10-observations-x-1-x-2-x-3-x-10-are-x-and-respectively-let-10-is-added-to-x-1-x-2-x-9-and-90-is-subtracted-from-x-

slug:
the-mean-and-standard-deviation-of-10-observations-x-1-x-2-x-3-x-10-are-x-and-respectively-let-10-is-added-to-x-1-x-2-x-9-and-90-is-subtracted-from-x-57387

question:
<div class="question">The mean and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> observations <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>3</mn></msub><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>10</mn></msub></math> are <math><mover accent="false"><mrow><mi>x</mi></mrow><mo>Â¯</mo></mover></math> and <math><mi>Ï</mi></math> respectively. Let <math><mn>10</mn></math> is added to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>9</mn></msub></math> and <math><mn>90</mn></math> is subtracted from <math><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>.</mo></math> If still, the standard deviation is the same, then <math><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>Â¯</mo></mover></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>35</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mn>45</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>55</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>50</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>45</mn></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mfrac><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup><mn>10</mn></mfrac><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup></math> … <math><mfenced separators="|"><mrow><mn>1</mn></mrow></mfenced></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>10</mn><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mn>10</mn></msub><mo>-</mo><mn>90</mn><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup></mrow><mn>10</mn></mfrac><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup></math> … <math><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced></math><br/>From <math><mfenced separators="|"><mrow><mn>1</mn></mrow></mfenced></math> and <math><mo>(</mo><mn>2</mn><mo>)</mo></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup></mrow><mn>10</mn></mfrac><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>20</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mo>-</mo><mn>180</mn><mfenced separators="|"><mrow><msub><mi>x</mi><mn>10</mn></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><mn>100</mn><mo>×</mo><mn>9</mn><mo>+</mo><mn>8100</mn></mrow><mn>10</mn></mfrac></math><br/><math><mn>2</mn><mo>×</mo><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>9</mn></mrow></munderover><mrow><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>-</mo><mn>20</mn><mfenced separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>10</mn></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced><mo>+</mo><mn>90</mn><mo>+</mo><mn>810</mn><mo>=</mo><mn>0</mn></mrow></mrow></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>900</mn><mo>=</mo><mn>20</mn><mfenced separators="|"><mrow><msub><mi>x</mi><mn>10</mn></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mo>⇒</mo><msub><mi>x</mi><mn>10</mn></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mn>4</mn><mn>5</mn></math></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2022 (28 Jun Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">17</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-15-observations-are-found-to-be-8-and-3-respectively-on-rechecking-it-was-found-that-in-the-observations-20-was-mis

slug:
the-mean-and-standard-deviation-of-15-observations-are-found-to-be-8-and-3-respectively-on-rechecking-it-was-found-that-in-the-observations-20-was-mis-52000

question:
<div class="question">The mean and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> observations are found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> respectively. On rechecking it was found that, in the observations, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> was misread as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>. Then, the correct variance is equal to _____.</div>

answer:
<div class="solution"><p>Old mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>14</mn></msub><mo>+</mo><mn>5</mn></mrow><mn>15</mn></mfrac><mo>=</mo><mn>8</mn></math> (given)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>14</mn></msub><mo>=</mo><mn>115</mn></math></p><p>Correct mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>x</mi><mn>14</mn></msub><mo>+</mo><mn>20</mn></mrow><mn>15</mn></mfrac><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mi>new</mi></msub><mo>⇒</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mi>new</mi></msub><mo>=</mo><mn>9</mn></math></p><p>Given, old variance<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>⇒</mo><mfrac><mrow><msup><msub><mi>x</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>x</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msup><msub><mi>x</mi><mn>14</mn></msub><mn>2</mn></msup><mo>+</mo><msup><mn>5</mn><mn>2</mn></msup></mrow><mn>15</mn></mfrac><mo>-</mo><msup><mn>8</mn><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>⇒</mo><msup><msub><mi>x</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>x</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msup><msub><mi>x</mi><mn>14</mn></msub><mn>2</mn></msup><mo>=</mo><mn>1070</mn></math></p><p>Now <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>var</mi><mi>new</mi></msub><mo>=</mo><mfrac><mrow><msup><msub><mi>x</mi><mn>1</mn></msub><mn>2</mn></msup><mo>+</mo><msup><msub><mi>x</mi><mn>2</mn></msub><mn>2</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><msup><msub><mi>x</mi><mn>14</mn></msub><mn>2</mn></msup><mo>+</mo><msup><mn>20</mn><mn>2</mn></msup></mrow><mn>15</mn></mfrac><mo>-</mo><msup><mn>9</mn><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>1070</mn><mo>+</mo><mn>400</mn></mrow><mn>15</mn></mfrac><mo>-</mo><mn>81</mn><mo>=</mo><mn>17</mn></math></p></div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-20-observations-are-found-to-be-10-and-2-respectively-on-rechecking-it-was-found-that-an-observation-8-was-incorrec

slug:
the-mean-and-standard-deviation-of-20-observations-are-found-to-be-10-and-2-respectively-on-rechecking-it-was-found-that-an-observation-8-was-incorrec-63740

question:
<div class="question">The mean and standard deviation of $\mathbf{2 0}$ observations are found to be 10 and 2 respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases :<br/>(i) If wrong item is omitted<br/>(ii) If it is replaced by 12</div>

answer:
<div class="solution">$\begin{aligned}<br/>&amp;\text { Mean }=\frac{\sum x_i}{n}=\frac{\sum_{i=1}^{20} x_i}{20}=10 \\<br/>&amp;\Rightarrow \sum_{i=1}^{20} x_i=20 \times 10=200<br/>\end{aligned}$<br/>(i) Since, the wrong item whose observation 8 has been omitted, the correct mean $=\frac{\sum_{i=1}^{20} x-8}{19}$ because on omitting the wrong item, the total number of observations left are 19.<br/>Now, $\frac{\sum_{i=1}^{20}\left(x_i-\bar{x}\right)^2}{n}=\frac{\sum_{i=1}^{20} x_i^2}{n}-\bar{x}^2$<br/>Since, incorrect mean and variance are given as 10 and 4 ,<br/>so $\frac{\sum_{i=1}^{20} x_i^2}{20}-10^2=4$<br/>i.e., $\sum_{i=1}^{20} x_i^2=(4+100) 20=104 \times 20=2080$<br/>Correct variance $=\frac{\sum_{i=1}^{19}\left(x_i-10.11\right)^2}{19}$<br/>i.e., $\frac{\sum_{i=1}^{20} x_i^2-8^2}{19}-(10.11)^2$<br/>$\frac{2080-64}{19}-(10.11)^2=\frac{2016}{19}-(10.11)^2$<br/>$=106.105-102.212=3.893$<br/>Correct S.D $=\sqrt{3.893}=1.97$<br/>(ii) The wrong item which was 8 had been replaced by 12 , as such the correct mean<br/>$=\frac{\sum_{i=1}^{20} x_i-8+12}{20}=\frac{\sum_{i=1}^{20} x_i}{20}+\frac{4}{20}$<br/>$=\frac{200}{20}+\frac{1}{5}=10+0.2=10.2$<br/>Since, incorrect mean and variance are given as 10 and 4 ,<br/>So<br/>$\begin{aligned}<br/>&amp;\frac{\sum_{i=1}^{20} x_i^2}{20}-10^2=4 \Rightarrow \frac{\sum_{i=1}^{20} x_i^2}{20}=4+100=104 \\<br/>&amp;\Rightarrow \sum_{i=1}^{20} x_i^2=20 \times 104 \quad \text { i.e } \sum_{i=1}^{20} x_i^2=2080 \\<br/>&amp;\Rightarrow \text { Correct value of } \sum_i x_i^2=2080-8^2+12^2 \\<br/>&amp;=2080-64+144=2160<br/>\end{aligned}$<br/>Correct variance $=\frac{\sum_{i=1}^{20} x_i^2}{20}-10.2^2$<br/>$=\frac{2160}{20}-(10.2)^2=108-104.04=3.96$<br/>$\therefore \quad$ Correct $\mathrm{SD}=\sqrt{\text { variance }}$<br/>or $\quad \sqrt{3.96}=1.99(\mathrm{appx})$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2024 (06 Apr Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-20-observations-are-found-to-be-10-and-2-respectively-on-rechecking-it-was-found-that-an-observation-by-mistake-was

slug:
the-mean-and-standard-deviation-of-20-observations-are-found-to-be-10-and-2-respectively-on-rechecking-it-was-found-that-an-observation-by-mistake-was-45405

question:
<div class="question">The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.94</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{3.96}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{3.86}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3.96}$</span> </div>

answer:
<div class="solution">$\begin{aligned}<br/>&amp; \text { Mean }(\overline{\mathrm{x}})=10 \\<br/>&amp; \Rightarrow \frac{\Sigma \mathrm{x}_{\mathrm{i}}}{20}=10 \\<br/>&amp; \Sigma \mathrm{x}_{\mathrm{i}}=10 \times 20=200<br/>\end{aligned}$<br/>If 8 is replaced by 12, then $\Sigma x_i=200-8+12=204$<br/>$\therefore$ Correct mean $(\overline{\mathrm{x}})=\frac{\Sigma \mathrm{x}_{\mathrm{i}}}{20}$<br/>$=\frac{204}{20}=10.2$<br/>$\because$ Standard deviation $=2$<br/>$\therefore$ Variance $=(\text { S.D. })^2=2^2=4$<br/>$\Rightarrow \frac{\Sigma \mathrm{x}_{\mathrm{i}}^2}{20}-\left(\frac{\Sigma \mathrm{x}_{\mathrm{i}}}{20}\right)^2=4$<br/>$\Rightarrow \frac{\Sigma \mathrm{x}_{\mathrm{i}}^2}{20}-(10)^2=4$<br/>$\Rightarrow \frac{\Sigma \mathrm{x}_{\mathrm{i}}^2}{20}=104$<br/>$\Rightarrow \Sigma x_i^2=2080$<br/>Now, replaced ' 8 ' observations by ' 12 '<br/>Then, $\Sigma \mathrm{x}_{\mathrm{i}}^2=2080-8^2+12^2=2160$<br/>$\therefore$ Variance of removing observations<br/>$\begin{aligned}<br/>&amp; \Rightarrow \frac{\Sigma x_i^2}{20}-\left(\frac{\Sigma x_i}{20}\right)^2 \\<br/>&amp; \Rightarrow \frac{2160}{20}-(10.2)^2 \\<br/>&amp; \Rightarrow 108-104.04 \\<br/>&amp; \Rightarrow 3.96<br/>\end{aligned}$<br/>Correct standard deviation<br/>$=\sqrt{3.96}$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-20-observations-were-calculated-as-10-and-2-5-respectively-it-was-found-that-by-mistake-one-data-value-was-taken-as-1

slug:
the-mean-and-standard-deviation-of-20-observations-were-calculated-as-10-and-2-5-respectively-it-was-found-that-by-mistake-one-data-value-was-taken-as-1-50033

question:
<div class="question">The mean and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> observations were calculated as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn></math> respectively. It was found that by mistake one data value was taken as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>35</mn><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>β</mi></msqrt></math> are the mean and standard deviation respectively for correct data, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math> is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>25</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>11</mn><mo>,</mo><mn>25</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>11</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> </div>

answer:
<div class="solution"><p>Mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>+</mo><mn>25</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mn>10</mn></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi mathvariant="normal">x</mi><mn>19</mn></msub><mo>+</mo><mn>25</mn><mo>=</mo><mn>200</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>=</mo><mn>175</mn></math></p><p>New mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>+</mo><mn>35</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mfrac><mrow><mn>175</mn><mo>+</mo><mn>35</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mn>10</mn><mo>.</mo><mn>5</mn></math></p><p>S.D. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>25</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>25</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mi mathvariant="normal">x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>19</mn><mn>2</mn></msubsup><mo>=</mo><mn>1500</mn></math></p><p>New <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>. </mo><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>35</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mn>1500</mn><mo>+</mo><mo>(</mo><mn>35</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>110</mn><mo>.</mo><mn>25</mn><mo>)</mo></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>26</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2021 (26 Aug Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-20-observations-were-calculated-as-10-and-2-5-respectively-it-was-found-that-by-mistake-one-data-value-was-taken-as

slug:
the-mean-and-standard-deviation-of-20-observations-were-calculated-as-10-and-2-5-respectively-it-was-found-that-by-mistake-one-data-value-was-taken-as-81158

question:
<div class="question">The mean and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> observations were calculated as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn></math> respectively. It was found that by mistake one data value was taken as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>35</mn><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>β</mi></msqrt></math> are the mean and standard deviation respectively for correct data, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math> is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>25</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>11</mn><mo>,</mo><mn>25</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>11</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></span> </div>

answer:
<div class="solution"><p>Mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>+</mo><mn>25</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mn>10</mn></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi mathvariant="normal">x</mi><mn>19</mn></msub><mo>+</mo><mn>25</mn><mo>=</mo><mn>200</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>=</mo><mn>175</mn></math></p><p>New mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msub><mi>x</mi><mn>19</mn></msub><mo>+</mo><mn>35</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mfrac><mrow><mn>175</mn><mo>+</mo><mn>35</mn></mrow><mn>20</mn></mfrac><mo>=</mo><mn>10</mn><mo>.</mo><mn>5</mn></math></p><p>S.D. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>25</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>25</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mi mathvariant="normal">x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi mathvariant="normal">x</mi><mn>19</mn><mn>2</mn></msubsup><mo>=</mo><mn>1500</mn></math></p><p>New <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>. </mo><mo>=</mo><msqrt><mfrac><mrow><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>x</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msubsup><mi>x</mi><mn>19</mn><mn>2</mn></msubsup><mo>+</mo><mo>(</mo><mn>35</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mn>1500</mn><mo>+</mo><mo>(</mo><mn>35</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>20</mn></mfrac><mo>-</mo><mo>(</mo><mn>110</mn><mo>.</mo><mn>25</mn><mo>)</mo></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>26</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>10</mn><mo>.</mo><mn>5</mn><mo>,</mo><mn>26</mn><mo>)</mo></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2550</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-50-observations-are-50-and-10-later-on-it-was-decided-to-omit-an-observation-that-was-incorrectly-recorded-as-99-if

slug:
the-mean-and-standard-deviation-of-50-observations-are-50-and-10-later-on-it-was-decided-to-omit-an-observation-that-was-incorrectly-recorded-as-99-if-55691

question:
<div class="question">The mean and standard deviation of <math><mn>50</mn></math> observations are <math><mn>50</mn></math> and <math><mn>10</mn><mo>.</mo></math> Later on, it was decided to omit an observation that was incorrectly recorded as <math><mn>99</mn><mo>.</mo></math> If the variance of the remaining <math><mn>49</mn></math> observations is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>λ</mi><mn>49</mn></mfrac><mo>,</mo></math> then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> is equal to</div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>-</mo></mover><mo>=</mo><mfrac><mrow><mo>â</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mn>50</mn></mfrac><mo>=</mo><mn>50</mn></math><br/><math><mo>â</mo><mrow><mo stretchy="false">â</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow><mo>=</mo><mn>2500</mn></math><br/>Correct <math><mrow><mo stretchy="false">â</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow><mo>=</mo><mn>2500</mn><mo>-</mo><mn>99</mn><mo>=</mo><mn>2401</mn></math><br/>Correct <math><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover><mo>=</mo><mfrac><mrow><mn>2401</mn></mrow><mrow><mn>49</mn></mrow></mfrac><mo>=</mo><mn>49</mn></math><br/><math><msup><mrow><mi>Ï</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>â</mo><mn>100</mn><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mn>50</mn></mrow></mfrac><mo>-</mo><mn>2500</mn></math><br/><math><mo>â</mo><mrow><mo stretchy="false">â</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>=</mo><mn>130000</mn></math></p><p>Correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>130000</mn><mo>-</mo><msup><mn>99</mn><mn>2</mn></msup><mo>=</mo><mn>120199</mn></math><br/>Correct <math><msup><mrow><mi>Ï</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">â</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>120199</mn><mn>49</mn></mfrac><mo>-</mo><msup><mfenced separators="|"><mn>49</mn></mfenced><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>120199</mn><mo>-</mo><mn>117649</mn></mrow><mn>49</mn></mfrac><mo>=</mo><mfrac><mn>2550</mn><mn>49</mn></mfrac><mo>=</mo><mfrac><mi>Î»</mi><mn>49</mn></mfrac></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Î»</mi><mo>=</mo><mn>2550</mn></math><br/> </p></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">44</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-combined-mean-of-all-250-items

slug:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-combined-mean-of-all-250-items-72666

question:
<div class="question">The mean and standard deviation of 100 items are 50,5 and that of $\begin{array}{ll}150 \text { items are } 40,6 \text { respectively. } &amp; [20] 4-1]\end{array}$<br/>What is the combined mean of all 250 items?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">43</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">44</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">45</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">46</span> </li> </ul>

answer:
<div class="solution">Mean of 100 items $=\overline{\mathrm{x}}_{100}=50$<br/>Mean of 150 items $=\overline{\mathrm{x}}_{150}=40$<br/>Standard deviation of 100 items $=\sigma_{100}=5$ Standard deviation of 150 items $=\sigma_{150}=6$<br/>$\begin{aligned} \bar{x}_{250}=\frac{\mathrm{n}_{1} \bar{x}_{100}+\mathrm{n}_{2} \cdot \bar{x}_{150}}{\mathrm{n}_{1}+\mathrm{n}_{2}}=\frac{(100 \times 50)+(150 \times 40)}{100+150} \\=&amp; \frac{11000}{250}=44 \end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">75</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-combined-standard-deviation-of-all

slug:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-combined-standard-deviation-of-all-20014

question:
<div class="question">The mean and standard deviation of 100 items are 50,5 and that of $\begin{array}{ll}150 \text { items are } 40,6 \text { respectively. } &amp; [20] 4-1]\end{array}$<br/>What is the combined standard deviation of all 250 items?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 7.1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">73</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">75</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.7$</span> </li> </ul>

