# questions

tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-leftx2frac1xright15-what-is-the-sum-of-the-coefficients-of-the-middle-terms-in-the-given-expansion

slug:
consider-the-expansion-leftx2frac1xright15-what-is-the-sum-of-the-coefficients-of-the-middle-terms-in-the-given-expansion-41307

question:
<div class="question">Consider the expansion \(\left(x^{2}+\frac{1}{x}\right)^{15}\). What is the sum of the coefficients of the middle terms in the given expansion?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{C}(15,9)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{C}(16,9)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{C}(16,8)$</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">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{C}(16,8)$</span> </div>

answer:
<div class="solution">Given $\left(x^{2}+\frac{1}{x}\right)^{15}$<br/>Since, $n$ is odd<br/>So, it has two middle terms $\mathrm{T}_{8}$ and $\mathrm{T}_{9}$.<br/>$$<br/>\begin{aligned}<br/>\therefore \mathrm{T}_{8}+\mathrm{T}_{9}={ }^{15} \mathrm{C}_{7}+{ }^{15} \mathrm{C}_{8}={ }^{16} \mathrm{C}_{8} \\<br/>&amp;\left(\because{ }^{n} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}\right)<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-of-1-x-2-n-1-if-the-coefficients-of-x-r-and-x-r-1-are-equal-in-the-expansion-then-r-is-equal-to

slug:
consider-the-expansion-of-1-x-2-n-1-if-the-coefficients-of-x-r-and-x-r-1-are-equal-in-the-expansion-then-r-is-equal-to-78516

question:
<div class="question">Consider the expansion of $(1+\mathrm{x})^{2 \mathrm{n}+1}$<br/>If the coefficients of $\mathrm{x}^{\mathrm{r}}$ and $\mathrm{x}^{\mathrm{r}+1}$ are equal in the expansion, then $\mathrm{r}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{n}$</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 \mathrm{n}-1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2 \mathrm{n}+1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n}+1$</span> </li> </ul>

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

answer:
<div class="solution">$(1+\mathrm{x})^{2 \mathrm{n}+1}={ }^{(2 \mathrm{n}+1)} \mathrm{C}_{0} \mathrm{x}^{0}+{ }^{(2 \mathrm{n}+1)} \mathrm{C}_{1} \mathrm{x}^{1}$<br/>$+\ldots+{ }^{(2 n+1)} C_{2 n+1}(x)^{2 n+1}$<br/>Coefficient of $\mathrm{x}^{\mathrm{r}}=(2 \mathrm{n}+1) \mathrm{Cr}$<br/>Coefficient of $\mathrm{x}^{\mathrm{r}+1}=(2 \mathrm{n}+1) \mathrm{Cr}+1$<br/>$(2 \mathrm{n}+1) \mathrm{C}_{\mathrm{r}}$=$(2 n+1) C r+1$<br/>$\Rightarrow \frac{(2 \mathrm{n}+1) !}{\mathrm{r} !(2 \mathrm{n}+1-\mathrm{r}) !}=\frac{(2 \mathrm{n}+1) !}{(\mathrm{r}+1) !(2 \mathrm{n}-\mathrm{r}) !}$<br/>$\Rightarrow \frac{(2 \mathrm{n}-\mathrm{r}) !}{(2 \mathrm{n}+1-\mathrm{r})(2 \mathrm{n}-\mathrm{r}) !}=\frac{\mathrm{r} !}{(\mathrm{r}+1) \mathrm{r} !}$<br/>$\Rightarrow(\mathrm{r}+1)=2 \mathrm{n}+1-\mathrm{r}$<br/>$\Rightarrow \mathrm{r}=\mathrm{n}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-of-1-x-2-n-1-the-average-of-the-coefficients-of-the-two-middle-terms-in-the-expansion-is

slug:
consider-the-expansion-of-1-x-2-n-1-the-average-of-the-coefficients-of-the-two-middle-terms-in-the-expansion-is-81615

question:
<div class="question">Consider the expansion of $(1+\mathrm{x})^{2 \mathrm{n}+1}$<br/>The average of the coefficients of the two middle terms in the expansion is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{2 \mathrm{n}+1} \mathrm{C}_{\mathrm{n}+}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">${ }^{2 n+1} C_{n}$</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><li class=""> <span class="option-label">C</span> <span class="option-data">${ }^{2 n+1} C_{n-1}{ }^{1+2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }^{2 n} C_{n+1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{2 n+1} C_{n}$</span> </div>