answer:
<div class="solution">Mean of 100 items $=\overline{\mathrm{x}}_{100}=50$<br/>Mean of 150 items $=\overline{\mathrm{x}}_{150}=40$<br/>Standard deviation of 100 items $=\sigma_{100}=5$ Standard deviation of 150 items $=\sigma_{150}=6$<br/>$\begin{array}{ll}\mathrm{d}_{1}=50-44=6 &amp; \mathrm{~d}_{1}^{2}=36 \\ \mathrm{~d}_{2}=40-44=-4 &amp; \mathrm{~d}_{2}^{2}=16\end{array}$<br/>$\sigma_{250}=\frac{\sqrt{\mathrm{n}_{1}\left(\sigma_{100}^{2}+\mathrm{d}_{1}^{2}\right)+\mathrm{n}_{2}\left(\sigma_{150}^{2}+\mathrm{d}_{2}^{2}\right)}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$<br/>$=\frac{\sqrt{390}}{5}=\frac{37.28}{5}=7.456=7.5$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$55.6$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-variance-of-all-250-items

slug:
the-mean-and-standard-deviation-of-100-items-are-505-and-that-of-150-items-are-40-6-respectively-20-4-1-what-is-the-variance-of-all-250-items-32439

question:
<div class="question">The mean and standard deviation of 100 items are 50,5 and that of $\begin{array}{ll}150 \text { items are } 40,6 \text { respectively. } &amp; [20] 4-1]\end{array}$<br/>What is the variance of all 250 items?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$50.6$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$53.3$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$55.6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$59.6$</span> </li> </ul>

answer:
<div class="solution">Mean of 100 items $=\overline{\mathrm{x}}_{100}=50$<br/>Mean of 150 items $=\overline{\mathrm{x}}_{150}=40$<br/>Standard deviation of 100 items $=\sigma_{100}=5$ Standard deviation of 150 items $=\sigma_{150}=6$<br/>Variance of all 250 items $=\left(\sigma_{250}\right)^{2}=(7.456)^{2}=55.6$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instead-of-40-for-on-1

slug:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instead-of-40-for-on-1-67908

question:
<div class="question">The mean and standard deviation of <math><mn>100</mn></math> observations were calculated as <math><mn>40</mn></math> and <math><mn>5.1</mn><mo>,</mo></math> respectively by a student who took by mistake <math><mn>50</mn></math> instead of <math><mn>40</mn></math> for one observation, then the correct standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>4</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>6</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p>Standard deviation <math><mfenced separators="|"><mrow><mi>σ</mi></mrow></mfenced><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math></p><p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>5.1</mn><mo>=</mo><msqrt><mfrac><mn>1</mn><mn>100</mn></mfrac><mo>×</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><msup><mfenced separators="|"><mn>40</mn></mfenced><mn>2</mn></msup></msqrt></math><br/>or, <math><mn>26.01</mn><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>×</mo></math> Incorrect <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><mn>1600</mn></math><br/>Therefore,<br/>Incorrect <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>100</mn><mo>(</mo><mn>26</mn><mo>.01</mo><mo>+</mo><mn>1600</mn><mo>)</mo><mo>=</mo><mn>162601</mn></math></p><p>Correct mean, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>x</mi><mo> </mo></mrow><mo>¯</mo></mover><mo>=</mo><mfrac><mn>3990</mn><mn>100</mn></mfrac><mo>=</mo><mn>39</mn><mo>.</mo><mn>9</mn></math><br/>Now, Correct <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>=</mo></math> Incorrect <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>50</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn>40</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>=</mo><mn>162601</mn><mo>-</mo><mn>2500</mn><mo>+</mo><mn>1600</mn><mo>=</mo><mn>161701</mn></math><br/>Therefore correct standard deviation<br/><math><mo>=</mo><msqrt><mfrac><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi></mi><mrow><mo stretchy="false">∑</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>161701</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>39.9</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt><mo>=</mo><msqrt><mn>1617.01</mn><mo>-</mo><mn>1592.01</mn></msqrt><mo>=</mo><msqrt><mn>25</mn></msqrt><mo>=</mo><mn>5</mn></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instead-of-40-for-on

slug:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instead-of-40-for-on-66622

question:
<div class="question">The mean and standard deviation of <math><mn>100</mn></math> observations were calculated as <math><mn>40</mn></math> and <math><mn>5.1</mn><mo>,</mo></math> respectively by a student who took by mistake <math><mn>50</mn></math> instead of <math><mn>40</mn></math> for one observation, then the correct standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>4</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>6</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p>Standard deviation <math><mfenced separators="|"><mrow><mi>σ</mi></mrow></mfenced><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math></p><p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>5.1</mn><mo>=</mo><msqrt><mfrac><mn>1</mn><mn>100</mn></mfrac><mo>×</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><msup><mfenced separators="|"><mn>40</mn></mfenced><mn>2</mn></msup></msqrt></math><br/>or, <math><mn>26.01</mn><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>×</mo></math> Incorrect <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><mn>1600</mn></math><br/>Therefore,<br/>Incorrect <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>100</mn><mo>(</mo><mn>26</mn><mo>.01</mo><mo>+</mo><mn>1600</mn><mo>)</mo><mo>=</mo><mn>162601</mn></math></p><p>Correct mean, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>x</mi><mo> </mo></mrow><mo>¯</mo></mover><mo>=</mo><mfrac><mn>3990</mn><mn>100</mn></mfrac><mo>=</mo><mn>39</mn><mo>.</mo><mn>9</mn></math><br/>Now, Correct <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>=</mo></math> Incorrect <math><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>50</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn>40</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math><mo>=</mo><mn>162601</mn><mo>-</mo><mn>2500</mn><mo>+</mo><mn>1600</mn><mo>=</mo><mn>161701</mn></math><br/>Therefore correct standard deviation<br/><math><mo>=</mo><msqrt><mfrac><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi></mi><mrow><mo stretchy="false">∑</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mi>n</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>161701</mn></mrow><mrow><mn>100</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn>39.9</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></msqrt><mo>=</mo><msqrt><mn>1617.01</mn><mo>-</mo><mn>1592.01</mn></msqrt><mo>=</mo><msqrt><mn>25</mn></msqrt><mo>=</mo><mn>5</mn></math></p></div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2023 (14 May Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">161701</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-later-on-it-was-found-that-one-of-the-observations-was-t

slug:
the-mean-and-standard-deviation-of-100-observations-were-calculated-as-40-and-51-respectively-later-on-it-was-found-that-one-of-the-observations-was-t-92783

question:
<div class="question">The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively. Later on it was found that one of the observations was taken as 50 in the place of 40 . If the wrong entry is replaced by the correct one, then the sum of the squares of all the observations is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">162701</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">163501</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">162601</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">161701</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\bar{x}=40$<br/>$\begin{aligned} &amp; \sigma=5.1 \\ &amp; n=100\end{aligned}$<br/>Mean $=\frac{\sum x_i}{n}=40$<br/>$\begin{aligned} &amp; \therefore \sum x_i=4000 \\ &amp; \sigma=5.1 \\ &amp; \operatorname{Var}(x)=(5.1)^2=26.01 \\ &amp; \frac{\sum x_i^2}{100}-(40)^2=26.01 \\ &amp; \therefore \sum x_i^2=162601\end{aligned}$<br/>Incorrect $\sum x_i^2=162601$<br/>Correct $\sum x_i^2=162601-50^2+40^2=161701$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2020 (14 Sep Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">161701</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-100-observations-x-1-x-2-x-100-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instea

slug:
the-mean-and-standard-deviation-of-100-observations-x-1-x-2-x-100-were-calculated-as-40-and-51-respectively-by-a-student-who-took-by-mistake-50-instea-82122

question:
<div class="question">The mean and standard deviation of 100 observations $x_1, x_2, \ldots, x_{100}$ were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. Then the correct value of $\sum_{i=1}^{100} x_i^2=$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3990</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">161701</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">162601</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4000</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \text { Given, } x=40, \sigma=5 \cdot 1, n=100 \\ &amp; \qquad \sigma^2=\frac{\Sigma x_i^2}{100}-(\bar{x})^2 \Rightarrow(5 \cdot 1)^2=\frac{\Sigma x_1^2}{100}-(40)^2\end{aligned}$<br/>$\Sigma x_1^2=(26.01+1600) \times 100=2601+1600 \times 100$<br/>The correct value of $\Sigma x_1^2=2601+160000-(50)^2$<br/>$+(40)^2$<br/>$=2601+160000-2500+1600=161701$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">18</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-binomial-distribution-are-12-and-2-respectively-what-is-the-number-of-trials

slug:
the-mean-and-standard-deviation-of-a-binomial-distribution-are-12-and-2-respectively-what-is-the-number-of-trials-31648

question:
<div class="question">The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">18</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">24</span> </li> </ul>

answer:
<div class="solution">Mean, $x=n p=12$ ... (i)<br/>Standard deviation<br/>$\Rightarrow \sqrt{\mathrm{npq}}=2$<br/>$\Rightarrow n p q=4$ ... (ii)<br/>$\frac{(\mathrm{ii})}{(\mathrm{i})} \Rightarrow \frac{\not \mathrm{h} \mathrm{pq}}{\not \mathrm{h} \not \mathrm{p}}=\frac{4}{12} \Rightarrow \mathrm{q}=\frac{1}{3}$<br/>$\mathrm{p}=1-\frac{1}{3}=\frac{2}{3}$<br/>Now, (i) $\Rightarrow \mathrm{np}=12$<br/>$\Rightarrow n\left(\frac{2}{3}\right)=12 \Rightarrow 2 n=36 \Rightarrow n=18$</div>


tags: ['Mathematics', 'Probability', 'AP EAMCET', 'AP EAMCET 2007']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-binomial-variate-x-are-4-and-3-respectively-then-p-x-1-is-equal-to-1

slug:
the-mean-and-standard-deviation-of-a-binomial-variate-x-are-4-and-3-respectively-then-p-x-1-is-equal-to-1-46304

question:
<div class="question">The mean and standard deviation of a binomial variate $X$ are 4 and $\sqrt{3}$ respectively. Then $P(X \geq 1)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1-\left(\frac{1}{4}\right)^{16}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1-\left(\frac{3}{4}\right)^{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1-\left(\frac{2}{3}\right)^{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1-\left(\frac{1}{3}\right)^{16}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1-\left(\frac{3}{4}\right)^{16}$</span> </div>

answer:
<div class="solution">Mean $=n p=4$, variance $=n q p=3$<br/>On solving, we get $q=\frac{3}{4}, n=16, p=\frac{1}{4}$<br/>Now, $\begin{aligned} p(X \geq 1) &amp; =1-p(X=0)=1-{ }^n C_0 p^0 q^{n-0} \\ &amp; =1-\left(\frac{3}{4}\right)^{16}\end{aligned}$</div>


tags: ['Mathematics', 'Probability', 'TS EAMCET', 'TS EAMCET 2007']
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-binomial-variate-x-are-4-and-3-respectively-then-p-x-1-is-equal-to

slug:
the-mean-and-standard-deviation-of-a-binomial-variate-x-are-4-and-3-respectively-then-p-x-1-is-equal-to-58727

question:
<div class="question">The mean and standard deviation of a binomial variate $X$ are 4 and $\sqrt{3}$ respectively. Then $P(X \geq 1)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1-\left(\frac{1}{4}\right)^{16}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1-\left(\frac{3}{4}\right)^{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1-\left(\frac{2}{3}\right)^{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1-\left(\frac{1}{3}\right)^{16}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1-\left(\frac{3}{4}\right)^{16}$</span> </div>

answer:
<div class="solution">Mean $=n p=4$, variance $=n q p=3$<br/>On solving, we get $q=\frac{3}{4}, n=16, p=\frac{1}{4}$<br/>Now, $\begin{aligned} p(X \geq 1) &amp; =1-p(X=0)=1-{ }^n C_0 p^0 q^{n-0} \\ &amp; =1-\left(\frac{3}{4}\right)^{16}\end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2018 (05 May Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">26.78</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-distribution-of-weights-of-a-group-of-20-boys-are-40-kg-and-5-kg-respectively-if-two-boys-of-weights-43-kg-and-37

slug:
the-mean-and-standard-deviation-of-a-distribution-of-weights-of-a-group-of-20-boys-are-40-kg-and-5-kg-respectively-if-two-boys-of-weights-43-kg-and-37-10372

question:
<div class="question">The mean and standard deviation of a distribution of weights of a group of 20 boys are $40 \mathrm{~kg}$ and $5 \mathrm{~kg}$ respectively.<br/>If two boys of weights $43 \mathrm{~kg}$ and $37 \mathrm{~kg}$ are excluded from this group, then the variance of the distribution of weights of the remaining group of boys is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">26.18</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5.27</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">26.78</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5.17</span> </li> </ul>

answer:
<div class="solution">Here, $n=20$<br/>Mean $=40$<br/>Mean $=\frac{\text { Sum of observation }}{n}$<br/>$\Rightarrow 40=\frac{\text { Sum of weights }}{20}$<br/>Sum of weights $=20 \times 40=800$<br/>Now, standard deviation, $\sigma=5$<br/>$\because$ variance $=\sigma^2=5^2=25$<br/>Now, two boys are excluded of weight $43 \mathrm{~kg}$ and $73 \mathrm{~kg}$.<br/>New mean $=\frac{\text { Remaining weight }}{20-2}=\frac{800-43-73}{18}$<br/>$$<br/>\bar{X}_{\text {new }}=\text { new mean }=38<br/>$$<br/><br/>Now, variance $\sigma^2=25$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\Sigma x^2}{n}-(\bar{X})^2=25 \Rightarrow \frac{\Sigma x^2}{20}-40^2=25 \\<br/>\Rightarrow \quad &amp; \frac{\Sigma x^2}{20}=1600+25 \Rightarrow \Sigma x^2=1625 \times 20 \\<br/>\Rightarrow \quad &amp; \Sigma x^2=32500<br/>\end{aligned}<br/>$$<br/><br/>When two boys are removed<br/>$$<br/>\Sigma x_{\text {new }}^2=32500-(43)^2-(73)^2=25322 .<br/>$$<br/><br/><br/>So, new variance<br/>$$<br/>\begin{aligned}<br/>&amp; \sigma_{\text {new }}^2=\frac{\sum x_{\text {new }}^2}{18}-\left(\bar{X}_{\text {new }}\right)^2=\frac{25322}{18}-(38)^2 \\<br/>&amp; \sigma_{\text {new }}^2=26.78<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-set-n-1-observations-are-x-1-and-s-1-respectively-while-the-mean-and-standard-deviation-of-another-set-of-n-2-obs

slug:
the-mean-and-standard-deviation-of-a-set-n-1-observations-are-x-1-and-s-1-respectively-while-the-mean-and-standard-deviation-of-another-set-of-n-2-obs-64175

question:
<div class="question">The mean and standard deviation of a set $n_1$ observations are $\bar{x}_1$ and $s_1$, respectively while the mean and standard deviation of another set of $n_2$ observations are $\overline{\mathrm{x}}_2$ and $s_2$, respectively. Show that the standard deviation of the<br/><br/><br/>combined set of $\left(n_1+n_2\right)$ observations is given by<br/>$$<br/>\text { S.D. }=\sqrt{\frac{n_1\left(s_1\right)^2+n_2\left(s_2\right)^2}{n_1+n_2}+\frac{n_1 n_2\left(\bar{x}-\bar{x}_2\right)^2}{\left(n_1+n_2\right)}}<br/>$$</div>

answer:
<div class="solution">Let $d_1=\left(\bar{x}_1-\bar{x}_{12}\right), d_2=\left(\bar{x}_2-\bar{x}_{12}\right)$<br/>$\therefore$ Standard deviation (combined set) is given as<br/>$\mathrm{SD}=\sqrt{\frac{n_1\left(s_1^2+d_1^2\right)+n_2\left(s_2^2+d_2^2\right)}{n_1+n_2}}$<br/>$\Rightarrow \quad \mathrm{SD}=\sqrt{\frac{\mathrm{n}_1 \mathrm{~s}_1^2+\mathrm{n}_2 \mathrm{~s}_2^2}{\mathrm{n}_1+\mathrm{n}_2}+\frac{\mathrm{n}_1 \mathrm{~d}_1^2+\mathrm{n}_2 \mathrm{~d}_2^2}{\mathrm{n}_1+\mathrm{n}_2}}$<br/>Now $\frac{n_1 d_1^2+n_2 d_2^2}{n_1+n_2}=\frac{n_1\left(\bar{x}_1-\bar{x}_{12}\right)^2+n_2\left(\bar{x}_2-\bar{x}_{12}\right)^2}{n_1+n_2}$<br/>$=\left(\frac{1}{n_1+n_2}\right)\left[n_1\left\{\bar{x}_1-\frac{n_1 \bar{x}_1+n_2 \bar{x}_2}{n_1+n_2}\right\}^2+n_2\left\{\bar{x}_2-\frac{n_1 \bar{x}_1+n_2 \bar{x}_2}{n_1+n_2}\right\}^2\right]$<br/>$=\left(\frac{1}{n_1+n_2}\right)\left[n_1\left\{\frac{n_2\left(\bar{x}_1-\bar{x}_2\right)}{n_1+n_2}\right\}^2+n_2\left\{\frac{n_1\left(\bar{x}_2-\bar{x}_1\right)}{n_1+n_2}\right\}^2\right]$<br/>$=\left(\frac{1}{n_1+n_2}\right)\left(\frac{\bar{x}_1-\bar{x}_2}{n_1+n_2}\right)^2\left(n_1 n_2^2+n_2 n_1^2\right)$<br/>$=\left(\frac{1}{n_1+n_2}\right)\left(\frac{\bar{x}_1-\bar{x}_2}{n_1+n_2}\right)^2\left(n_1 n_2\right)\left(n_2+n_1\right)$<br/><br/>$=\frac{n_1 n_2\left(\bar{x}_1-\bar{x}_2\right)^2}{\left(n_1+n_2\right)^2}$, put in (i) we have :<br/>$$<br/>\mathrm{SD}=\sqrt{\frac{n_1 s_1^2+n_2 s_2^2}{n_1+n_2}+\frac{n_1 n_2\left(\bar{x}_1-\bar{x}_2\right)^2}{\left(n_1+n_2\right)}}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-a-set-of-values-are-5-and-2-respectively-if-5-is-added-to-each-value-then-what-is-the-coefficient-of-variation-for-