answer:
<div class="solution">Total no. of terms in the expansion is $2 \mathrm{n}+2$. The middle two terms will be $\mathrm{n}^{\text {th }},(\mathrm{n}+1)^{\text {th }}$ term. So.<br/>Average $=\frac{(2 \mathrm{n}+1)}{\mathrm{C}_{\mathrm{n}}+{ }^{(2 \mathrm{n}+1)} \mathrm{C}_{\mathrm{n}+1}}{2}$<br/>$=\left[\frac{(2 \mathrm{n}+1) !}{\mathrm{n} !(\mathrm{n}+1) !}+\frac{(2 \mathrm{n}+1) !}{(\mathrm{n}+1) ! \mathrm{n} !}\right] / 2$<br/>$=\frac{(2 \mathrm{n}+1) !}{\mathrm{n} !(\mathrm{n}+1) !}={ }^{(2 \mathrm{n}+1)} \mathrm{C}_{\mathrm{n}}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-of-1-x-2-n-1-the-sum-of-the-coefficients-of-all-the-terms-in-the-expansion-is

slug:
consider-the-expansion-of-1-x-2-n-1-the-sum-of-the-coefficients-of-all-the-terms-in-the-expansion-is-15052

question:
<div class="question">Consider the expansion of $(1+\mathrm{x})^{2 \mathrm{n}+1}$<br/>The sum of the coefficients of all the terms in the expansion is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{2 n-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4^{\mathrm{n}-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \times 4^{\text {n }}$</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><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \times 4^{\text {n }}$</span> </div>

answer:
<div class="solution">Sum of all coefficient<br/>$={ }^{(2 \mathrm{n}+1)} \mathrm{C}_{0}+{ }^{(2 \mathrm{n}+1)} \mathrm{C}_{1}+\ldots \ldots .+{ }^{(2 \mathrm{n}+1)} \mathrm{C}_{2 \mathrm{n}+1}$<br/>$=(1+1)^{2 \mathrm{n}+1}=2^{(2 \mathrm{n}+1)}=2.2^{2 \mathrm{n}}=2.4^{\mathrm{n}}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-x-2-x-1-15-consider-the-following-statements-1-the-term-containing-x-2-does-not-exist-in-the-given-expansion-2-the-sum-of-the-c

slug:
consider-the-expansion-x-2-x-1-15-consider-the-following-statements-1-the-term-containing-x-2-does-not-exist-in-the-given-expansion-2-the-sum-of-the-c-19075

question:
<div class="question">Consider the expansion $\left(x^{2}+\frac{1}{x}\right)^{15}$. <br/>Consider the following statements:<br/>1- The term containing $\mathrm{x}^{2}$ does not exist in the given expansion.<br/>2- The sum of the coefficients of all the terms in the given expansion is $2^{15}$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">1- For coefficient of $x^{2}$,<br/>$30-3 \mathrm{r}=2 \Rightarrow \mathrm{r}=\frac{28}{3}, \mathrm{r} \notin \mathrm{N}$<br/>So, $x^{2}$ does not exist in the expansion Hence, Statement 1 is correct.<br/>2- Now<br/>$$<br/>\begin{array}{r}<br/>\left(x^{2}+\frac{1}{4}\right)^{15}={ }^{15} C_{0}\left(x^{2}\right)^{15}+{ }^{15} C_{1}\left(x^{2}\right)^{14}\left(\frac{1}{x}\right) \\<br/>+\ldots+{ }^{15} C_{15}\left(\frac{1}{x}\right)^{15}<br/>\end{array}<br/>$$<br/>Put $\mathrm{x}=1$ both sides, we get $(1+1)^{15}={ }^{15} \mathrm{C}_{0}+{ }^{15} \mathrm{C}_{1}+\ldots .+{ }^{15} \mathrm{C}_{15}$<br/>$\Rightarrow 2^{15}={ }^{15} \mathrm{C}_{0}+{ }^{15} \mathrm{C}_{1}+\ldots .+{ }^{15} \mathrm{C}_{15}$<br/>Hence, Statement 2 is correct</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-x-2-x-1-15-consider-the-following-statements-1-there-are-15-terms-in-the-given-expansion-2-the-coefficient-of-x-12-is-equal-to-

slug:
consider-the-expansion-x-2-x-1-15-consider-the-following-statements-1-there-are-15-terms-in-the-given-expansion-2-the-coefficient-of-x-12-is-equal-to-37324