slug:
the-mean-and-standard-deviation-of-a-set-of-values-are-5-and-2-respectively-if-5-is-added-to-each-value-then-what-is-the-coefficient-of-variation-for-64991

question:
<div class="question">The mean and standard deviation of a set of values are 5 and 2 respectively. If 5 is added to each value, then what is the coefficient of variation for the new set of values</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">70</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">40</span> </li> </ul>

answer:
<div class="solution">Given, Mean $=5$ Standard deviation $=2$ If 5 is added to each value, mean $=5+5=10$. Standard deviation will not change.<br/>$=\frac{2}{10} \times 100=2 \times 10$<br/>$=20$</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-group-of-100-observations-were-found-to-be-20-and-3-respectively-later-on-it-was-found-that-three-observations-were

slug:
the-mean-and-standard-deviation-of-group-of-100-observations-were-found-to-be-20-and-3-respectively-later-on-it-was-found-that-three-observations-were-43944

question:
<div class="question">The mean and standard deviation of group of 100 observations were found to be 20 and 3 , respectively, Later on it was found that three observations were incorrect, which were recorded as 21,21 and 18 . Find the mean and standard deviation if the incorrect observations are omitted.</div>

answer:
<div class="solution">$\operatorname{Mean} \bar{x}=\frac{\sum x_i}{n}, \bar{x}=20$<br/>$\therefore \sum x_i=n \times \bar{x}=100 \times 20=2000$<br/>Incorrect value of $\sum x_i=2000$<br/>Correct value of $\sum x_i=2000-21-21-18$<br/>$=1940$<br/>Correct value of mean $=\frac{1940}{97}=20$<br/>$\sigma^2($ incorrect $)=\frac{\sum x_i^2}{n}-(\bar{x})^2$<br/>$\begin{aligned}<br/>&amp;9=\frac{\sum x_i}{100}-(20)^2 \\<br/>&amp;\therefore \sum x_i=409 \times 100=40900<br/>\end{aligned}$<br/>Incorrect value of $\sum x_i=40900$<br/>Correct value of $\sum x_i=40900-(21)^2-(21)^2$<br/>$-(18)^2=40900-441-441-324=39694$<br/>$\begin{aligned}<br/>\therefore \text { Correct } &amp; \sigma^2=\frac{39694}{97}-(20)^2 \\<br/>&amp;=409.22-400=9.22<br/>\end{aligned}$<br/>Correct value of standard deviation $\sigma=3.04$, Thus correct mean $=20$<br/>Correct value of $\sigma=3.04$</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-marks-obtained-by-50-students-of-a-class-in-three-subjects-mathematics-physics-and-chemistry-are-given-below-which-

slug:
the-mean-and-standard-deviation-of-marks-obtained-by-50-students-of-a-class-in-three-subjects-mathematics-physics-and-chemistry-are-given-below-which-25040

question:
<div class="question">The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below :<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/6xV-EisUM0hmtV-wv3s6Vf-MxL-nKcZfoEXDf5lme_8.original.fullsize.png"/><br/><br/>Which of the three subjects shows the height variability in marks and which shows the lowest ? Subject Mathematics</div>

answer:
<div class="solution">Subject Mathematics<br/>Mean $\bar{x}=42$, standard deviation $\sigma=12$<br/>$\therefore$ Coefficient of variaion $=\frac{\sigma}{\bar{x}} \times 100$ or $\frac{12}{42} \times 100=28.56$<br/>Subject Physics<br/>Mean $\bar{x}=32$<br/>Standard deviation, $\sigma=15$<br/>$\therefore$ Coefficient of variation $=\frac{\sigma}{\bar{x}} \times 100$<br/>$=\frac{15}{32} \times 100=46.875$<br/>Subject Chemistry<br/>Mean $\bar{x}=40.9$, Standard deviation $\sigma=20$,<br/>$\therefore \quad$ Coefficient of variation<br/>$=\frac{\sigma}{\bar{x}} \times 100=\frac{20}{40.9} \times 100=48.9$<br/>Highest variability of marks is in Chemistry and it is least in Mathematics.</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-six-observations-are-8-and-4-respectively-if-each-observation-is-multiplied-by-3-find-the-new-mean-and-new-standard

slug:
the-mean-and-standard-deviation-of-six-observations-are-8-and-4-respectively-if-each-observation-is-multiplied-by-3-find-the-new-mean-and-new-standard-95308

question:
<div class="question">The mean and standard deviation of six observations are 8 and 4 , respectively. If each observation is multiplied by 3 , find the new mean and new standard deviation of the resulting observations.</div>

answer:
<div class="solution">Let the observation be $x_1, x_2, x_3, x_4, x_5$ and $x_6$<br/>So, their mean $\bar{x}=\frac{\sum_{i=1}^6 x_i}{6}=8 \Rightarrow \sum_{i=1}^6 x_i=48$.<br/>and variance $=\frac{\sum_{i=1}^6\left(x_i-8\right)^2}{6}=(4)^2=16$<br/>$\frac{\sum_{i=1}^6\left(x_i-8\right)^2}{6}=96$<br/>On multiplying each observation by 3 , we get the new observations as $3 x_1, 3 x_2, 3 x_3, 3 x_4, 3 x_5$, and $3 x_6$, Now, their mean<br/>$\bar{x}=\frac{\sum_{i=1}^6 3 x_i}{6}=\frac{3 \sum_{i=1}^6 x_i}{6} \text { i.e. } \bar{x}=\frac{3 \times 48}{6}=24$<br/>Variance of new observations $=\frac{\sum_{i=1}^6\left(3 x_i-24\right)^2}{6}$<br/>$=\frac{3^2 \sum_{i=1}^6\left(x_i-8\right)^2}{6}=\frac{9 \times 96}{6}=144$<br/>Hence, Standard deviation of new observations $=\sqrt{\text { Variance }}=\sqrt{144}=12$</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-some-data-for-the-time-taken-to-complete-a-test-are-calculated-with-the-following-results-number-of-observations-25

slug:
the-mean-and-standard-deviation-of-some-data-for-the-time-taken-to-complete-a-test-are-calculated-with-the-following-results-number-of-observations-25-16311

question:
<div class="question">The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results :<br/>Number of observations $=25$, mean $=18.2$ seconds, standard deviation $=3.25$ seconds.<br/>Further, another set of 15 observations $\mathrm{x}_1, \mathrm{x}_2, \ldots . ., \mathrm{x}_{15}$, also in seconds, is now available and we have $\sum_{i=1}^{15} x_i=279$ and $\sum_{i=1}^{15} x_i^2=5524$. Calculate the standard deviation based on all 40 observations.</div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp;\mathrm{n}_1=25, \overline{\mathrm{x}}_1=18.2, \sigma_1=3.25 \\<br/>&amp;\mathrm{n}_2=15, \overline{\mathrm{x}}_2=\frac{279}{15}=18.6 \\<br/>&amp;\Sigma \mathrm{x}_2=279, \Sigma \mathrm{x}_2^2=5524 \\<br/>&amp;\overline{\mathrm{x}}_{12}=\frac{\mathrm{n}_1 \overline{\mathrm{x}}_1+\mathrm{n}_2 \overline{\mathrm{x}}_2}{\mathrm{n}_1+\mathrm{n}_2} \\<br/>&amp;=\frac{25(18.2)+15(18.6)}{40}=\frac{455+279}{40} \\<br/>&amp;=\frac{734}{40}=18.35 \\<br/>&amp;\sigma_2=\sqrt{\frac{5524}{15}-(18.6)^2}<br/>\end{aligned}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>&amp;=\sqrt{368.266-345.960}=\sqrt{22.306}\\<br/>&amp;=4.7229=4.723\\<br/>&amp;\text { Let } \mathrm{d}_1=\overline{\mathrm{x}}_1-\overline{\mathrm{x}}_{12}=18.2-18.35=(-0.15)\\<br/>&amp;\mathrm{d}_2=\overline{\mathrm{x}}_2-\overline{\mathrm{x}}_{12}=18.6-18.35=(0.25)\\<br/>&amp;\therefore \sigma_{12}=\sqrt{\frac{\mathrm{n}_1\left(\sigma_1^2+\mathrm{d}_1^2\right)+\mathrm{n}_2\left(\sigma_2^2+\mathrm{d}_2^2\right)}{\mathrm{n}_1+\mathrm{n}_2}}\\<br/>&amp;=\sqrt{\frac{25\left\{(3.25)^2+(-0.15)^2\right\}+15\left\{(4.723)^2+(0.25)^2\right\}}{40}}\\<br/>&amp;=\sqrt{\frac{25(10.5625+0.0225)+15(22.3067+0.0625)}{40}}\\<br/>&amp;=\sqrt{\frac{264.625+335.538}{40}}=\sqrt{\frac{600.163}{40}}\\<br/>&amp;=\sqrt{15.004}=3.873<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2023 (13 Apr Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">269</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-standard-deviation-of-the-marks-of-10-students-were-found-to-be-50-and-12-respectively-later-it-was-observed-that-two-marks-20-and-25-wer

slug:
the-mean-and-standard-deviation-of-the-marks-of-10-students-were-found-to-be-50-and-12-respectively-later-it-was-observed-that-two-marks-20-and-25-wer-42575

question:
<div class="question">The mean and standard deviation of the marks of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> students were found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> respectively. Later, it was observed that two marks <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> were wrongly read as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> respectively. Then the correct variance is</div>

answer:
<div class="solution"><blockquote><p>Let the observations be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>3</mn></msub><mo>,</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>8</mn></msub><mo>,</mo><mo> </mo><mn>45</mn><mo>,</mo><mo> </mo><mn>50</mn></math>,</p><p>Mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>50</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>x</mi><mn>8</mn></msub><mo>+</mo><mn>45</mn><mo>+</mo><mn>50</mn></mrow><mn>10</mn></mfrac><mo>=</mo><mn>50</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>x</mi><mn>8</mn></msub><mo>=</mo><mn>405</mn><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Hence, new mean</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mover><mi>X</mi><mo>¯</mo></mover></mfenced><mi>new</mi></msub><mo>=</mo><mfrac><mrow><mn>405</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>25</mn></mrow><mn>10</mn></mfrac><mo>=</mo><mn>45</mn></math></p><p>Now,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>=</mo><msqrt><mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>8</mn></munderover><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><msup><mn>45</mn><mn>2</mn></msup><mo>+</mo><msup><mn>50</mn><mn>2</mn></msup></mstyle><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mn>50</mn></mfenced><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>12</mn><mo>=</mo><msqrt><mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>8</mn></munderover><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><mn>4525</mn></mstyle><mn>10</mn></mfrac><mo>-</mo><mn>2500</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>8</mn></munderover><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>=</mo><mn>21915</mn></math></p><p>Now,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mi>Variance</mi></mfenced><mi>new</mi></msub><mo>=</mo><mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mn>8</mn></munderover><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><msup><mn>20</mn><mn>2</mn></msup><mo>+</mo><msup><mn>25</mn><mn>2</mn></msup></mstyle><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mn>45</mn></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mfenced><mi>Variance</mi></mfenced><mi>new</mi></msub><mo>=</mo><mfrac><mstyle displaystyle="true"><mn>21915</mn><mo>+</mo><msup><mn>20</mn><mn>2</mn></msup><mo>+</mo><msup><mn>25</mn><mn>2</mn></msup></mstyle><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mn>45</mn></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mfenced><mi>Variance</mi></mfenced><mi>new</mi></msub><mo>=</mo><mn>2294</mn><mo>-</mo><mn>2025</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mfenced><mi>Variance</mi></mfenced><mi>new</mi></msub><mo>=</mo><mn>269</mn></math></p><p>Hence this is the correct answer.</p></blockquote></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">29</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-standard-deviation-of-a-data-of-8-items-are-25-and-5-respectively-if-two-items-15-and-25-are-added-to-this-data-then-the-variance-of--1

slug:
the-mean-and-the-standard-deviation-of-a-data-of-8-items-are-25-and-5-respectively-if-two-items-15-and-25-are-added-to-this-data-then-the-variance-of-1-22646

question:
<div class="question">The mean and the standard deviation of a data of 8 items are 25 and 5 respectively. If two items 15 and 25 are added to this data, then the variance of the new data is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">29</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">26</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{29}$</span> </li> </ul>

answer:
<div class="solution">We have, $n=8, \bar{x}=25$ and $\sigma=5$<br/>$$<br/>\begin{aligned}<br/>&amp; \bar{x}=\frac{\Sigma x_i}{n} \\<br/>&amp; \Rightarrow \quad \Sigma x_i=n \bar{x}=8 \times 25=200 \\<br/>&amp; \Rightarrow \text { In corrected } \Sigma x_i=200 \\<br/>&amp; \text { and } \quad \sigma=5 \\<br/>&amp; \Rightarrow \quad \sigma^2=25 \\<br/>&amp; \Rightarrow \frac{1}{n} \Sigma x_i^2-(\text { mean })^2=25 \\<br/>&amp; \Rightarrow \quad \frac{\Sigma x_i^2}{8}-625=25 \\<br/>&amp; \Rightarrow \quad \Sigma x_i^2=5200 \\<br/>&amp; \text { Corrected } \Sigma x_i^2=5200+225+625=6050 \\<br/>&amp; \text { and corrected mean }=200+15+25=240 \\<br/>&amp; \therefore \text { Corrected variance }=\frac{6050}{10}-\left(\frac{240}{10}\right)^2 \\<br/>&amp; =605-(24)^2=605-576=29 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2018 (22 Apr Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">29</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-standard-deviation-of-a-data-of-8-items-are-25-and-5-respectively-if-two-items-15-and-25-are-added-to-this-data-then-the-variance-of-

slug:
the-mean-and-the-standard-deviation-of-a-data-of-8-items-are-25-and-5-respectively-if-two-items-15-and-25-are-added-to-this-data-then-the-variance-of-11087

question:
<div class="question">The mean and the standard deviation of a data of 8 items are 25 and 5 respectively. If two items 15 and 25 are added to this data, then the variance of the new data is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">29</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">26</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{29}$</span> </li> </ul>

answer:
<div class="solution">We have, $n=8, \bar{x}=25$ and $\sigma=5$<br/>$$<br/>\begin{aligned}<br/>&amp; \bar{x}=\frac{\Sigma x_i}{n} \\<br/>&amp; \Rightarrow \quad \Sigma x_i=n \bar{x}=8 \times 25=200 \\<br/>&amp; \Rightarrow \text { In corrected } \Sigma x_i=200 \\<br/>&amp; \text { and } \quad \sigma=5 \\<br/>&amp; \Rightarrow \quad \sigma^2=25 \\<br/>&amp; \Rightarrow \frac{1}{n} \Sigma x_i^2-(\text { mean })^2=25 \\<br/>&amp; \Rightarrow \quad \frac{\Sigma x_i^2}{8}-625=25 \\<br/>&amp; \Rightarrow \quad \Sigma x_i^2=5200 \\<br/>&amp; \text { Corrected } \Sigma x_i^2=5200+225+625=6050 \\<br/>&amp; \text { and corrected mean }=200+15+25=240 \\<br/>&amp; \therefore \text { Corrected variance }=\frac{6050}{10}-\left(\frac{240}{10}\right)^2 \\<br/>&amp; =605-(24)^2=605-576=29 \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2018 (16 Apr Online)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-standard-deviation-s-d-of-five-observations-are-9-and-0-respectively-if-one-of-the-observation-is-increased-such-that-the-mean-of-the

slug:
the-mean-and-the-standard-deviation-s-d-of-five-observations-are-9-and-0-respectively-if-one-of-the-observation-is-increased-such-that-the-mean-of-the-53375

question:
<div class="question">The mean and the standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>.</mo></mrow></mfenced></math> of five observations are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>, respectively. If one of the observation is increased such that the mean of the new set of five observations becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>, then their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>.</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac><mo>=</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">s</mi><mo>.</mo><mi mathvariant="normal">d</mi><mo>=</mo><msqrt><mo>∑</mo><mfrac><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced><mn>2</mn></msup><mi>n</mi></mfrac></msqrt><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mover><mi>x</mi><mo>¯</mo></mover><mo>∀</mo><mi>i</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> each term in the original observation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> observation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mo>{</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>}</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi>new</mi><mo> </mo></mrow></msub><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac><mo>=</mo><mfrac><mrow><mn>9</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>9</mn><mo>+</mo><msub><mi>x</mi><mn>5</mn></msub></mrow><mn>5</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>10</mn><mo>=</mo><mfrac><mrow><mn>36</mn><mo>+</mo><msub><mi>x</mi><mn>5</mn></msub></mrow><mn>5</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>5</mn></msub><mo>=</mo><mn>14</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> new set of observations <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>{</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>14</mn><mo>}</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">s</mi><mo>.</mo><msub><mi mathvariant="normal">d</mi><mrow><mi>new</mi><mo> </mo></mrow></msub><mo>=</mo><msqrt><mfrac><mrow><mo>∑</mo><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mi>new</mi></msub></mrow></mfenced><mn>2</mn></msup></mrow><mi>n</mi></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mo>(</mo><mn>9</mn><mo>-</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>9</mn><mo>-</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>9</mn><mo>-</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>14</mn><mo>-</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>5</mn></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></mrow><mn>5</mn></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mn>20</mn><mn>5</mn></mfrac></msqrt><mo>=</mo><mn>2</mn></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2020 (08 Jan Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-standard-deviation-sd-of-10-observations-are-20-and-2-respectively-each-of-these-10-observations-is-multiplied-by-p-and-then-reduced-