question:
<div class="question">Consider the expansion $\left(x^{2}+\frac{1}{x}\right)^{15}$. <br/>Consider the following statements :<br/>1- There are 15 terms in the given expansion.<br/>2- The coefficient of $\mathrm{x}^{12}$ is equal to that of $\mathrm{x}^{3}$ Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 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">C</span> <span class="option-data">Both 1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">1- We know that, $(\mathrm{a}+\mathrm{b})^{\mathrm{n}}$ have total $(\mathrm{n}+1)$ number of terms<br/>So, $\left(x^{2}+\frac{1}{x}\right)^{15}$ haver 16 terms.<br/>Hence, Statement 1 is false.<br/>2- For coefficient of $\mathrm{x}^{12}$ $30-3 \mathrm{r}=12 \Rightarrow \mathrm{r}=6 \Rightarrow{ }^{15} \mathrm{C}_{6}$<br/>and for coefficient of $\mathrm{x}^{3}$ $30-3 r=3 \Rightarrow r=9 \Rightarrow{ }^{15} C_{9}$<br/>${ }^{15} \mathrm{C}_{6}={ }^{15} \mathrm{C}_{9}$<br/>Hence, statement 2 is correct.</div>


title: consider-the-expansion-x-2-x-1-15-what-is-the-independent-term-in-the-given-expansion
slug: consider-the-expansion-x-2-x-1-15-what-is-the-independent-term-in-the-given-expansion-85132
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3003</span> </div>
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the expansion $\left(x^{2}+\frac{1}{x}\right)^{15}$. <br/>What is the independent term in the given expansion?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2103</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3003</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">4503</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\left(x^{2}+\frac{1}{x}\right)^{15}$<br/>$$<br/>\begin{array}{rl}T_{r+1} &amp; ={ }^{15} C_{r}\left(x^{2}\right)^{15-r}\left(\frac{1}{x}\right)^{r} \\ &amp; =15 \\ C_{r} x &amp; 30-2 r-r=15\end{array}<br/>$$<br/>$C_{r} x^{30-3 r}$<br/>For independent term, $30-3 r=0 \Rightarrow r=10$<br/>Put $\mathrm{r}=10$, we get<br/>$\mathrm{T}_{10+1}={ }^{15} \mathrm{C}_{10}=\frac{15 !}{10 ! 5 !}$<br/>$=\frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 !}{10 ! \times 1 \times 2 \times 3 \times 4 \times 5}=3003$</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-expansion-x-2-x-1-15-what-is-the-ratio-of-coefficient-of-x-15-to-the-term-independent-of-x-in-the-given-expansion

slug:
consider-the-expansion-x-2-x-1-15-what-is-the-ratio-of-coefficient-of-x-15-to-the-term-independent-of-x-in-the-given-expansion-44954

question:
<div class="question">Consider the expansion $\left(x^{2}+\frac{1}{x}\right)^{15}$. <br/>What is the ratio of coefficient of $\mathrm{x}^{15}$ to the term independent of $\mathrm{x}$ in the given expansion?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</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">$1 / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 / 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 / 4$</span> </li> </ul>

answer:
<div class="solution">For coefficient of $\mathrm{x}^{15}$,<br/>$30-3 r=15$<br/>$\Rightarrow \mathrm{r}=5$<br/>$\therefore$ the coefficient of $\mathrm{x}^{15} \mathrm{is}^{15} \mathrm{C}_{5}$<br/>and coefficient of independent of $\mathrm{x}$ is<br/>$30-3 \mathrm{r}=0$<br/>$\Rightarrow \mathrm{r}=10$<br/>So, coefficient of independent of $\mathrm{x}$ is ${ }^{15} \mathrm{C}_{10}$.<br/>$\therefore$ Required ratio $=\frac{15 \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{10}}=\frac{{ }^{15} \mathrm{C}_{5}}{{ }^{15} \mathrm{C}_{5}}=1$<br/>$\left(\because{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-\mathrm{r}}\right)$</div>


title: consider-the-following-1-0-2-2-2-0-which-of-the-above-isare-correct
slug: consider-the-following-1-0-2-2-2-0-which-of-the-above-isare-correct-39358
tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the following:<br/>1.$\quad \alpha+\beta+\alpha \beta&gt;0$<br/>2.$\quad \alpha^{2} \beta+\beta^{2} \alpha&gt;0$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">Sum of roots $=\alpha+\beta=-b$ Multiplication of roots $=\alpha \beta=c$ Hence $\begin{aligned} \alpha+\beta+\alpha \beta=-b+c &amp; \alpha^{2} \beta+\beta^{2}=&amp; \alpha \beta(\alpha+\beta) \\ &amp;=-b c \end{aligned}$<br/>$\because \mathrm{b}&gt;0 \&amp; \mathrm{c}   &lt; 0$<br/>$\therefore-\mathrm{b}+\mathrm{c} &lt; 0 \&amp;-\mathrm{bc}&gt;0$</div>


title: consider-the-following-1-4-i-3-i-0-2-3-i-4-i-3-4-which-of-the-above-isare-correct
slug: consider-the-following-1-4-i-3-i-0-2-3-i-4-i-3-4-which-of-the-above-isare-correct-55364
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the following :<br/>$1.$ $\quad 4 \hat{\mathrm{i}} \times 3 \hat{\mathrm{i}}=\overrightarrow{0}$<br/>$2.$ $\frac{4 \hat{\mathrm{i}}}{3 \hat{\mathrm{i}}}=\frac{4}{3}$<br/>Which of the above is/are correct ?</div>