slug:
the-mean-and-the-standard-deviation-sd-of-10-observations-are-20-and-2-respectively-each-of-these-10-observations-is-multiplied-by-p-and-then-reduced-83015

question:
<div class="question">The mean and the standard deviation (s.d.) of <math><mn>10</mn></math> observations are <math><mn>20</mn></math> and <math><mn>2</mn></math> respectively. Each of these <math><mn>10</mn></math> observations is multiplied by <math><mi>p</mi></math> and then reduced by <math><mi>q</mi><mo>,</mo></math> where <math><mi>p</mi><mo>≠</mo><mn>0</mn></math> and <math><mi>q</mi><mo>≠</mo><mn>0</mn><mo>.</mo></math> If the new mean and new s.d. become half of their original values, then <math><mi>q</mi></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mo>-</mo><mn>5</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>10</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mo>-</mo><mn>20</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mo>-</mo><mn>10</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mo>-</mo><mn>20</mn></math></span> </div>

answer:
<div class="solution">If each observation is multiplied with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> &amp; then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> is subtracted, then<br/>New mean <math><msub><mrow><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>p</mi><mover accent="true"><mrow><mi>x</mi></mrow><mo>-</mo></mover><mo>-</mo><mi>q</mi></math><br/><math><mo>⇒</mo><mo> </mo><mn>10</mn><mo>=</mo><mi>p</mi><mfenced separators="|"><mrow><mn>20</mn></mrow></mfenced><mo>-</mo><mi>q</mi><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced separators="|"><mrow><mn>1</mn></mrow></mfenced></math><br/>and new standard deviation<br/><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfenced close="|" open="|" separators="|"><mrow><mi>p</mi></mrow></mfenced><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo> </mo><mo>⇒</mo><mo> </mo><mn>1</mn><mo>=</mo><mfenced close="|" open="|" separators="|"><mrow><mi>p</mi></mrow></mfenced><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo>⇒</mo><mo> </mo><mo> </mo><mfenced close="|" open="|" separators="|"><mrow><mi>p</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo> </mo><mo> </mo><mo>⇒</mo><mo> </mo><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math><br/>If <math><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math><br/>then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>  (from equation (1))<br/>If <math><mi>p</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math><br/><math><mi>q</mi><mo>=</mo><mo>-</mo><mn>20</mn></math></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-variance-10-observations-are-given-to-be-4-and-2-respectively-if-every-observation-is-multiplied-by-2-the-mean-and-the-variance-of-th

slug:
the-mean-and-the-variance-10-observations-are-given-to-be-4-and-2-respectively-if-every-observation-is-multiplied-by-2-the-mean-and-the-variance-of-th-71917

question:
<div class="question">The mean and the variance 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2 , the mean and the variance of the new series will be respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">8 and 20</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">8 and 4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8 and 8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">80 and 40</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">8 and 8</span> </div>

answer:
<div class="solution">$\quad$ Mean $(\bar{X})=4$<br/>Variance $(\mathrm{X})=2$<br/>Number of observations $=10$<br/>New average $=\frac{(4 \times 10) \times 2}{10}=8$<br/>New variance $=(2)^{2} \times$ Variance $(\mathrm{X})$ $=4 \times 2=8$<br/>Option (c) is correct.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-and-the-variance-in-a-binomial-distribution-are-found-to-be-2-and-1-respectively-the-probability-p-x-0-is

slug:
the-mean-and-the-variance-in-a-binomial-distribution-are-found-to-be-2-and-1-respectively-the-probability-p-x-0-is-38360

question:
<div class="question">The mean and the variance in a binomial distribution are found to be 2 and 1 respectively. The probability $P(X=0)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{8}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{16}$</span> </div>

answer:
<div class="solution">$\mathrm{nP}=2$ $\mathrm{n} \mathrm{Pq}=1$<br/>$\mathrm{q}=\frac{1}{2}$<br/>$\mathrm{P}+\mathrm{q}=1$<br/>$\mathrm{P}=1-\frac{1}{2}=\frac{1}{2}$<br/>$\mathrm{n}\left(\frac{1}{2}\right)=2$<br/>$\therefore \quad \mathrm{n}=4$<br/>${ }^{4} \mathrm{C}_{0}=\left(\frac{1}{2}\right)^{4-0}=\frac{1}{16} .$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-variance-of-10-observations-were-calculated-as-15-and-15-respectively-by-a-student-who-took-by-mistake-25-instead-of-15-for-one-observati-1

slug:
the-mean-and-variance-of-10-observations-were-calculated-as-15-and-15-respectively-by-a-student-who-took-by-mistake-25-instead-of-15-for-one-observati-1-29632

question:
<div class="question">The mean and variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> observations were calculated as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> respectively by a student who took by mistake <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> for one observation. Then, the correct standard deviation is _______.</div>

answer:
<div class="solution"><blockquote><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>10</mn></math> and mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>10</mn></mfrac><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>150</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>25</mn><mo>=</mo><mn>150</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>125</mn></math></p><p>Now new sum will be <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>15</mn><mo>=</mo><mn>140</mn></math></p><p>So, actual mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>140</mn><mn>10</mn></mfrac><mo>=</mo><mn>14</mn><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi>new</mi><mo> </mo></mrow></msub></math></p><p>Now finding variance </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>2400</mn></math></p><p>Removing observation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mn>625</mn><mo>=</mo><mn>2400</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>1775</mn></math></p><p>Now adding observation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><mn>2000</mn><mo>=</mo><msub><mfenced><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub></math></p><p>Now calculating actual standard deviation we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>actual</mi><mo> </mo></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><msub><mfenced><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi>new</mi><mo> </mo></mrow></msub></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2000</mn><mn>10</mn></mfrac><mo>-</mo><msup><mn>14</mn><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>200</mn><mo>-</mo><mn>196</mn><mo>=</mo><mn>4</mn></math></p><p>Calculating standard deviation, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mrow><mi>S</mi><mo>.</mo><mi>D</mi></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></math></p></blockquote></div>


tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2022 (27 Jul Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-and-variance-of-10-observations-were-calculated-as-15-and-15-respectively-by-a-student-who-took-by-mistake-25-instead-of-15-for-one-observati

slug:
the-mean-and-variance-of-10-observations-were-calculated-as-15-and-15-respectively-by-a-student-who-took-by-mistake-25-instead-of-15-for-one-observati-16475

question:
<div class="question">The mean and variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> observations were calculated as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> respectively by a student who took by mistake <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> for one observation. Then, the correct standard deviation is _______.</div>

answer:
<div class="solution"><blockquote><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>10</mn></math> and mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mn>10</mn></mfrac><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>150</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>25</mn><mo>=</mo><mn>150</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>125</mn></math></p><p>Now new sum will be <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>15</mn><mo>=</mo><mn>140</mn></math></p><p>So, actual mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>140</mn><mn>10</mn></mfrac><mo>=</mo><mn>14</mn><mo>=</mo><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi>new</mi><mo> </mo></mrow></msub></math></p><p>Now finding variance </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><mover><mi>x</mi><mo>¯</mo></mover></mfenced><mn>2</mn></msup><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>10</mn></mfrac><mo>-</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><mn>15</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>2400</mn></math></p><p>Removing observation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math> we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mn>625</mn><mo>=</mo><mn>2400</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>1775</mn></math></p><p>Now adding observation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>9</mn></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msup><mn>15</mn><mn>2</mn></msup><mo>=</mo><mn>2000</mn><mo>=</mo><msub><mfenced><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub></math></p><p>Now calculating actual standard deviation we get,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><mi>actual</mi><mo> </mo></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><msub><mfenced><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub><mn>10</mn></mfrac><mo>-</mo><msup><mfenced><msub><mover><mi>x</mi><mo>¯</mo></mover><mrow><mi>new</mi><mo> </mo></mrow></msub></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2000</mn><mn>10</mn></mfrac><mo>-</mo><msup><mn>14</mn><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>200</mn><mo>-</mo><mn>196</mn><mo>=</mo><mn>4</mn></math></p><p>Calculating standard deviation, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mrow><mi>S</mi><mo>.</mo><mi>D</mi></mrow></mfenced><mrow><mi>actual</mi><mo> </mo></mrow></msub><mo>=</mo><msqrt><mn>4</mn></msqrt><mo>=</mo><mn>2</mn></math></p></blockquote></div>


title: the-mean-and-variance-of-a-binomial-distribution-are-8-and-4-respectively-what-is-p-x-1-equal-to
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
the-mean-and-variance-of-a-binomial-distribution-are-8-and-4-respectively-what-is-p-x-1-equal-to-93733

question:
<div class="question">The mean and variance of a binomial distribution are 8 and 4 respectively. What is $P(X=1)$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2^{12}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2^{8}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2^{6}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2^{4}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2^{12}}$</span> </div>

answer:
<div class="solution">We know that mean and variance of Binomial distribution are $n p$ and $n p q$ respectively therefore $n p=$ 8 and $n p q=4$ On dividing we get<br/>$q=\frac{n p q}{n p}=\frac{4}{8}=\frac{1}{2}$ and $p+q=1 \Rightarrow p=1-\frac{1}{2}=\frac{1}{2}$<br/>$\Rightarrow n\left(\frac{1}{2}\right)=8 \Rightarrow n=16$<br/>We know that $P(X=r)=n_{C_{r}} p^{r} q^{n-r}$<br/>therefore<br/>$P(X=1)={ }^{16} C_{1}\left(\frac{1}{2}\right)^{16-1}\left(\frac{1}{2}\right)^{1}$<br/>$=\frac{16}{2^{15} \cdot 2}=\frac{1}{2^{12}}$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2018 (05 May Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.733</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-deviation-about-the-mean-for-the-following-data-is-beginarraycccccc-hline-boldsymbolxi-2-4-5-7-9-hline-boldsymbolfi-2-4-10-8-6-hline-endarray

slug:
the-mean-deviation-about-the-mean-for-the-following-data-is-beginarraycccccc-hline-boldsymbolxi-2-4-5-7-9-hline-boldsymbolfi-2-4-10-8-6-hline-endarray-60029

question:
<div class="question">The mean deviation about the mean for the following data, is<br/>\begin{array}{c|c|c|c|c|c}<br/>\hline \boldsymbol{x}_i &amp; 2 &amp; 4 &amp; 5 &amp; 7 &amp; 9 \\<br/>\hline \boldsymbol{f}_i &amp; 2 &amp; 4 &amp; 10 &amp; 8 &amp; 6 \\<br/>\hline<br/>\end{array}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6.3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2.83</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1.733</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We have, the following data.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/zRlxs61xS38rCkltzqK8U2ytBRL4JNJyFnX3hT5l2nE.original.fullsize.png"/><br/><br/>Here, mean<br/>$$<br/>\text { (à¤°) }=\frac{\sum x_1 f_i}{\sum f_i}=\frac{4+16+50+56+54}{2+4+10+8+6}=\frac{180}{30}=6<br/>$$<br/>$\therefore$ Mean deviation about mean $=\frac{\sum f_i\left|x_{\dot{i}}-\bar{x}\right|}{\sum f_{\dot{i}}}$<br/>$$<br/>\begin{aligned}<br/>&amp; =\frac{2 \cdot(4)+4 \cdot(2)+10 \cdot(1)+8 \cdot(1)+6 \cdot(3)}{30} \\<br/>&amp; =\frac{8+8+10+8+18}{30}=\frac{52}{30} \approx \frac{17.33}{10}=1.733<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2018 (24 Apr Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch2_P3.db
title:
the-mean-deviation-of-the-following-data-about-its-mean-is-beginarrayllllll-hlinexi-5-15-25-35-45-hlinefi-8-48-56-32-16-hline-endarray

slug:
the-mean-deviation-of-the-following-data-about-its-mean-is-beginarrayllllll-hlinexi-5-15-25-35-45-hlinefi-8-48-56-32-16-hline-endarray-77364

question:
<div class="question">The mean deviation of the following data about its mean is<br/>\begin{array}{|l|l|l|l|l|l|}<br/>\hlinex_i &amp; 5 &amp; 15 &amp; 25 &amp; 35 &amp; 45 \\<br/>\hlinef_i &amp; 8 &amp; 48 &amp; 56 &amp; 32 &amp; 16 \\<br/>\hline<br/>\end{array}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">25</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>


tags: ['Physics', 'Thermodynamics', 'TS EAMCET', 'TS EAMCET 2018 (05 May Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-kinetic-energy-of-monoatomic-gas-molecules-under-standard-conditions-is-e-1-if-the-gas-is-compressed-adiabatically-8-times-to-its-initial-vol

slug:
the-mean-kinetic-energy-of-monoatomic-gas-molecules-under-standard-conditions-is-e-1-if-the-gas-is-compressed-adiabatically-8-times-to-its-initial-vol-68853

question:
<div class="question">The mean kinetic energy of monoatomic gas molecules under standard conditions is $\left\langle E_1\right\rangle$. If the gas is compressed adiabatically 8 times to its initial volume, the mean kinetic energy of gas molecules changes to $\left\langle E_2\right\rangle$. The ratio $\frac{\left\langle E_2\right\rangle}{\left\langle E_1\right\rangle}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">In adiabatic process,<br/>$$<br/>T V^{\gamma-1}=\text { constant }<br/>$$<br/>In given conditions,<br/>$$<br/>V_1=V\left(\text { say), so } V_2=\left(\frac{V}{8}\right)\right.<br/>$$<br/>and $\gamma=\frac{5}{3}$ (gas is monoatomic)<br/>So, $T_1 V^{\frac{5}{3}-1}=T_2\left(\frac{V}{8}\right)^{\frac{5}{3}-1} \Rightarrow \frac{T_2}{T_1}=\frac{V^{\frac{2}{3}}}{\left(\frac{V}{8}\right)^{\frac{2}{3}}}=8^{\frac{2}{3}}=4$<br/>So, temperature of gas increases by 4 times.<br/>As mean KE of gas molecule is proportional to its temperature, then $\mathrm{KE}$ also increases by 4 times.</div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
the-mean-life-of-a-sample-of-60-bulbs-was-650-hours-and-the-standard-deviation-was-8-hours-a-second-sample-of-80-bulbs-has-a-mean-life-of-660-hours-an

slug:
the-mean-life-of-a-sample-of-60-bulbs-was-650-hours-and-the-standard-deviation-was-8-hours-a-second-sample-of-80-bulbs-has-a-mean-life-of-660-hours-an-26964

question:
<div class="question">The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.</div>

answer:
<div class="solution">Here, $\mathrm{n}_1=60, \overline{\mathrm{x}}_1=650, \sigma_1=8$<br/>$$<br/>\begin{aligned}<br/>\mathrm{n}_2 &amp;=80, \overline{\mathrm{x}}_2=660, \sigma_2=7 \\<br/>\overline{\mathrm{x}}_{12} &amp;=\frac{\mathrm{n}_1 \overline{\mathrm{x}}_1+\mathrm{n}_2 \overline{\mathrm{x}}_2}{\mathrm{n}_1+\mathrm{n}_2} \\<br/>&amp;=\frac{60(650)+80(660)}{60+80}=\frac{91800}{140}=655.71 \\<br/>\mathrm{~d}_1 &amp;=\overline{\mathrm{x}}_1-\overline{\mathrm{x}}_{12}=650-655.71=-5.714 \\<br/>\mathrm{~d}_2 &amp;=\overline{\mathrm{x}}_2-\overline{\mathrm{x}}_{12}=660-655.71=4.29 \\<br/>\sigma_{12} &amp;=\sqrt{\frac{\mathrm{n}_1\left(\sigma_1^2+\mathrm{d}_1^2\right)+\mathrm{n}_2\left(\sigma_2^2+\mathrm{d}_2^2\right)}{\mathrm{n}_1+\mathrm{n}_2}} \\<br/>&amp;=\sqrt{\frac{60(64+32.65)+80(49+18.40)}{60+80}} \\<br/>&amp;=\sqrt{\frac{11191}{140}=\sqrt{79.9}=8.94}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$8.5$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-of-7-observations-is-10-and-that-of-3-observations-is-5-what-is-the-mean-of-all-the-10-observations

slug:
the-mean-of-7-observations-is-10-and-that-of-3-observations-is-5-what-is-the-mean-of-all-the-10-observations-94039

question:
<div class="question">The mean of 7 observations is 10 and that of 3 observations is $5 .$ What is the mean of all the 10 observations?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$8.5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.5$</span> </li> </ul>

answer:
<div class="solution">Given mean of 7 observations is $\underline{10}$.<br/>$$<br/>\frac{x_{1}+x_{2}+\ldots .+x_{7}}{7}=10<br/>$$<br/>$$<br/>\Rightarrow x_{1}+x_{2}+\ldots .+x_{7}=70<br/>$$<br/>$(1)$<br/>Also, mean of 3 observations<br/>$$<br/>\frac{x_{8}+x_{9}+x_{10}}{3}=5<br/>$$<br/>$\Rightarrow x_{8}+x_{9}+x_{10}=15$<br/>(2)<br/>So, from $(1)$ and $(2)$ Required mean<br/>$\frac{x_{1}+x_{2}+\ldots+x_{7}+x_{8}+x_{9}+x_{10}}{10}$<br/>$=\frac{70+15}{10}=\frac{85}{10}=8.5$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">21</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-of-10-observations-is-5-if-2-is-added-to-each-observation-and-then-multiplied-by-3-then-what-will-be-the-new-mean

slug:
the-mean-of-10-observations-is-5-if-2-is-added-to-each-observation-and-then-multiplied-by-3-then-what-will-be-the-new-mean-81020

question:
<div class="question">The mean of 10 observations is 5 . If 2 is added to each observation and then multiplied by 3 , then what will be the new mean</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \quad$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">21</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given : Mean of 10 observations is 5 .<br/>Mean of 10 observations is 5<br/>$\Rightarrow \frac{\sum_{i=1}^{10} x_{i}}{10}=5$<br/>According to the Question<br/>$\frac{\sum_{i=1}^{10} 3\left(x_{i}+2\right)}{10}=$ New mean<br/>$\Rightarrow \frac{3 \sum_{i=1}^{10} x_{i}}{10}+\frac{3 \times 2 \times 10}{10}=$ New mean<br/>$\Rightarrow 3 \times 5+6=$ New mean $\quad \Rightarrow 21=$ Newmean</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
the-mean-of-20-observations-is-15-on-checking-it-was-found-that-two-observations-were-wrongly-copied-as-3-and-7-wrong-observations-are-replaced-by-cor

slug:
the-mean-of-20-observations-is-15-on-checking-it-was-found-that-two-observations-were-wrongly-copied-as-3-and-7-wrong-observations-are-replaced-by-cor-64378

question:
<div class="question">The mean of 20 observations is $15 .$ On checking, it was found that two observations were wrongly copied as 3 and 7 wrong observations are replaced by correct value 8 and 4, then the correct mean is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">15</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$15.15$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad 15.35$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">16</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15.15$</span> </div>

answer:
<div class="solution">Sum of all observations $=20 \times 15=300$ Sum of correct observations $=300-(3+6)+(8+4)=303$<br/>Correct mean $=\frac{303}{20}=15.1$ ?</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">17</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-of-40-observations-is-20-and-their-standard-deviation-is-5-if-the-sum-of-the-squares-of-the-observations-is-k-then-the-value-of-k-1000-is

slug:
the-mean-of-40-observations-is-20-and-their-standard-deviation-is-5-if-the-sum-of-the-squares-of-the-observations-is-k-then-the-value-of-k-1000-is-21175

question:
<div class="question">The mean of <math><mn>40</mn></math> observations is <math><mn>20</mn></math> and their standard deviation is <math><mn>5</mn><mo>.</mo></math> If the sum of the squares of the observations is <math><mi>k</mi></math>, then the value of <math><mfrac><mrow><mi>k</mi></mrow><mrow><mn>1000</mn></mrow></mfrac></math> is</div>

answer:
<div class="solution"><math><mi>n</mi><mo>=</mo><mn>4</mn><mn>0</mn></math><br/><math><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>=</mo><mn>2</mn><mn>0</mn></math><br/><math><mi>σ</mi><mo>=</mo><mn>0</mn><mn>5</mn></math><br/><math><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mrow><mrow><mn>40</mn></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>=</mo><mfrac><mrow><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mn>40</mn></mfrac><mo>-</mo><mn>400</mn><mo>⇒</mo><mo>∑</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>1700</mn><mn>0</mn></math></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
the-mean-of-100-observations-is-50-and-the-standard-deviation-is-10-if-5-is-subtracted-from-each-observation-and-then-it-is-divided-by-4-then-what-wil

slug:
the-mean-of-100-observations-is-50-and-the-standard-deviation-is-10-if-5-is-subtracted-from-each-observation-and-then-it-is-divided-by-4-then-what-wil-93224

question:
<div class="question">The mean of 100 observations is 50 and the standard deviation is 10 . If 5 is subtracted from each observation and then it is divided by 4 , then what will be the new mean and the new standard deviation respectively ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">45,5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$11.25,1.25$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$11.25,2.5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12.5,2.5$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$11.25,2.5$</span> </div>

answer:
<div class="solution">Number of observations $(\mathrm{N})=100$<br/>Mean $(\bar{x})=50$<br/>Standard deviation $(\sigma)=10$<br/>New mean, $\bar{x}_{1}=\frac{\Sigma \frac{x-5}{4}}{N}=\frac{1}{4}\left(\frac{\Sigma x}{N}-5\right)=\frac{1}{4}(50-5)$<br/>$=\frac{45}{4}=11.25$<br/>New standard deviation $\left(\sigma_{1}\right)=\sigma\left(\frac{x-5}{4}\right)$<br/>$=\frac{1}{4} \sigma(x-5)=\frac{1}{4} \sigma(x)=\frac{10}{4}=2.5$</div>


tags: ['Mathematics', 'Statistics']
source_db: MarksBatch2_P3.db
title:
the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-the-sum-of-squares-of-all-observation-is

slug:
the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-the-sum-of-squares-of-all-observation-is-17316

question:
<div class="question">The mean of 100 observations is 50 and their standard deviation is 5 . The sum of squares of all observation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>50000<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>250000<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>252500<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>255000</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>252500<br/></span> </div>

answer:
<div class="solution">We have $\mathrm{n}=100, \overline{\mathrm{x}}=50, \mathrm{~s}=5, \mathrm{~s}^2=25$<br/>We know that<br/>$$<br/>\begin{aligned}<br/>&amp; \sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{1}{n} \sum x_i\right)^2 \Rightarrow 25=\frac{\sum x_i^2}{100}-(50)^2 \\<br/>\Rightarrow &amp; 2500=\sum x_i^2-250000 \\<br/>\Rightarrow &amp; \sum x_i^2=252500<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Statistics', 'KCET', 'KCET 2023']
source_db: MarksBatch2_P3.db
title:
the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-then-the-sum-of-squares-of-all-observations-is

slug:
the-mean-of-100-observations-is-50-and-their-standard-deviation-is-5-then-the-sum-of-squares-of-all-observations-is-19527

question:
<div class="question">The mean of 100 observations is 50 and their standard deviation is 5 . Then, the sum of squares of all observations is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$252500$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$250000$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$255000$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$50000$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$252500$</span> </div>

answer:
<div class="solution">We know, standard deviation,<br/>$\begin{aligned} P^2 &amp; =\frac{\Sigma x^2}{n}-(\bar{x})^2 \\ P &amp; =5, \Sigma x=50, n=100\end{aligned}$<br/>$\begin{aligned} &amp; \Rightarrow \quad 25=\frac{\Sigma x^2}{100}-(50)^2 \\ &amp; \Rightarrow \quad \Sigma x^2=252500\end{aligned}$<br/>Thus, sum of squares of all observation in 252500 .</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-of-a-group-of-100-observations-was-found-to-be-20-later-it-was-found-that-four-observations-were-incorrect-which-were-recorded-as-21-21-18-an

slug:
the-mean-of-a-group-of-100-observations-was-found-to-be-20-later-it-was-found-that-four-observations-were-incorrect-which-were-recorded-as-21-21-18-an-44357

question:
<div class="question">The mean of a group of 100 observations was found to be 20 . Later it was found that four observations were incorrect, which were recorded as $21,21,18$ and 20 . What is the mean if the incorrect observations are omitted? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">18</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">21</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">22</span> </li> </ul>

answer:
<div class="solution">Mean of 100 observations $=20$<br/>Required mean $=\frac{20 \times 100-(21+21+18+20)}{96}$<br/>$=\frac{1920}{96}=20$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">38</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-of-five-numbers-is-30-if-one-number-is-excluded-their-mean-becomes-28-the-excluded-number-is

slug:
the-mean-of-five-numbers-is-30-if-one-number-is-excluded-their-mean-becomes-28-the-excluded-number-is-15924

question:
<div class="question">The mean of five numbers is 30 . If one number is excluded their mean becomes 28 . The excluded number is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">28</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">35</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">30</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">38</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Mean of 5 numbers $=30$ Total sum of 5 numbers $=30 \times 5=150$ After excluded one number<br/>Mean of 4 numbers will be $=28$ Total sum of 4 numbers $=4 \times 28=112$ Thus, excluded number $=$ (sum of 5 numbers $-$ sum of 4 numbers) $=150-112=38$<br/>Option (d) is correct.</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2014']
source_db: MarksBatch2_P3.db
title:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is-1

slug:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is-1-65700

question:
<div class="question">The mean of four observations is 3 . If the sum of the squares of these observations is 48 , then their standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3}$</span> </div>

answer:
<div class="solution">Let four observations are be $x_1, x_2, x_3$ and $x_4$<br/>Given, mean<br/>and<br/>$$<br/>\begin{aligned}<br/>(\bar{x}) &amp; =3 \\<br/>\Sigma x_i^2 &amp; =48<br/>\end{aligned}<br/>$$<br/>$\begin{aligned} \therefore \quad \mathrm{SD} &amp; =\sqrt{\frac{\sum x_i^2}{n}-(\bar{x})^2} \\ =\sqrt{\frac{48}{4}-(3)^2}=\sqrt{12-9} &amp; =\sqrt{3}\end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is-2

slug:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is-2-73212

question:
<div class="question">The mean of four observations is 3 . If the sum of the squares of these observations is 48 , then their standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3}$</span> </div>

answer:
<div class="solution">Let four observations are be $x_1, x_2, x_3$ and $x_4$<br/>Given, mean<br/>and<br/>$$<br/>\begin{aligned}<br/>(\bar{x}) &amp; =3 \\<br/>\Sigma x_i^2 &amp; =48<br/>\end{aligned}<br/>$$<br/>$\begin{aligned} \therefore \quad \mathrm{SD} &amp; =\sqrt{\frac{\sum x_i^2}{n}-(\bar{x})^2} \\ =\sqrt{\frac{48}{4}-(3)^2}=\sqrt{12-9} &amp; =\sqrt{3}\end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2014']
source_db: MarksBatch2_P3.db
title:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is

slug:
the-mean-of-four-observations-is-3-if-the-sum-of-the-squares-of-these-observations-is-48-then-their-standard-deviation-is-80727

question:
<div class="question">The mean of four observations is 3 . If the sum of the squares of these observations is 48 , then their standard deviation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5}$</span> </li> </ul>

answer:
<div class="solution">Let four observations are be $x_1, x_2, x_3$ and $x_4$<br/>Given, mean<br/>and<br/>$$<br/>\begin{aligned}<br/>(\bar{x}) &amp; =3 \\<br/>\Sigma x_i^2 &amp; =48<br/>\end{aligned}<br/>$$<br/>$\begin{aligned} \therefore \quad \mathrm{SD} &amp; =\sqrt{\frac{\sum x_i^2}{n}-(\bar{x})^2} \\ =\sqrt{\frac{48}{4}-(3)^2}=\sqrt{12-9} &amp; =\sqrt{3}\end{aligned}$</div>


title: the-mean-of-the-series-x-1-x-2-x-n-is-x-if-x-2-is-replaced-by-then-what-is-the-new-mean
slug: the-mean-of-the-series-x-1-x-2-x-n-is-x-if-x-2-is-replaced-by-then-what-is-the-new-mean-92793
tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The mean of the series $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{\mathrm{n}}$ is $\overline{\mathrm{X}}$. If $\mathrm{x}_{2}$ is replaced by $\lambda$, then what is the new mean? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overline{\mathrm{X}}-\mathrm{x}_{2}+\lambda$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\overline{\mathrm{X}}-\mathrm{x}_{2}-\lambda}{\mathrm{n}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\overline{\mathrm{X}}-\mathrm{x}_{2}+\lambda}{\mathrm{n}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{n} \overline{\mathrm{X}}-\mathrm{x}_{2}+\lambda}{\mathrm{n}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{n} \overline{\mathrm{X}}-\mathrm{x}_{2}+\lambda}{\mathrm{n}}$</span> </div>

answer:
<div class="solution">Mean of series $\left(x_{1}, x_{2}, x_{3}, \ldots . x_{n}\right)$<br/>$\bar{x}=\frac{x_{1}+x_{2}+x_{3}+\ldots \ldots x_{n}}{n}$<br/>$\Rightarrow x_{1}+x_{2}+x_{3}+\ldots \ldots x_{n}=n \bar{x}$<br/>Now we will replace $x$, by $\lambda$ so no. of elements in series will not change. New series will include $\lambda$ and exclude $x_{2}$ Hence new series sum<br/>$\left(x_{1}+x_{2}+\ldots . x_{n}\right)-x_{2}+\lambda=n \bar{x}+\lambda-x_{2}$<br/>Now new mean</div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-square-deviation-of-a-set-of-n-observation-x-1-x-2-2-x-n-about-a-point-c-is-defined-as-n-1-i-1-n-x-i-c-2-the-mean-square-deviations-about-2-a

slug:
the-mean-square-deviation-of-a-set-of-n-observation-x-1-x-2-2-x-n-about-a-point-c-is-defined-as-n-1-i-1-n-x-i-c-2-the-mean-square-deviations-about-2-a-33914

question:
<div class="question">The mean square deviation of a set of n observation $\mathrm{x}_{1}, \mathrm{x}_{2^{2}}, \ldots \mathrm{x}_{\mathrm{n}}$ about a point $\mathrm{c}$ is defined as $\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-c\right)^{2}$
<br/> <br/>The mean square deviations about -2 and 2 are 18 and 10 respectively, the standard deviation of this set of observations is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">We have $\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}=18$ and
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=10 \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}=18 n \text { and }
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=10 n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}+\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=28 n
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>and $\sum_{i=1}^{n}\left(x_{i}+2\right)^{2}-\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=8 n$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow 2 \sum_{i=1}^{n}\left(x_{i}+4\right)^{2}=28 n \text { and } 2 \sum_{i=1}^{n} 4 x_{i}=8 n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n} x_{1}^{2}+4 n=14 n \text { and } \sum_{i=1}^{n} x_{i}=n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n} x_{i}^{2}=10 n \text { and } \sum_{i=1}^{n} x_{i}=n \\
<br/> <br/>\therefore \sigma=\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_{i}^{2}-\left(\frac{1}{n} \sum_{i=1^{\prime}}^{n} x_{i}\right)^{2}}=\sqrt{\frac{10 n}{n}-\left(\frac{n}{n}\right)^{2}}=3
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-mean-square-deviation-of-a-set-of-observations-x-1-x-2-x-n-about-a-point-m-is-defined-as-1-n-i-1-n-x-i-m-2-if-the-mean-square-deviations-about-1-a

slug:
the-mean-square-deviation-of-a-set-of-observations-x-1-x-2-x-n-about-a-point-m-is-defined-as-1-n-i-1-n-x-i-m-2-if-the-mean-square-deviations-about-1-a-17940

question:
<div class="question">The mean square deviation of a set of observations <math><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">n</mi></mrow></msub></math> about a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi></math> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi mathvariant="normal">n</mi></mfrac><munderover><mo stretchy="true">â</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">n</mi></munderover><msup><mfenced separators="|"><mrow><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi></msub><mo>-</mo><mi mathvariant="normal">m</mi></mrow></mfenced><mn>2</mn></msup></math>. If the mean square deviations about <math><mo>â</mo><mn>1</mn></math> and <math><mn>1</mn></math> of a set of observations are <math><mn>7</mn></math> and <math><mn>3</mn></math> respectively. The standard deviation of those observations is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><msqrt><mn>2</mn></msqrt></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>2</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>5</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><msqrt><mn>3</mn></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msqrt><mn>3</mn></msqrt></math></span> </div>

answer:
<div class="solution"><math><mfrac><mrow><mrow><munderover><mo stretchy="true">∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">n</mi></mrow></munderover><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mi mathvariant="normal">n</mi></mrow></mfrac><mo>=</mo><mn>7</mn></math>(given)<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mo>∑</mo><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi></msub><mo>+</mo><mi mathvariant="normal">n</mi><mo>=</mo><mn>7</mn><mi mathvariant="normal">n</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∑</mo><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><mo>∑</mo><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi></msub><mo>=</mo><mn>6</mn><mi mathvariant="normal">n</mi></math> … (1)<br/>Also, <math><mfrac><mrow><mrow><mo stretchy="false">∑</mo><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow><mrow><mi mathvariant="normal">n</mi></mrow></mfrac><mo>=</mo><mn>2</mn></math> (given)<br/><math><mo>⇒</mo><mo>∑</mo><msubsup><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>-</mo><mn>2</mn><mo>∑</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">n</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">n</mi></math><br/><math><mo>⇒</mo><mo>∑</mo><msubsup><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>-</mo><mn>2</mn><mo>∑</mo><msub><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mi mathvariant="normal">i</mi></mrow></msub><mo>=</mo><mn>2</mn><mi mathvariant="normal">n</mi></math> … (2)<br/>From <math><mo>(</mo><mn>1</mn><mo>)</mo></math> and <math><mo>(</mo><mn>2</mn><mo>)</mo></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup><mo>=</mo><mn>4</mn><mi mathvariant="normal">n</mi><mo>,</mo><mo> </mo><mo>∑</mo><msub><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi></msub><mo>=</mo><mi mathvariant="normal">n</mi></math><br/>Standard deviation<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mo>∑</mo><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mn>2</mn></msubsup></mrow><mi mathvariant="normal">n</mi></mfrac><mo>-</mo><msup><mfenced separators="|"><mfrac><mrow><mo>∑</mo><msubsup><mi mathvariant="normal">x</mi><mi mathvariant="normal">i</mi><mrow></mrow></msubsup></mrow><mi mathvariant="normal">n</mi></mfrac></mfenced><mn>2</mn></msup></msqrt></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn><mi mathvariant="normal">n</mi></mrow><mi mathvariant="normal">n</mi></mfrac><mo>-</mo><mn>1</mn></msqrt><mo>=</mo><msqrt><mn>3</mn></msqrt></math></div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">50</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-weight-of-150-students-in-a-certain-class-is-60-kg-the-mean-weight-of-boys-in-the-class-is-70-kg-and-that-of-girls-is-55-kg-what-is-the-numbe

slug:
the-mean-weight-of-150-students-in-a-certain-class-is-60-kg-the-mean-weight-of-boys-in-the-class-is-70-kg-and-that-of-girls-is-55-kg-what-is-the-numbe-20095

question:
<div class="question">The mean weight of 150 students in a certain class is $60 \mathrm{~kg}$. The mean weight of boys in the class is $70 \mathrm{~kg}$ and that of girls is $55 \mathrm{~kg}$. What is the number of boys in the class?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">50</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">55</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">60</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">100</span> </li> </ul>