answer:
<div class="solution">Statement $(1): 4 \hat{\mathrm{i}} \times 3 \hat{\mathrm{i}}=0$<br/>It is a true statement<br/>$(\because \hat{\mathrm{i}} \times \hat{\mathrm{i}}=0)$<br/>Statement $(2): \frac{4 \hat{\mathrm{i}}}{3 \hat{\mathrm{i}}}=\frac{4}{3}$<br/>It is a incorrect statement<br/>$\because \quad \frac{4 \hat{\mathrm{i}}}{3 \hat{\mathrm{i}}}=\frac{4}{3} \hat{\mathrm{i}}$</div>


title: consider-the-following-1-a-b-c-a-b-a-c-2-a-b-c-a-b-a-c-which-of-the-above-isare-correct
slug: consider-the-following-1-a-b-c-a-b-a-c-2-a-b-c-a-b-a-c-which-of-the-above-isare-correct-52106
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the following: <br/>1.$A \cup(B \cap C)=(A \cap B) \cup(A \cap C)$<br/>2.$\quad A \cap(B \cup C)=(A \cup B) \cap(A \cup C)$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Neither 1 nor 2</span> </div>

answer:
<div class="solution">According to 'Distribution law in set theorry the given<br/>both statements are wrong. 1.A $\cup(\mathrm{B} \cap \mathrm{C})=(\mathrm{A} \cup \mathrm{B}) \cap(\mathrm{A} \cup \mathrm{C})$<br/>2. A\cap(B\cupC) $=(A \cap B) \cup(A \cap C)$</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-1-lim-x-0-x-1-exists-2-lim-x-0-e-x-1-does-not-exist-which-of-the-above-isare-correct

slug:
consider-the-following-1-lim-x-0-x-1-exists-2-lim-x-0-e-x-1-does-not-exist-which-of-the-above-isare-correct-83674

question:
<div class="question">Consider the following :<br/>1- $\lim _{x \rightarrow 0} \frac{1}{x}$ exists.<br/>2- $\lim _{x \rightarrow 0} \mathrm{e}^{\frac{1}{x}}$ does not exist.<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{1}{x}=\frac{1}{0}=\infty$ which does not exist.<br/>Hence, statement- 1 is incorrect. Now, $\lim _{x \rightarrow 0} \mathrm{e}^{1 / \mathrm{x}}=\mathrm{e}^{\infty}$ which also does not exist.<br/>Hence, statement- 2 is correct.</div>


tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-1-ln-10-dx-x-c-2-1-0-x-dx-1-0-x-c-where-c-is-the-constant-of-integration-which-of-the-above-isare-correct

slug:
consider-the-following-1-ln-10-dx-x-c-2-1-0-x-dx-1-0-x-c-where-c-is-the-constant-of-integration-which-of-the-above-isare-correct-24244

question:
<div class="question">Consider the following:<br/>$1. \int \ln 10 \mathrm{dx}=\mathrm{x}+\mathrm{c}$<br/>$2. \int 10^{\mathrm{x}} \mathrm{dx}=10^{\mathrm{x}}+\mathrm{c}$<br/>where $\mathrm{c}$ is the constant of integration. Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 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> </ul>

answer:
<div class="solution">(1) Let $\mathrm{I}=\int \ell \mathrm{n} 10 \mathrm{dx}=\ell \mathrm{n} 10 \int \mathrm{d} \mathrm{x}=[\ell \mathrm{n} 10] \mathrm{x}+\mathrm{c}$<br/>(2) Let $\mathrm{I}=\int 10^{\mathrm{x}} \mathrm{dx}=\frac{10^{\mathrm{x}}}{\log _{\mathrm{e}} 10}+\mathrm{c}=\frac{10^{\mathrm{x}}}{\ln 10}+\mathrm{c}$</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-1-sin-1-5-4-sin-1-5-3-2-2-tan-1-3-tan-1-1-tan-1-2-3-which-of-the-above-isare-correct

slug:
consider-the-following-1-sin-1-5-4-sin-1-5-3-2-2-tan-1-3-tan-1-1-tan-1-2-3-which-of-the-above-isare-correct-21549

question:
<div class="question">Consider the following : <br/>1) $\quad \sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}=\frac{\pi}{2}$<br/>2) $\tan ^{-1} \sqrt{3}+\tan ^{-1} 1=-\tan ^{-1}(2+\sqrt{3})$<br/>which of the above is/are correct?</div>

answer:
<div class="solution">$\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)$<br/>$\operatorname{If}(-1 \leq x, y \leq 1) \&amp;\left(x^{2}+y^{2} \leq 1\right)$<br/>$\Rightarrow \sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1} \frac{3}{5}$<br/>$=\sin ^{-1}\left[\frac{4}{5} \sqrt{1-\left(\frac{3}{5}\right)^{2}}+\frac{3}{5} \sqrt{1-\left(\frac{4}{5}\right)^{2}}\right]$<br/>$=\sin ^{-1}\left[\frac{16}{25}+\frac{9}{25}\right]=\sin ^{-1}(1)=\frac{\pi}{2}$<br/>$\therefore$ Statement $(1)$ is correct<br/>Again, $\tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1}\left[\frac{x+y}{1-x y}\right]$<br/>$\begin{aligned} \text { If } ;(x&gt;0),(y&gt;0) \text { and }(x y&gt;1) \\ \tan ^{-1}(\sqrt{3})+\tan ^{-1}(1)=\pi+\tan ^{-1}\left[\frac{\sqrt{3}+1}{1-\sqrt{3}}\right] \\=\pi+\tan ^{-1}\left[\frac{(\sqrt{3}+1)(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})}\right] \\=\pi+\tan ^{-1}\left(\frac{4+2 \sqrt{3}}{-2}\right)=\pi+\tan ^{-1}[-(2+\sqrt{3})] \\=&amp; \pi-\tan ^{-1}(2+\sqrt{3}) \quad \because \tan ^{-1}(-x)=-\tan ^{-1} x \end{aligned}$<br/>$\therefore$ Statement $(2)$ is incorrect</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-1-x-x-2-is-continuous-at-x-0-2-x-cos-x-1-is-discontinuous-at-x-0-3-x-2-cos-x-1-is-continuous-at-x-0-which-of-the-above-are-corr

slug:
consider-the-following-1-x-x-2-is-continuous-at-x-0-2-x-cos-x-1-is-discontinuous-at-x-0-3-x-2-cos-x-1-is-continuous-at-x-0-which-of-the-above-are-corr-50773

question:
<div class="question">Consider the following :<br/>1- $\mathrm{x}+\mathrm{x}^{2}$ is continuous at $\mathrm{x}=0$<br/>2- $\mathrm{x}+\cos \frac{1}{\mathrm{x}}$ is discontinuous at $\mathrm{x}=0$<br/>3- $\mathrm{x}^{2}+\cos \frac{1}{\mathrm{x}}$ is continuous at $\mathrm{x}=0$<br/>Which of the above are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 and 2 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">2 and 3 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 2 only</span> </div>

answer:
<div class="solution">$\mathrm{x}+\mathrm{x}^{2}$ is continuous at $\mathrm{x}=0$<br/>$\mathrm{x}+\cos \frac{1}{\mathrm{x}}$ is discontinuous at $\mathrm{x}=0$<br/>$\mathrm{x}^{2}+\cos \frac{1}{\mathrm{x}}$ is not continuous at $\mathrm{x}=0$</div>


tags: ['Chemistry', 'Surface Chemistry', 'TS EAMCET', 'TS EAMCET 2023 (12 May Shift 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-about-the-tyndall-effect-i-it-is-used-to-distinguish-between-a-true-and-colloidal-solution-ii-it-is-possible-only-when-the-disp

slug:
consider-the-following-about-the-tyndall-effect-i-it-is-used-to-distinguish-between-a-true-and-colloidal-solution-ii-it-is-possible-only-when-the-disp-61956

question:
<div class="question">Consider the following about the Tyndall effect.<br/>(I) It is used to distinguish between a true and colloidal solution<br/>(II) It is possible only when the dispersed medium and dispersed phase differ much in their refractive indices<br/>(III) It is observed only when the size of colloidal particles is much smaller than the wavelength of the light used<br/><br/>The correct statements are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I &amp; III only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II &amp; III only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">I, II &amp; III</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">I &amp; II 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I &amp; II only</span> </div>

answer:
<div class="solution">Tyndall effect is observed in large-sized colloidal particles due to scattering of light by them.<br/><br/>This happens when the size of the particles and the wavelength of the light are of similar magnitude.<br/><br/>Thus, Statement (III) is incorrect while (I) and (II) are correct.</div>


tags: ['Chemistry', 'General Organic Chemistry', 'NEET']
source_db: MarksBatch1_P2.db
title:
consider-the-following-compounds-1-c-6-h-5-n-h-c-h-3-2-c-6-h-5-n-h-c-6-h-5-3-c-6-h-5-3-n-4-c-h-3-c-h-2-n-h-2

slug:
consider-the-following-compounds-1-c-6-h-5-n-h-c-h-3-2-c-6-h-5-n-h-c-6-h-5-3-c-6-h-5-3-n-4-c-h-3-c-h-2-n-h-2-94843