answer:
<div class="solution">Number of students $=150$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/VmZTTBGyiC087zp_YSkt3NKOYljP61lltSyFHK9tIS4.original.fullsize.png"/><br/><br/>By aligation, ratio $=1: 2$<br/>$\therefore$ No. of boys $=\frac{1}{3} \times 150=50$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 2$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-weight-of-all-the-students-in-a-certain-class-is-60-kg-the-mean-weight-of-the-boys-from-the-class-is-70-kg-while-that-of-the-girls-is-55-kg-w

slug:
the-mean-weight-of-all-the-students-in-a-certain-class-is-60-kg-the-mean-weight-of-the-boys-from-the-class-is-70-kg-while-that-of-the-girls-is-55-kg-w-31694

question:
<div class="question">The mean weight of all the students in a certain class is $60 \mathrm{~kg}$. The mean weight of the boys from the class is $70 \mathrm{~kg}$. while that of the girls is $55 \mathrm{~kg}$. What is the ratio of number of boys to that of girls?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2: 1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1: 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1: 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4: 1$</span> </li> </ul>

answer:
<div class="solution">Let there be $x$ number of boys and $y$ number of girls. Total students $=\mathrm{x}+\mathrm{y}$<br/>Total weight of the students $=(\mathrm{x}+\mathrm{y}) 60$ Total weight for boys $=x \times 70$ Total weight for girls $=\mathrm{y} \times 55$ Hence, $(\mathrm{x}+\mathrm{y}) 60=70 \mathrm{x}+55 \mathrm{y}$<br/>$60 x+60 y=70 x+55 y$<br/>$5 y=10 x \Rightarrow y=2 x$<br/>$\frac{\mathrm{x}}{\mathrm{y}}=\frac{1}{2} \Rightarrow \mathrm{x}: \mathrm{y}=1: 2$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2020 (09 Sep Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-means-of-two-groups-of-observations-a-and-b-are-x-y-respectively-and-their-standard-deviations-are-respectively-2-and-3-in-order-that-the-group-a-

slug:
the-means-of-two-groups-of-observations-a-and-b-are-x-y-respectively-and-their-standard-deviations-are-respectively-2-and-3-in-order-that-the-group-a-75310

question:
<div class="question">The means of two groups of observations <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>→</mo></mover><mo>,</mo><mo> </mo><mover><mi>y</mi><mo>→</mo></mover></math> respectively and their standard deviations are respectively <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo></math> In order that the group <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is to be more consistent than the group <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mover><mi>y</mi><mo>→</mo></mover><mover><mi>x</mi><mo>→</mo></mover></mfrac><mo>&lt;</mo></math><gwmw style="display:none;"></gwmw></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>1</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>6</mn><mn>5</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac></math></span> </div>

answer:
<div class="solution">We know, coefficient of variation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>σ</mi><mover><mi>x</mi><mo>→</mo></mover></mfrac><mo>×</mo><mn>100</mn></math><p>Given that, the group <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo></math>is more consistent than the group <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo></math></p><p>So, coefficient of variation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>&lt;</mo></math>coefficient of variation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo> </mo></math></p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mmultiscripts><mi>A</mi><mprescripts></mprescripts><none></none><mi>σ</mi></mmultiscripts><mover><mi>x</mi><mo>→</mo></mover></mfrac><mo>&lt;</mo><mfrac><mmultiscripts><mi>B</mi><mprescripts></mprescripts><none></none><mi>σ</mi></mmultiscripts><mover><mi>y</mi><mo>→</mo></mover></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mover><mi>y</mi><mo>→</mo></mover><mover><mi>x</mi><mo>→</mo></mover></mfrac><mo>&lt;</mo><mfrac><mmultiscripts><mi>B</mi><mprescripts></mprescripts><none></none><mi>σ</mi></mmultiscripts><mmultiscripts><mi>A</mi><mprescripts></mprescripts><none></none><mi>σ</mi></mmultiscripts></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p></div>


tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-means-of-two-samples-of-size-40-and-50-were-found-to-be-54-and-63-respectively-their-standard-deviations-were-6-and-9-respectively-the-variance-of

slug:
the-means-of-two-samples-of-size-40-and-50-were-found-to-be-54-and-63-respectively-their-standard-deviations-were-6-and-9-respectively-the-variance-of-67490

question:
<div class="question">The means of two samples of size <math><mn>40</mn></math> and <math><mn>50</mn></math> were found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>54</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn></math> respectively. Their standard deviations were <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> respectively. The variance of the combined sample of size <math><mn>90</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>90</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>7</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>9</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>81</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>81</mn></math></span> </div>

answer:
<div class="solution"><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>40</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub><mo>=</mo><mn>50</mn></math><br/><math><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>54</mn></math>, <math><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>63</mn></math><br/><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>6</mn></math>, <math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>9</mn></math><br/><math><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>40</mn><mo>×</mo><mn>54</mn><mo>+</mo><mn>50</mn><mo>×</mo><mn>63</mn></mrow><mrow><mn>90</mn></mrow></mfrac><mo>=</mo><mn>5</mn><mn>9</mn></math><br/><math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>-</mo><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>5</mn></math><br/><math><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><mover accent="false"><mrow><mi>x</mi></mrow><mo>¯</mo></mover><mo>=</mo><mn>4</mn></math><br/>Combined <math><mi mathvariant="normal">S</mi><mo>.</mo><mi mathvariant="normal">D</mi><mo>=</mo><msqrt><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><msqrt><mfrac><mrow><mn>40</mn><mo>×</mo><mn>25</mn><mo>+</mo><mn>50</mn><mo>×</mo><mn>16</mn><mo>+</mo><mn>40</mn><mo>×</mo><mn>36</mn><mo>+</mo><mn>50</mn><mo>×</mo><mn>81</mn></mrow><mrow><mn>90</mn></mrow></mfrac></msqrt></math><br/><math><mo>=</mo><mn>9</mn></math><br/>Variance <math><mo>=</mo><mn>81</mn></math></div>


title: the-measure-of-dispersion-is
slug: the-measure-of-dispersion-is-68343
question: <div class="question">The measure of dispersion is</div>
tags: ['Mathematics', 'Statistics', 'JEE Main']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Mean deviation</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">S.D.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Quartile deviation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">All of these</span> </div>

answer:
<div class="solution">Measures of dispersion include:<br/>1)Sample standard deviation<br/>2)Interquartile range (IQR) or Interdecile range<br/>3)Range<br/>4)Mean difference<br/>5)Median absolute deviation (MAD)<br/>6)Average absolute deviation (or simply called average deviation)<br/>7)Distance standard deviation</div>


title: the-measure-of-the-angle-between-the-lines-x-2-2-x-y-cosec-y-2-0-is
slug: the-measure-of-the-angle-between-the-lines-x-2-2-x-y-cosec-y-2-0-is-59542
tags: ['Mathematics', 'Pair of Lines', 'MHT CET', 'MHT CET 2020 (20 Oct Shift 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The measure of the angle between the lines $x^{2}+2 x y \operatorname{cosec} \alpha+y^{2}=0$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{2}-\alpha$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}+\alpha$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi-\alpha$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{2}-\alpha$</span> </div>

answer:
<div class="solution">We have $x^{2}+2 x y \operatorname{cosec} \alpha+y^{2}=0$<br/>Comparing it with standard form, we get $\mathrm{a}=1, \mathrm{~h}=\operatorname{cosec} \alpha, \mathrm{b}=1$<br/>Let $\theta$ be the angle between the lines.<br/>$$<br/>\begin{aligned}<br/>\tan \theta &amp;=\left|\frac{2 \sqrt{\mathrm{h}^{2}-\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}\right| \\<br/>&amp;=\left|\frac{2 \sqrt{\operatorname{cosec}^{2} \alpha-1}}{1+1}\right|=\left|\frac{2 \sqrt{\cot ^{2} \alpha}}{2}\right| \\<br/>\tan \theta &amp;=\cot \alpha \Rightarrow \tan \theta=\tan \left(\frac{\pi}{2}-\alpha\right) \\<br/>\therefore \theta=\frac{\pi}{2}-\alpha &amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Current Electricity']
source_db: MarksBatch2_P3.db
title:
the-measurement-of-an-unknown-resistance-r-is-to-be-carried-out-using-wheatstones-bridge-as-given-in-the-figure-below-two-students-perform-an-experime

slug:
the-measurement-of-an-unknown-resistance-r-is-to-be-carried-out-using-wheatstones-bridge-as-given-in-the-figure-below-two-students-perform-an-experime-22628

question:
<div class="question">The measurement of an unknown resistance $\mathrm{R}$ is to be carried out using. Wheatstones bridge as given in the figure below. Two students perform an experiment in two ways. Thefirst studentstakes $R_2=10 \Omega$ and $R_1=5 \Omega$. The other student takes $\mathrm{R}_2=1000 \Omega$ and $\mathrm{R}_1=500 \Omega$. In the standard arm, both take $\mathrm{R}_3=5 \Omega$.<br/>Both find $\mathrm{R}=\frac{\mathrm{R}_2}{\mathrm{R}_1}, \mathrm{R}_3=10 \Omega$ within errors.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>The errors of measurement of the two students are the same<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>Errors of measurement do depend on the accuracy with which $\mathrm{R}_2$ and $\mathrm{R}_1$ can be measured<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>If the student uses large values of $R_2$ and $R_1$ the currents through the arms will be feeble. This will make determination of null point accurately more difficult<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Wheatstone bridge is a very accurate instrument and has no errors of measurement</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><br/>Errors of measurement do depend on the accuracy with which $\mathrm{R}_2$ and $\mathrm{R}_1$ can be measured<br/>, <br/>If the student uses large values of $R_2$ and $R_1$ the currents through the arms will be feeble. This will make determination of null point accurately more difficult<br/></span> </div>

answer:
<div class="solution">As given that,<br/>Resistance for first student, $\mathrm{R}_1=5 \Omega, \mathrm{R}_2=10 \Omega, \mathrm{R}_3=5 \Omega$<br/>Resistance for second student, $\mathrm{R}_1=500 \Omega$, $\mathrm{R}_2=1000 \Omega, \mathrm{R}_3=5 \Omega$<br/>Now, according to Wheatstone bridge balanced condition, $\frac{\mathrm{R}_2}{\mathrm{R}}=\frac{\mathrm{R}_1}{\mathrm{R}_3} \quad$ or $\quad \mathrm{R}=\mathrm{R}_3 \times \frac{\mathrm{R}_2}{\mathrm{R}_1}$<br/><img src="https://cdn-question-pool.getmarks.app/modules/cbse/U4W1Lkt-Br3CvRWZNB-sdTOEMWBx_r-hwIqXNQCK1fY.original.fullsize.png"/><br/><br/>Now putting all the values in Eq. (i), we get $R=10 \Omega$ for both students. Thus, we can analyse that the Wheatstone bridge is most sensitive and accurate if resistances are of same value. So the results of both students depend on the accuracy of resistances used.<br/>When $R_2$ and $R_1$ are larger, the currents through galvanometer becomes very weak. It can make the determination of null point accurately more complex.</div>


title: the-mechanism-of-action-of-a-steroidal-hormone-on-target-cell-is-established-through-the
slug: the-mechanism-of-action-of-a-steroidal-hormone-on-target-cell-is-established-through-the-27507
tags: ['Biology', 'Chemical Coordination and Integration', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">The mechanism of action of a steroidal hormone on target cell is established through the</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">formation of secondary messengers.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">activation of transcriptional apparatus.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">attachment with cytosolic receptor protein.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">regulation of gene expression.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">regulation of gene expression.</span> </div>

answer:
<div class="solution">The mechanism of action of steroid hormones like estrogen, testosterone, etc. is through regulation of gene expression. These hormones interact with the intercellular receptors. They form a hormone-receptor complex with the genome.<br/><br/>Hormones like FSH do not enter the target cell. They interact with the membrane-bound receptors and generate secondary messengers like cyclic AMP for their action.</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
the-median-ad-of-a-triangle-abc-is-bisected-at-f-and-bf-is-produced-to-meet-the-side-a-c-in-p-if-a-p-a-c-then-what-is-the-value-of

slug:
the-median-ad-of-a-triangle-abc-is-bisected-at-f-and-bf-is-produced-to-meet-the-side-a-c-in-p-if-a-p-a-c-then-what-is-the-value-of-88637

question:
<div class="question">The median AD of a triangle $\mathrm{ABC}$ is bisected at $\mathrm{F}$, and $\mathrm{BF}$ is produced to meet the side $A C$ in $P$. If $A P=\lambda A C$, then what is the value of $\lambda$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $\mathrm{AD}$ is the median of $\triangle \mathrm{ABC} . \mathrm{F}$ is mid-point of AD and $\mathrm{BF}$ is produced to meet the side $\mathrm{AC}$ in $\mathrm{P}$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/3GPDQMm_I-AFIj2JFYKJwjJoiPLCRLGb2r0ungg2a3o.original.fullsize.png"/><br/><br/>Draw DQ $\| \mathrm{FP}$ In $\Delta \mathrm{ADQ}, \mathrm{F}$ is mid-point of $\mathrm{AD}$ and $\mathrm{FP} \| \mathrm{DQ}$. Pis mid-point of $A Q$ (converse of mid-point theorem) $\Rightarrow \mathrm{AP}=\mathrm{PQ}$ ...(1)<br/>In $\Delta \mathrm{BCP}, \mathrm{D}$ is mid-point of $\mathrm{BC}$ and $\mathrm{DQ} \| \mathrm{BP}$.<br/>$\therefore$ Q is midpoint of $P C$. $\Rightarrow \mathrm{PQ}=\mathrm{QC}$ ...(2)<br/>From $(1),(2)$ we get, $A P=P Q=Q C$ From figure, $\mathrm{AP}+\mathrm{PQ}+\mathrm{QC}=\mathrm{AC}$<br/>$\Rightarrow \mathrm{AP}+\mathrm{AP}+\mathrm{AP}=\mathrm{AC}$<br/>$\Rightarrow 3 \times \mathrm{AP}=\mathrm{AC}$<br/>$\Rightarrow \mathrm{AP}=\frac{1}{3} \times \mathrm{AC} \quad \therefore \quad \lambda=\frac{1}{3}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">19</span> </div>
source_db: MarksBatch2_P3.db
title:
the-median-of-27-observations-of-a-variable-is-18-three-more-observations-are-made-and-the-values-of-these-observations-are-1618-and-50-what-is-the-me

slug:
the-median-of-27-observations-of-a-variable-is-18-three-more-observations-are-made-and-the-values-of-these-observations-are-1618-and-50-what-is-the-me-80905

question:
<div class="question">The median of 27 observations of a variable is 18 . Three more observations are made and the values of these observations are 16,18 and 50 . What is the median of these 30 observations ? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">18</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">19</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$25.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Can not be determined due to insufficient data</span> </li> </ul>

answer:
<div class="solution">Median is middle of data. Observations are 27 and median is $18 .$ So, sum of all the observation are $18 \times 27$ $=486$ Now, 16,18 and 50 are additional three observations. So, Total $=486+16+18+50=570$.<br/>and number of obs. are 30 .<br/>Median $=\frac{570}{30}=19$</div>


tags: ['Biology', 'Animal Kingdom', 'NEET']
source_db: MarksBatch2_P3.db
title:
the-member-of-this-phylum-are-terrestrial-or-aquatic-having-an-organ-system-level-organisation-bilaterally-symmetrical-triploblastic-segmented-and-euc

slug:
the-member-of-this-phylum-are-terrestrial-or-aquatic-having-an-organ-system-level-organisation-bilaterally-symmetrical-triploblastic-segmented-and-euc-46423

question:
<div class="question">The member of this Phylum are terrestrial or aquatic having an organ system level organisation bilaterally symmetrical, triploblastic, segmented and eucoelomate animals?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Mollusca</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Arthropoda</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Platyhelminthes</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Echinodermata</span> </li> </ul>

answer:
<div class="solution">The body od arthropoda is Bilateral, Triploblastic with organ system level of organization. Head is distinct  with high degree of caphalization, consists of many fused segments bears well developed sense organ such as eyes, compound eyes, antennae. Arthropoda have various shapes with externally segmented body. Segmentation is either two region head and trunk or cephalothorax and abdomen or three region with head, thorax and abdomen. Some or all segments bear jointed appendages. Hence name Arthro - joints, poda - foot. No internal segmentation like Annelida. Body wall consists of cuticle composed of protein and polysaccharide chitin which is further strengthened by deposition of mineral (calcium phosphate and carbonate).<br/></div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
title:
the-meristem-that-occurs-in-the-mature-regions-of-root-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis-and-appear-later-than-prim

slug:
the-meristem-that-occurs-in-the-mature-regions-of-root-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis-and-appear-later-than-prim-49418

question:
<div class="question">The meristem that occurs in the mature regions of root and shoots of many plants, particularly those that produce woody axis and appear later than primary meristem is called</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">secondary meristem</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">lateral meristem</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">both a and b</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">permanent tissue</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">both a and b</span> </div>

answer:
<div class="solution">The meristematic cell present on mature root and shoot which produce a woody axis known as lateral meristem. They are cylindrical meristem and are the secondary meristems that induce the growth of the plant with increasing girth.</div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
title:
the-meristem-that-occurs-in-the-mature-regions-of-roots-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis-and-appear-later-than-pri

slug:
the-meristem-that-occurs-in-the-mature-regions-of-roots-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis-and-appear-later-than-pri-46641

question:
<div class="question">The meristem that occurs in the mature regions of roots and shoots of many plants, particularly those that produce woody axis and appear later than primary meristem, is called</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Intercalary meristem</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Apical meristem</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Secondary meristem</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Secondary meristem</span> </div>

answer:
<div class="solution">The meristem that occurs in the mature regions of roots and shoots of many plants, particularly those that produce woody axis and appear later than primary meristem, is called secondary or lateral meristem. They are cylindrical meristems and are responsible for producing the secondary tissue.</div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
title:
the-meristem-that-occurs-in-the-mature-regions-of-roots-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis

slug:
the-meristem-that-occurs-in-the-mature-regions-of-roots-and-shoots-of-many-plants-particularly-those-that-produce-woody-axis-56488