question:
<div class="question">Consider the following compounds<br/> <math><mo>(</mo><mn>1</mn><mo>)</mo></math> <math><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>-</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub></math> <math><mo>(</mo><mn>2</mn><mo>)</mo></math> <math><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>â</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub></math><br/> <math><mo>(</mo><mn>3</mn><mo>)</mo></math> <math><mo>(</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><msub><mrow><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msub><mi mathvariant="normal">N</mi></math> <math><mo>(</mo><mn>4</mn><mo>)</mo></math> <math><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 &gt; 2 &gt; 3 &gt; 4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4 &gt; 1 &gt; 2 &gt; 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">C</span> <span class="option-data">3 &gt; 2 &gt; 1 &gt; 4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4 &gt; 3 &gt; 2 &gt; 1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">4 &gt; 1 &gt; 2 &gt; 3</span> </div>

answer:
<div class="solution">Here the basic nature order is given here<br/><math><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>-</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>&gt;</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>â</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>â</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>&gt;</mo><mo>(</mo><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><msub><mrow><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msub><mi mathvariant="normal">N</mi></math><br/>(i) Aliphatic amine is more basic then aromatic amines.<br/>(ii) Among aromatic amines, primary is more basic then secondary which is more basic than tertiary due to delocalisation of lone pair of nitrogen.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-data-2010-ii-beginarraycccccc-hline-mathrmx-5-7-8-4-6-hline-mathrmy-2-4-3-2-4-hline-endarray-what-is-the-regression-equation-of

slug:
consider-the-following-data-2010-ii-beginarraycccccc-hline-mathrmx-5-7-8-4-6-hline-mathrmy-2-4-3-2-4-hline-endarray-what-is-the-regression-equation-of-86209

question:
<div class="question">Consider the following data: $\quad[2010-\mathrm{II}]$ \begin{array}{|c|c|c|c|c|c|}<br/>\hline \mathrm{x} &amp; 5 &amp; 7 &amp; 8 &amp; 4 &amp; 6 \\<br/>\hline \mathrm{y} &amp; 2 &amp; 4 &amp; 3 &amp; 2 &amp; 4 \\<br/>\hline<br/>\end{array} What is the regression equation of $y$ on $x ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y=0.6+0.4 x$</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">$y=0.7+0.3 x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=6+5 x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=4+9 x$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=0.6+0.4 x$</span> </div>

answer:
<div class="solution">Given table can be rewritten as<br/>\begin{array}{|c|c|c|c|c|}<br/>\hline \mathbf{x} &amp; \mathbf{y} &amp; \mathbf{x}^{2} &amp; \mathbf{y}^{2} &amp; \mathbf{x y} \\<br/>\hline 5 &amp; 2 &amp; 25 &amp; 4 &amp; 10 \\<br/>\hline 7 &amp; 4 &amp; 49 &amp; 16 &amp; 28 \\<br/>\hline 8 &amp; 3 &amp; 64 &amp; 9 &amp; 24 \\<br/>\hline 4 &amp; 2 &amp; 16 &amp; 4 &amp; 8 \\<br/>\hline 6 &amp; 4 &amp; 36 &amp; 16 &amp; 24 \\<br/>\hline\Sigma x=30 &amp; \sum y=15 &amp; \sum x^{2}=190 &amp; \sum y^{2}=49 &amp; \sum x y=94 \\<br/>\hline<br/>\end{array}<br/>$\overline{\mathrm{x}}=\frac{30}{5}=6$ and $\overline{\mathrm{y}}=\frac{15}{5}=3$<br/>$b_{y x}=\frac{n \sum x y-\left(\sum x\right)\left(\sum y\right)}{n \sum x^{2}-\left(\sum x\right)^{2}}$<br/>$=\frac{5 \times 94-30 \times 15}{5 \times 190-(30)^{2}}=\frac{2}{5}=0.4$<br/>Hence, line of regression is $y-3=0.4(x-6) \Rightarrow y=0.4 x+0.6$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-data-beginarraylcc-hline-factory-a-factory-b-hline-mean-wage-of-workers-540-620-hline-standard-deviation-of-wages-4050-31-hline

slug:
consider-the-following-data-beginarraylcc-hline-factory-a-factory-b-hline-mean-wage-of-workers-540-620-hline-standard-deviation-of-wages-4050-31-hline-78385