question:
<div class="question">The meristem that occurs in the mature regions of roots and shoots of many plants, particularly those that produce woody axis</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary meristem</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Secondary meristem</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Intercalary meristem</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both (B) and (C)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Secondary meristem</span> </div>

answer:
<div class="solution">The meristem that occurs in the mature regions of roots and shoots of many plants, particularly those that produce woody axis and appear later than primary meristem is called the secondary or lateral meristem. They are cylindrical meristems. Fascicular vascular cambium, interfascicular cambium, and cork-cambium are examples of lateral meristems. These are responsible for producing the secondary tissues.</div>


title: the-meristem-which-occurs-in-grasses-and-regenerates-parts-removed-by-the-grazing-herbivores-is
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
slug:
the-meristem-which-occurs-in-grasses-and-regenerates-parts-removed-by-the-grazing-herbivores-is-86971

question:
<div class="question"><p>The meristem which occurs in grasses and regenerates parts removed by the grazing herbivores is</p> </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Apical meristem</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Intercalary meristem</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Secondary meristem</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cambium</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Intercalary meristem</span> </div>

answer:
<div class="solution"><p>The meristem which occurs between mature tissues is known as intercalary meristem. They occur in grasses and regenerate parts removed by the grazing herbivores.</p></div>


title: the-metal-that-cannot-be-obtained-by-the-electrolysis-of-an-aqueous-solution-of-its-salt-is
slug: the-metal-that-cannot-be-obtained-by-the-electrolysis-of-an-aqueous-solution-of-its-salt-is-14065
tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2014 (06 Apr)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The metal that cannot be obtained by the electrolysis of an aqueous solution of its salt is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ag</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ca</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Cu</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Cr</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ca</mi></math></span> </div>

answer:
<div class="solution">For the reduction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math> at cathode,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>+</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mo>-</mo></msup><mo>→</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>+</mo><mn>2</mn><msup><mi>OH</mi><mo>-</mo></msup><mo>;</mo><mo> </mo><msubsup><mi mathvariant="normal">E</mi><mi>red</mi><mo>∘</mo></msubsup><mo>=</mo><mo>-</mo><mn>1.23</mn><mi mathvariant="normal">V</mi></math><br/><br/>For <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Ca</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mo>-</mo></msup><mo>→</mo><mi>Ca</mi><mo>;</mo><mo> </mo><msubsup><mi mathvariant="normal">E</mi><mi>red</mi><mo>∘</mo></msubsup><mo>=</mo><mo>-</mo><mn>2.87</mn><mi mathvariant="normal">V</mi></math><br/><br/>i.e., water will be reduced at the cathode, in preference to the liberation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ca</mi></math>.<br/><br/>However, in the case of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ag</mi><mo>,</mo><mo> </mo><mi>Cu</mi><mo> </mo><mi>and</mi><mo> </mo><mi>Cr</mi></math>, their standard reduction potentials are higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>.</mo><mn>23</mn><mo> </mo><mi mathvariant="normal">V</mi></math> and will be liberated/ deposited at the cathode and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math> will not be reduced.</div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2024 (31 Jan Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-metals-that-are-employed-in-the-battery-industries-are-a-fe-b-mn-c-ni-d-cr-e-cd-choose-the-correct-answer-from-the-options-given-below

slug:
the-metals-that-are-employed-in-the-battery-industries-are-a-fe-b-mn-c-ni-d-cr-e-cd-choose-the-correct-answer-from-the-options-given-below-81836

question:
<div class="question">The metals that are employed in the battery industries are A. Fe,   B. Mn,   C. Ni,   D. Cr,   E. Cd<p>Choose the correct answer from the options given below:</p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">B, C and E only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A, B, C, D and E</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A, B, C and D only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">B, D and E only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">B, C and E only</span> </div>

answer:
<div class="solution"><p>A rechargeable nickel-cadmium cell in a jelly roll arrangement and separated by a layer soaked in moist sodium or potassium hydroxide. Another important secondary cell is the nickel-cadmium cell which has longer life than the lead storage cell but more expensive to manufacture. We shall not go into details of working of the cell and the electrode reactions during charging and discharging. The overall reaction during discharge is.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cd</mi><mfenced><mi mathvariant="normal">s</mi></mfenced></msub><mo>+</mo><mn>2</mn><mi>Ni</mi><msub><mfenced><mi>OH</mi></mfenced><mn>3</mn></msub><mo>â</mo><msub><mi>CdO</mi><mfenced><mi mathvariant="normal">s</mi></mfenced></msub><mo>+</mo><mn>2</mn><mi>Ni</mi><msub><msub><mfenced><mi>OH</mi></mfenced><mn>2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></msub><mo>+</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mfenced><mi mathvariant="normal">l</mi></mfenced></msub></math></p><p>The primary cell LeclanchÃ© cell cell is represented asÂ </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Zn</mi><mo>|</mo><msup><mi>Zn</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>â¥</mo><msub><mi>MnO</mi><mn>2</mn></msub><mo>|</mo><msub><mi>Mn</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub><mo>|</mo><mi mathvariant="normal">C</mi></math></p><p>Mn, Ni and Cd metals used in battery industries.</p></div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 unit</span> </div>
source_db: MarksBatch2_P3.db
title:
the-middle-point-of-a-1-2-and-b-x-y-is-c-2-4-if-bd-is-perpendicular-to-ab-such-that-cd-3-umit-then-what-is-the-length-bd

slug:
the-middle-point-of-a-1-2-and-b-x-y-is-c-2-4-if-bd-is-perpendicular-to-ab-such-that-cd-3-umit-then-what-is-the-length-bd-30536

question:
<div class="question">The middle point of $\mathrm{A}(1,2)$ and $\mathrm{B}(\mathrm{x}, \mathrm{y})$ is $\mathrm{C}(2,4)$. If BD is perpendicular to $\mathrm{AB}$ such that $\mathrm{CD}=3$ umit, then what is the length BD?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \sqrt{2}$ wit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3 unit</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \sqrt{2}$ unit</span> </li> </ul>

answer:
<div class="solution">Given that mid point of $\mathrm{A}(1,2)$ and $\mathrm{B}(\mathrm{x}, \mathrm{y})$ is $\mathrm{C}(2,4)$,<br/>$\frac{1+x}{2}=2$ and $\frac{2+y}{2}=4$<br/>$\Rightarrow \quad x=3$ and $y=6$<br/>So, coordinates of B are $(3,6)$. Given that<br/>$\mathrm{BD} \perp \mathrm{AB}$ and $\mathrm{CD}=3$ unit<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/zVmvnhpGFvdwQh_dBoWGhHY6PgaPRfxhehxPyTXvSM0.original.fullsize.png"/><br/><br/>$\mathrm{BC}=\sqrt{(2-3)^{2}+(4-6)^{2}}=\sqrt{1+4}=\sqrt{5}$<br/>In right angled $\Delta \mathrm{BCD}, \mathrm{CD}^{2}=\mathrm{BC}^{2}+\mathrm{BD}^{2}$<br/>$\Rightarrow 9=5+\mathrm{BD}^{2} \Rightarrow \mathrm{BD}^{2}=4 \Rightarrow \mathrm{BD}=2 \mathrm{unit}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
the-middle-point-of-the-segment-of-the-straight-line-joining-the-points-p-q-and-q-p-is-r-2-s-2-what-is-the-length-of-the-segment

slug:
the-middle-point-of-the-segment-of-the-straight-line-joining-the-points-p-q-and-q-p-is-r-2-s-2-what-is-the-length-of-the-segment-37707

question:
<div class="question">The middle point of the segment of the straight line joining the points $(p, q)$ and $(q,-p)$ is $(r / 2, s / 2)$. What is the length of the segment?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\left(s^{2}+r^{2}\right)^{1 / 2}\right] / 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\left(s^{2}+r^{2}\right)^{1 / 2}\right] / 4$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(s^{2}+r^{2}\right)^{1 / 2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$s+r$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(s^{2}+r^{2}\right)^{1 / 2}$</span> </div>

answer:
<div class="solution">Two joining points are $(p, q)$ and $(q,-p)$ Mid point of $(p, q)$ and $(q,-p)$ is $\left(\frac{p+q}{2}, \frac{q-p}{2}\right)$<br/>But it is given that the mid-point is $\left(\frac{r}{2}, \frac{s}{2}\right)$.<br/>$\therefore \frac{p+q}{2}=\frac{r}{2}$ and $\frac{q-p}{2}=\frac{s}{2}$<br/>$\Rightarrow p+q=r$ and $q-p=s$<br/>Now, length of segment $=\sqrt{(p-q)^{2}+(q+p)^{2}}$ (by distance formula) $=\sqrt{s^{2}+r^{2}}=\left(s^{2}+r^{2}\right)^{1 / 2}$</div>


title: the-minimum-distance-from-the-point-4-2-to-y-2-8-x-is-equal-to
slug: the-minimum-distance-from-the-point-4-2-to-y-2-8-x-is-equal-to-69749
question: <div class="question">The minimum distance from the point $(4,2)$ to $y^{2}=8 x$ is equal to</div>
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \sqrt{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{2}$</span> </div>

answer:
<div class="solution">Given equation is $\mathrm{y}^{2}-8 \mathrm{x}$ Take an arbitrary point on this curve. If we take y as $\mathrm{P}$.<br/>then point is $\left(\frac{P^{2}}{8}, P\right)$.<br/>The distance between $\left(\frac{P^{2}}{8}, P\right)$ and $(4,2)$ is<br/>$d^{2}=\left(\frac{P^{2}}{8}-4\right)^{2}+(P-2)^{2}$ ...(1)<br/>$=\frac{1}{64}\left(P^{2}-32\right)^{2}+(P-2)^{2}$<br/>$\Rightarrow 2 d \cdot \frac{d d}{d p}=\frac{1}{64} \times 2\left(P^{2}-32\right) \times 2 P+2(P-2)$<br/>$=\frac{1}{16}\left(P^{2}-32\right) P+2(P-2)$<br/>for minimum distance, $\frac{d d}{d p}=0$ $\Rightarrow \mathrm{P}^{3}-32 \mathrm{P}+32 \mathrm{P}-64=0$<br/>$\Rightarrow \mathrm{P}^{3}=64$<br/>$\Rightarrow \mathrm{P}=4$<br/>$\therefore(1) \Rightarrow d^{2}=\left(\frac{16}{8}-4\right)^{2}+(4-2)^{2}$<br/>$=(-2)^{2}+(2)^{2}=8$<br/>$\Rightarrow d=\sqrt{8}=2 \sqrt{2}$</div>


title: the-minimum-value-of-the-function-f-x-x-4-exists-at
slug: the-minimum-value-of-the-function-f-x-x-4-exists-at-37783
question: <div class="question">The minimum value of the function $f(x)=|x-4|$ exists at</div>
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=4$</span> </div>
answer: <div class="solution">$\quad$ At $x=4, f(x)=0$</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x=4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=-4$</span> </li> </ul>


tags: ['Chemistry', 'Coordination Compounds', 'JEE Main', 'JEE Main 2023 (13 Apr Shift 1)']
source_db: MarksBatch2_P3.db
title:
the-mismatched-combinations-are-a-chlorophyll-co-b-water-hardness-edta-c-photography-ag-cn-2-d-wilkinson-catalyst-ph-3-p-3-rhcl-e-chelating-ligand-d-p

slug:
the-mismatched-combinations-are-a-chlorophyll-co-b-water-hardness-edta-c-photography-ag-cn-2-d-wilkinson-catalyst-ph-3-p-3-rhcl-e-chelating-ligand-d-p-30221

question:
<div class="question"><p>The mismatched combinations are</p><p>A. Chlorophyll<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mi>Co</mi></math></p><p>B. Water hardness<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mi>EDTA</mi></math></p><p>C. Photography<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced close="]" open="["><mrow><mi>Ag</mi><mo>(</mo><mi>CN</mi><msub><mo>)</mo><mn>2</mn></msub></mrow></mfenced><mo>-</mo></msup></math></p><p>D. Wilkinson catalyst<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfenced close="]" open="["><mrow><msub><mfenced><mrow><msub><mi>Ph</mi><mn>3</mn></msub><mi mathvariant="normal">P</mi></mrow></mfenced><mn>3</mn></msub><mi>RhCl</mi></mrow></mfenced></math></p><p>E. Chelating ligand<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mi mathvariant="normal">D</mi><mo>-</mo></math>Penicillamine</p><p>Choose the correct answer from the options given below.</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><p>A, C and E only</p></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><p>A and C only</p></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><p>A and E only</p></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><p>D and E only</p></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><p>A and C only</p></span> </div>

answer:
<div class="solution"><p>The chlorophyll molecule has a central magnesium atom engulfed by a nitrogen-containing structure called a porphyrin ring; attached to the ring is a long carbon-hydrogen side chain.</p><p>Hardness of water is determined by titrating with a standard solution of ethylene diamine tetra acetic acid (EDTA) which is a complexing agent. Since EDTA is insoluble in water, the disodium salt of EDTA is taken for this experiment. EDTA can form four or six coordination bonds with a metal ion.</p><p>Silver bromide is used in photography as a component of an emulsion that helps develop a photographic image. Silver bromide is sensitive to light, and when suspended in gelatin, silver bromide's grains create a photographic emulsion.</p><p>Wilkinson's catalyst is the common name for chlorido­tris(triphenylphosphine)­rhodium(I), a coordination complex of rhodium with the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msub><mfenced><mrow><msub><mi>Ph</mi><mn>3</mn></msub><mi mathvariant="normal">P</mi></mrow></mfenced><mn>3</mn></msub><mi>RhCl</mi></mrow></mfenced></math></p><p>D-Penicillamine is chelating ligand.</p></div>


title: the-modulus-and-principal-argument-of-the-complex-number-1-1-i-2-1-2-i-are-respectively
slug: the-modulus-and-principal-argument-of-the-complex-number-1-1-i-2-1-2-i-are-respectively-28193
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1,0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The modulus and principal argument of the complex number<br/>$\frac{1+2 i}{1-(1-i)^{2}}$ are respectively</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1,0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1,1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2,0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2,1</span> </li> </ul>

answer:
<div class="solution">We know, if $Z=x+$ iy,$|z|=\sqrt{x^{2}+y^{2}}$<br/>Given, $\frac{1+2 i}{1-(1-i)^{2}}=\frac{1+2 i}{1-\left(1-2 i+i^{2}\right)}=\frac{1+2 i}{1+2 i}=1$<br/>Since, it is purelyreal number, modulus $=1$ and principal argument $=0$</div>


title: the-modulusamplitude-form-of-3-i-where-i-1-is
slug: the-modulusamplitude-form-of-3-i-where-i-1-is-47775
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The modulus-amplitude form of $\sqrt{3}+\mathrm{i}$, where $\mathrm{i}=\sqrt{-1}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$</span> </div>

answer:
<div class="solution">Let $x+$ iy $=\sqrt{3}+$ i<br/>Comparing real and imaginary parts, $\mathrm{x}=\sqrt{3}, \mathrm{y}=1$ modulus-amplitude of $\mathrm{x}+$ iy is $\mathrm{r}(\cos \theta+\sin \theta)$, where<br/>$r=\sqrt{x^{2}+y^{2}}$ and $\theta=\tan ^{-1}\left(\frac{y}{x}\right)$<br/>$\mathrm{r}=\sqrt{(\sqrt{3})^{2}+(1)^{2}}=\sqrt{4}=2$<br/>$\theta=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}$<br/>$\therefore$ modulus-amplitude is $2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 2)']
answer: <div class="solution">Required expenditure $=9000 \times \frac{10}{100}$<br/>$=â¹ 900 .$</div>
source_db: MarksBatch2_P3.db
title:
the-monthly-family-expenditure-in-percentage-on-different-items-are-as-follows-beginarraycccccc-hline-food-rent-cloth-transport-education-others-hline

slug:
the-monthly-family-expenditure-in-percentage-on-different-items-are-as-follows-beginarraycccccc-hline-food-rent-cloth-transport-education-others-hline-77609

question:
<div class="question">The monthly family expenditure (in percentage) on different items are as follows:<br/>\begin{array}{|c|c|c|c|c|c|}<br/>\hline Food &amp; Rent &amp; Cloth &amp; Transport &amp; Education &amp; Others \\<br/>\hline 38 &amp; 19 &amp; 18 &amp; - &amp; 9 &amp; 6 \\<br/>\hline<br/>\end{array} If the total monthly expenditure is $Ã¢âÂ¹ 9000$, then what is the benditure on transport</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$Ã¢âÂ¹ 180$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Ã¢âÂ¹1000</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad$ Ã¢âÂ¹900</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Ã¢âÂ¹ 360</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad$ Ã¢âÂ¹900</span> </div>


tags: ['Chemistry', 'Redox Reactions']
source_db: MarksBatch2_P3.db
title:
the-more-positive-the-value-of-e-the-greater-is-the-tendency-of-the-species-to-get-reduced-using-the-standard-electrode-potential-of-redox-couples-giv

slug:
the-more-positive-the-value-of-e-the-greater-is-the-tendency-of-the-species-to-get-reduced-using-the-standard-electrode-potential-of-redox-couples-giv-61105

question:
<div class="question">The more positive the value of $E^{\ominus}$, the greater is the tendency of the species to get reduced. Using the standard electrode potential of redox couples given below find out which of the following is the strongest oxidising agent. <br/>$\mathrm{E}^{\ominus}$ values $: \mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}=+0.77 \mathrm{~V} ; \mathrm{I}_2(\mathrm{s}) / \mathrm{I}^{-}=+0.54$; $\mathrm{Cu}^{2+} / \mathrm{Cu}=+0.34 ; \mathrm{Ag}^{+} / \mathrm{Ag}=0.80 \mathrm{~V}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$\mathrm{Fe}^{3+}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$\mathrm{I}_2(\mathrm{s})$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\mathrm{Cu}^{2+}$<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>$\mathrm{Ag}^{+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$\mathrm{Ag}^{+}$</span> </div>

answer:
<div class="solution">Since the redox couple $\mathrm{Ag}^{+} / \mathrm{Ag}$ has the highest positive value of $\mathrm{E}^0$ i.e., $0.80 \mathrm{~V}$, therefore, $\mathrm{Ag}^{+}$is the strongest oxidising agent.</div>


title: the-morphological-nature-of-the-edible-part-of-coconut-is
slug: the-morphological-nature-of-the-edible-part-of-coconut-is-49995
question: <div class="question">The morphological nature of the edible part of coconut is</div>
tags: ['Biology', 'Plant Kingdom', 'NEET', 'NEET 2017']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Perisperm</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Cotyledon</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Endosperm</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Pericarp</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Endosperm</span> </div>

answer:
<div class="solution">The edible part of coconut is endosperm. Coconut water is free nuclear endosperm and the white kernel is the cellular endosperm.<br/><br/>Initially, the coconut is a nuclear liquid endosperm. it is a coenocytic liquid tissue with lots of rapidly dividing nuclei. At some point, gelification of the soil surrounding the nuclei occurs and cellular boundaries start to appear. These cells and nuclei start to settle down on the endothelium (inner lining of the integuments) and the cells divide to produce a columnar sort of a covering over it. This is what forms the characteristic of coconut meat. Development of the cellular endosperm is by coalescence of these deposited cells and nuclei at the antipodal end and continues towards the top. The cells right above the endothelium stay meristematic for a long time while the columnar cells on top of this meristematic layer grow radially inward.</div>


title: the-most-abundant-antibody-in-colostrum-is
slug: the-most-abundant-antibody-in-colostrum-is-75136
question: <div class="question">The most abundant antibody in colostrum is</div>
tags: ['Biology', 'Human Health and Diseases', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Immunoglobulin A</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Immunoglobulin G</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Immunoglobulin M</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Immunoglobulin E</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Immunoglobulin A</span> </div>

answer:
<div class="solution">Colostrum contains large numbers of antibodies called "secretory immunoglobulin"  <math><mfenced separators="|"><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">A</mi></mrow></mfenced></math>  that help protect the mucous membranes in the throat, lungs, and intestines of the infant.</div>


title: the-most-abundant-chemical-in-living-organisms-could-be
slug: the-most-abundant-chemical-in-living-organisms-could-be-27902
question: <div class="question">The most abundant chemical in living organisms could be</div>
tags: ['Biology', 'Biomolecules']
answer: <div class="solution">Approx $70 \%$ of living orgamisms'body weight is water.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>protein<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>water<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>sugar<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>nucleic acid.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>water<br/></span> </div>


title: the-most-abundant-component-of-living-organisms-is
slug: the-most-abundant-component-of-living-organisms-is-35989
question: <div class="question">The most abundant component of living organisms is</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Water</span> </div>
answer: <div class="solution">The most abundant component of living organisms is water.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Protein</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Water</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Sugar</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Nucleic acid</span> </li> </ul>


title: the-most-abundant-element-on-earth-crust-is
slug: the-most-abundant-element-on-earth-crust-is-56297
question: <div class="question">The most abundant element on earth crust is</div>
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Oxygen</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Hydrogen</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Oxygen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Silicon</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Carbon</span> </li> </ul>

answer:
<div class="solution">$\begin{array}{lc}<br/>\text { Element } &amp; \text { % abundance by weight } \\<br/>\mathrm{O} &amp; 46.6 \\<br/>\mathrm{Si} &amp; 27.7 \\<br/>\mathrm{Al} &amp; 8.3 \\<br/>\mathrm{Fe} &amp; 5.1 \\<br/>\mathrm{Ca} &amp; 3.6<br/>\end{array}$</div>


title: the-most-abundant-element-present-in-the-plants-is
slug: the-most-abundant-element-present-in-the-plants-is-38093
question: <div class="question">The most abundant element present in the plants is:</div>
tags: ['Biology', 'Mineral Nutrition', 'NEET', 'NEET 2004']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">carbon</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">carbon</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">nitrogen</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">manganese</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">iron.</span> </li> </ul>

answer:
<div class="solution">Carbon is the most abundant element in living matter.<br/>Related Theory<br/>The six most common elements of life on Earth (including more than $97 \%$ of the mass of $a$ human body) are carbon, hydrogen, nitrogen, oxygen, sulphur and phosphorus.</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Main', 'JEE Main 2017 (02 Apr)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7.5 kg</span> </div>
source_db: MarksBatch2_P3.db
title:
the-most-abundant-elements-by-mass-in-the-body-of-a-healthy-human-adult-are-oxygen-614-carbon-229-hydrogen-100-and-nitrogen-26-the-weight-which-a-75kg

slug:
the-most-abundant-elements-by-mass-in-the-body-of-a-healthy-human-adult-are-oxygen-614-carbon-229-hydrogen-100-and-nitrogen-26-the-weight-which-a-75kg-76993

question:
<div class="question">The most abundant elements by mass in the body of a healthy human adult are:<br/><br/>Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%); and Nitrogen (2.6%). The weight which a 75kg person would gain if all <math><msup><mrow><mi> </mi></mrow><mrow><mn>1</mn></mrow></msup><mi>H</mi></math> atoms are replaced by <math><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><mi>H</mi></math> atoms is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">37.5 kg</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">7.5 kg</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">10 kg</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">15 kg</span> </li> </ul>

answer:
<div class="solution">Weight of Hydrogen in body of 75 kg person <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>75</mn><mo>×</mo><mfrac><mn>10</mn><mn>100</mn></mfrac><mo>=</mo><mn>7</mn><mo>.5</mo><mi>kg</mi></math><br/><br/>On replacing <math><msup><mrow><mi> </mi></mrow><mrow><mn>1</mn></mrow></msup><mi>H</mi></math> by <math><msup><mrow><mi> </mi></mrow><mrow><mn>2</mn></mrow></msup><mi>H</mi></math> , the new weight of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi><mo>=</mo><mn>7</mn><mo>.5</mo><mo>×</mo><mn>2</mn><mo>=</mo><mn>15</mn><mi>kg</mi></math><br/><br/>Weight increase = <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>5</mn><mi>kg</mi></math></div>


title: the-most-abundant-enzyme-in-the-biosphere-is
slug: the-most-abundant-enzyme-in-the-biosphere-is-26430
question: <div class="question">The most abundant enzyme in the biosphere is</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Collagen</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">RuBisCO</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Trypsin</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Insulin</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">RuBisCO</span> </div>

answer:
<div class="solution">The world's most abundant and most important enzyme is RuBisCo. It's the most abundant because it's present in relatively large quantities in every photosynthetic organism on the planet - from microscopic cyanobacteria and phytoplankton in the oceans to the leaves of giant-sized trees in the tropics.</div>


title: the-most-abundant-intracellular-cation-is
slug: the-most-abundant-intracellular-cation-is-62036
question: <div class="question">The most abundant intracellular cation is</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET', 'NEET 2013 (All India)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Na}^{+}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Ca}^{2+}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}^{+}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{K}^{+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{K}^{+}$</span> </div>

answer:
<div class="solution">Major intracellular cation is $\mathrm{K}^{+}$.<br/>Major extracellular cation is $\mathrm{Na}^{+}$.<br/>$\mathrm{Ca}^{+}$play major role in signal transduction pathways.<br/>$\mathrm{H}^{+}$is found in aqueous solution of all acids.</div>


title: the-most-abundant-metal-in-the-earth-crust-is
slug: the-most-abundant-metal-in-the-earth-crust-is-39744
question: <div class="question">The most abundant metal in the earth crust is</div>
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mtext> </mtext><mtext>N</mtext><mtext>a</mtext></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mtext>Â </mtext><mtext>M</mtext><mtext>g</mtext></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mtext> </mtext><mtext>A</mtext><mtext>l</mtext></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mtext>Â </mtext><mtext>F</mtext><mtext>e</mtext></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mtext> </mtext><mtext>A</mtext><mtext>l</mtext></math></span> </div>

answer:
<div class="solution">Aluminium is the most abundant metal in the Earthâs crust, and the third most abundant element there is after oxygen and silicon. It makes up about 8% by weight of the Earthâs solid surface. Earthâs crust occupies less than 1% of Earthâs volume. Oxygen is 46.6%, Silicon 27.77%, Aluminum is 8.09% and Iron is 5%.</div>


title: the-most-abundant-metal-ion-present-in-the-human-body-is
slug: the-most-abundant-metal-ion-present-in-the-human-body-is-31658
question: <div class="question">The most abundant metal ion present in the human body is</div>
tags: ['Chemistry', 's Block Elements', 'KVPY', 'KVPY 2017 (19 Nov SB/SX)']
answer: <div class="solution">Calcium Mainly present in Bones all other are required in lower amount</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Zn}^{2+}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Ca}^{2+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Na}^{+}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Fe}^{2+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Ca}^{2+}$</span> </div>


title: the-most-abundant-noble-gas-in-the-atmosphere-is
slug: the-most-abundant-noble-gas-in-the-atmosphere-is-83837
question: <div class="question">The most abundant noble gas in the atmosphere is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'MHT CET', 'MHT CET 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Argon</span> </div>
answer: <div class="solution">Argon is the most abundant noble gas present in the atmosphere.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Neon</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Argon</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Xenon</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Krypton</span> </li> </ul>


tags: ['Biology', 'Microbes in Human Welfare', 'NEET', 'NEET 2012 (Screening)']
source_db: MarksBatch2_P3.db
title:
the-most-abundant-prokaryotes-helpful-to-human-in-making-curd-from-milk-and-in-production-of-antibiotics-are-the-ones-categorized-as

slug:
the-most-abundant-prokaryotes-helpful-to-human-in-making-curd-from-milk-and-in-production-of-antibiotics-are-the-ones-categorized-as-64333

question:
<div class="question">The most abundant prokaryotes helpful to human in making curd from milk and in production of antibiotics are the ones categorized as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">cyanobacteria</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">archaebacteria</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">chemosynthetic autotrophs</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">heterotrophic bacteria</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">heterotrophic bacteria</span> </div>

answer:
<div class="solution">Heterotrophic bacteria are most abundant in nature. Many of them have a significant impact on human affairs. These are helpful in making curd from milk (e.g., Lactobacillus sp.), production of antibiotics (e.g., Streptomyces sp.) and fixing nitrogen in legume roots (e.g., Rhizobium spe).</div>


tags: ['Biology', 'Microbes in Human Welfare', 'NEET']
source_db: MarksBatch2_P3.db
title:
the-most-abundant-prokaryotes-helpful-to-humans-in-making-curd-from-milk-and-in-production-of-antibiotics-are-the-ones-categorised-as

slug:
the-most-abundant-prokaryotes-helpful-to-humans-in-making-curd-from-milk-and-in-production-of-antibiotics-are-the-ones-categorised-as-82939

question:
<div class="question">The most abundant prokaryotes helpful to humans in making curd from milk and in production of antibiotics are the ones categorised as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Cyanobacteria</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Archaebacteria</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Chemosynthetic autotrophs</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Heterotrophic bacteria</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Heterotrophic bacteria</span> </div>

answer:
<div class="solution">A maximum number of antibiotics are produced by mycelial bacteria known as actinomycetes, and most of the actinomycetes are saprotrophic (heterotrophic). Lactic acid bacteria that are used in the preparation of curd are also heterotrophic.</div>


title: the-most-abundant-protein-in-the-animal-world-is
slug: the-most-abundant-protein-in-the-animal-world-is-93225
question: <div class="question">The most abundant protein in the animal world is:</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Trypsin</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Haemoglobin</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Collagen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Insulin</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Collagen</span> </div>

answer:
<div class="solution">Collagen is an insoluble fibrous protein found extensively in the connective tissue of skin, tendons and bone. Collagen accounts for over 30% of the total body proteins of mammals and it is the most abundant animal protein.</div>


title: the-most-abundant-protein-of-the-biosphere-is-found-in
slug: the-most-abundant-protein-of-the-biosphere-is-found-in-19505
question: <div class="question">The most abundant protein of the biosphere is found in:</div>
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Chloroplasts</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Mitochondria</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Ribosomes</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Dictyosomes</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Chloroplasts</span> </div>

answer:
<div class="solution">Collagen is the most abundant protein in the animal world, and Ribulose bisphosphate Carboxylase-Oxygenase (RuBisCO) is the most abundant protein in the whole of the biosphere.<br/>Carboxylation is the fixation of CO<sub>2</sub> into a stable organic intermediate. Carboxylation is the most crucial step of the Calvin cycle where CO<sub>2</sub> is utilised for the carboxylation of RuBP. This reaction is catalysed by the enzyme RuBP carboxylase which results in the formation of two molecules of 3-PGA. Since this enzyme also has an oxygenation activity, it would be correct to call it RuBP carboxylase-oxygenase or RuBisCO.</div>


title: the-most-abundant-transition-metal-in-human-body-is
slug: the-most-abundant-transition-metal-in-human-body-is-39835
question: <div class="question">The most abundant transition metal in human body is</div>
tags: ['Chemistry', 'd and f Block Elements', 'KVPY', 'KVPY 2015 (SB/SX)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">iron</span> </div>
answer: <div class="solution">Most abundant element (transition) in human body is Iron.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">copper</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">iron</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">zinc</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">manganese</span> </li> </ul>


title: the-most-active-phagocytic-white-blood-cells-are-3
slug: the-most-active-phagocytic-white-blood-cells-are-3-90534
question: <div class="question">The most active phagocytic white blood cells are</div>
tags: ['Biology', 'Body Fluids and Circulation', 'JIPMER', 'JIPMER 2012']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">neutrophils and eosinophils</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">lymphocytes and macrophages</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">eosinophils and lymphocytes</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">neutrophils and monocytes</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">neutrophils and monocytes</span> </div>

answer:
<div class="solution">Neutrophils are most abundant granulecytes which engulf microbes by phagocytosis. These have 2-7 lobulated nucleus and do not stain any basic or acidic dye monocytes are large sized leucocytes, which are highly motile and phagocytic in nature.</div>


title: the-most-common-or-abundant-connective-tissue-of-body-is
slug: the-most-common-or-abundant-connective-tissue-of-body-is-92329
question: <div class="question">The most common or abundant connective tissue of body is</div>
tags: ['Biology', 'Structural Organisation in Animals', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Areolar connective tissue</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Fluid  connective tissue</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Dense connective tissue</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Adipose tissue</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Areolar connective tissue</span> </div>

answer:
<div class="solution">The areolar connective tissue is a tissue with 'small open spaces' (areola) and with small airy pockets between the network of cells and fibers. Within this tissue there are three main types of cells (fibroblasts, white blood cells, and mast cells) and three types of fibers (collagen fibers, elastin fibers, and reticular fibers), a fluid matrix called ground substance.</div>


title: the-most-reactive-amine-towards-reactions-with-dil-hcl-is
slug: the-most-reactive-amine-towards-reactions-with-dil-hcl-is-35208
tags: ['Chemistry', 'Amines', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The most reactive amine towards reactions with dil. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCl</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msub><mi>CH</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msub><mi>NH</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>CH</mi><mn>3</mn></msub></mfenced><mn>3</mn></msub><mo>·</mo><mi mathvariant="normal">N</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>PhNH</mi><mn>2</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msub><mi>CH</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msub><mi>NH</mi></math></span> </div>

answer:
<div class="solution"><p>More be the ability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>-atom of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math> group to donate lone pair of electrons to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCl</mi></math> (dil.), more be the reactivity of amine.</p><p>In (B) two alkyl groups are attached to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math> (having electron releasing nature) which increases the electron donating ability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>-atom of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math>. In aqueous medium, secondary amine is most basic. Hence, it is most reactive.  In (D), due to resonance, lone pair of electrons get delocalised and its basic strength is very less.</p></div>


title: the-most-reactive-molecules-towards-s-n-1-reaction-is
slug: the-most-reactive-molecules-towards-s-n-1-reaction-is-53345
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'TS EAMCET', 'TS EAMCET 2021 (05 Aug Shift 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The most reactive molecules towards <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi mathvariant="normal">S</mi><mi mathvariant="normal">N</mi></msub><mn>1</mn></msup></math> reaction is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/efee2d0c-65af-4152-bc4f-11c60e72dba3-image.png" style="width: 397px; height: 564px;"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(I), (IV) and (VI)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(I), (II) and (IV)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(II), (III) and (V)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(IV), (V) and (VI)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(I), (IV) and (VI)</span> </div>

answer:
<div class="solution"><blockquote><p>According to the question, The rate of the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>SN</mi><mn>1</mn></msup></math> (nucleophilic substitution unimolecular reaction) reaction depends only on the concentration of alkyl halide. Relative reactivity of different alkyl halides in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>SN</mi><mn>1</mn></msup></math> Reaction is as follows: Tertiary alkyl halide&gt; secondary alkyl halide &gt; Primary alkyl halide. </p><p>Product formation takes place by the formation of carbocation as reaction intermediate and their stability. Stability of various alkyl carbocation are- </p><p>Tertiary alkyl carbocation &gt;secondary alkyl carbocation &gt; Primary alkyl carbocation. </p><p>In option (a) carbocation is formed, due to hetrolytic bond fission of C-Cl bond is present in the conjugation with double bond of two benzene rings, here methoxy group show +R (maximum) and -I (minimum) effect. Due to this stability of carbocation is increases. So it is capable to show <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>SN</mi><mn>1</mn></msup></math> Reaction. </p><p>Option (d) is tertiary butyl chloride (tertiary alkyl halide, which show <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>SN</mi><mn>1</mn></msup></math> Reaction in maximum extant.</p><p>In option (f), due to hetrolytic bond fission of C-Cl bond. Tertiary carbocation is formed, which is stabilised by resonance effect of phenyl ring and +I effect of methyl group. So it also capable to show<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>SN</mi><mn>1</mn></msup></math>Reaction.  </p></blockquote></div>