question:
<div class="question">Consider the following data: $\quad\mathrm{}$ \begin{array}{|l|c|c|}<br/>\hline &amp; Factory - A &amp; Factory - B \\<br/>\hline Mean wage of workers &amp; Ã¢âÂ¹ 540 &amp; Ã¢âÂ¹ 620 \\<br/>\hline Standard deviation of wages &amp; Ã¢âÂ¹ 40.50 &amp; Ã¢âÂ¹ 31 \\<br/>\hline<br/>\end{array} What is the variability in the wages of the workers in Factory $-A ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \%$ more than the variability in the wages of the workers in Factory $-B$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$50 \%$ more than the variability in the wages of the workers in Factory $-B$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$50 \%$ less than the variability in the wages of the workers in Factory-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">$150 \%$ more than the variability in the wages of the workers in Factory- $B$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$50 \%$ less than the variability in the wages of the workers in Factory-B</span> </div>

answer:
<div class="solution">Coefficient of variation $=\frac{\text { S.D }}{\text { Mean }} \times 100$<br/>For factory $\mathrm{A}=\frac{40 \cdot 50}{540} \times 100=7.5$<br/>For factory $\mathrm{B}=\frac{31}{620} \times 100=5$<br/>Variability in wages of $\mathrm{A}$ is $50 \%$ more than the variability in wages of B.</div>


tags: ['Mathematics', 'Statistics', 'AP EAMCET', 'AP EAMCET 2017 (26 Apr Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">26.91</span> </div>
answer: <div class="solution">No solution. Refer to answer key.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-data-the-coefficient-of-variation-of-the-above-distribution-of-wages-if-its-standard-deviation-is-1472-is

slug:
consider-the-following-data-the-coefficient-of-variation-of-the-above-distribution-of-wages-if-its-standard-deviation-is-1472-is-46069

question:
<div class="question">Consider the following data:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/3dV5rJtbDVFyAbmME-dXUjZzzwVXcfAtzscJ8MtAr3s.original.fullsize.png"/><br/><br/>The coefficient of variation of the above distribution of wages, if its standard deviation is 14.72 , is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">14.73</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">23.73</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">26.91</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">20.82</span> </li> </ul>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.91 V</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-e-o-values-e-f-e-3-f-e-2-o-077-v-e-s-n-2-s-n-o-014-v-under-standard-conditions-the-potential-for-the-reaction-s-n-s-2-f-e-3-a-q

slug:
consider-the-following-e-o-values-e-f-e-3-f-e-2-o-077-v-e-s-n-2-s-n-o-014-v-under-standard-conditions-the-potential-for-the-reaction-s-n-s-2-f-e-3-a-q-10568

question:
<div class="question">Consider the following <math><msup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msup></math> values<br/> <math><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mrow><mrow><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></mrow><mo>/</mo><mrow><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup></mrow></mrow></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup><mo>=</mo><mn>0.77</mn><mi mathvariant="normal">V</mi></math><br/> <math><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">S</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup><mo>=</mo><mo>-</mo><mn>0.14</mn><mi mathvariant="normal">V</mi></math><br/>
Under standard conditions the potential for the reaction<br/> <math><mi mathvariant="normal">S</mi><mi mathvariant="normal">n</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>â</mo><mn>2</mn><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">S</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo></math><br/>
 </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.68 V</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0.91 V</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">0.63 V</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.46 V</span> </li> </ul>

answer:
<div class="solution"><math><mi mathvariant="normal">S</mi><mi mathvariant="normal">n</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>â</mo><mn>2</mn><mi mathvariant="normal">F</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">S</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo></math><br/><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mo>=</mo><msub> <mrow> <mfenced> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>reduction</mtext></mrow> <mtext>o</mtext> </msubsup> </mrow> </mfenced></mrow> <mrow> <mtext>cathode</mtext></mrow> </msub> <mo>−</mo><msub> <mrow> <mfenced> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>reduction</mtext></mrow> <mtext>o</mtext> </msubsup> </mrow> </mfenced></mrow> <mrow> <mtext>anode</mtext></mrow> </msub> </mrow></math><br/><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mo>=</mo><mn>0.77</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>0.14</mn><mo stretchy="false">)</mo></mrow></math><br/><math> <mrow> <mo>∴</mo><mtext> </mtext><msubsup> <mtext>E</mtext> <mrow> <mtext>cell</mtext></mrow> <mtext>o</mtext> </msubsup> <mo>=</mo><mo>−</mo><mn>0.91</mn><mtext> </mtext><mtext>V</mtext></mrow></math></div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-following-e-values-e-fe-3-fe-2-077-v-e-sn-2-sn-014-v-u-n-d-ers-t-an-d-a-r-d-co-n-d-i-t-i-o-n-s-t-h-e-p-o-t-e-n-t-ia-l-f-or-t-h-ere-a-c-t--1

slug:
consider-the-following-e-values-e-fe-3-fe-2-077-v-e-sn-2-sn-014-v-u-n-d-ers-t-an-d-a-r-d-co-n-d-i-t-i-o-n-s-t-h-e-p-o-t-e-n-t-ia-l-f-or-t-h-ere-a-c-t-1-55506

question:
<div class="question">Consider the following $\mathrm{E}^{\circ}$ values<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{E}_{\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}}^{\circ}=0.77 \mathrm{~V} \\<br/>&amp; \mathrm{E}_{\mathrm{Sn}^{2+} / \mathrm{Sn}}^{\circ}=-0.14 \mathrm{~V}<br/>\end{aligned}<br/>$$<br/>Under standard conditions the potential for the reaction<br/>$\mathrm{Sn}(\mathrm{s})+2 \mathrm{Fe}^{3+}(\mathrm{aq}) \longrightarrow 2 \mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{Sn}^{2+}(\mathrm{aq})$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.68 \mathrm{~V}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$0.63 \mathrm{~V}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$0.91 \mathrm{~V}$<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/>$1.40 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$0.91 \mathrm{~V}$<br/></span> </div>

answer:
<div class="solution">$\mathrm{E}_{\text {cell }}=\mathrm{E}_{\mathrm{RHS}}^{\circ}-\mathrm{E}_{\mathrm{LHS}}^{\circ}$<br/>$=(0.77)-(-0.14)$<br/>$=0.91 \mathrm{~V}$</div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2004']
source_db: MarksBatch1_P2.db
title:
consider-the-following-e-values-e-fe-3-fe-2-077-v-e-sn-2-sn-014-v-u-n-d-ers-t-an-d-a-r-d-co-n-d-i-t-i-o-n-s-t-h-e-p-o-t-e-n-t-ia-l-f-or-t-h-ere-a-c-t-

slug:
consider-the-following-e-values-e-fe-3-fe-2-077-v-e-sn-2-sn-014-v-u-n-d-ers-t-an-d-a-r-d-co-n-d-i-t-i-o-n-s-t-h-e-p-o-t-e-n-t-ia-l-f-or-t-h-ere-a-c-t-48406

question:
<div class="question">Consider the following $\mathrm{E}^{\circ}$ values<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{E}_{\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}}^{\circ}=0.77 \mathrm{~V} \\<br/>&amp; \mathrm{E}_{\mathrm{Sn}^{2+} / \mathrm{Sn}}^{\circ}=-0.14 \mathrm{~V}<br/>\end{aligned}<br/>$$<br/>Under standard conditions the potential for the reaction<br/>$\mathrm{Sn}(\mathrm{s})+2 \mathrm{Fe}^{3+}(\mathrm{aq}) \longrightarrow 2 \mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{Sn}^{2+}(\mathrm{aq})$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$1.68 \mathrm{~V}$<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>$0.63 \mathrm{~V}$<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>$0.91 \mathrm{~V}$<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/>$1.40 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$0.91 \mathrm{~V}$<br/></span> </div>

answer:
<div class="solution">$\mathrm{E}_{\text {cell }}=\mathrm{E}_{\mathrm{RHS}}^{\circ}-\mathrm{E}_{\mathrm{LHS}}^{\circ}$<br/>$=(0.77)-(-0.14)$<br/>$=0.91 \mathrm{~V}$</div>


tags: ['Chemistry', 'Electrochemistry', 'KCET', 'KCET 2021']
source_db: MarksBatch1_P2.db
title:
consider-the-following-electrodes-p-zn-2-00001-m-zn-q-zn-2-01-m-zn-r-zn-2-001-m-zn-s-zn-2-0001-m-zn-e-zn-zn-2-076-v-electrode-potentials-of-the-above-

slug:
consider-the-following-electrodes-p-zn-2-00001-m-zn-q-zn-2-01-m-zn-r-zn-2-001-m-zn-s-zn-2-0001-m-zn-e-zn-zn-2-076-v-electrode-potentials-of-the-above-13398

question:
<div class="question">Consider the following electrodes $P=\mathrm{Zn}^{2+}(0.0001 \mathrm{M}) / \mathrm{Zn}, Q=\mathrm{Zn}^{2+}(0.1$ M) $/ \mathrm{Zn}$ $R=\mathrm{Zn}^{2+}(0.01 \mathrm{M}) / \mathrm{Zn}, S=\mathrm{Zn}^{2+}(0.001 \mathrm{M}) / \mathrm{Zn}$ $E^{\circ}\left(\mathrm{Zn} / \mathrm{Zn}^{2+}\right)=-0.76 \mathrm{~V}$ electrode potentials of the above electrodes in volts are in the order</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$P&gt;S&gt;R&gt;Q$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$S&gt;R&gt;Q &lt; P$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$Q&gt;R&gt;S&gt;P$</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">D</span> <span class="option-data">$P&gt;Q&gt;R&gt;S$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Q&gt;R&gt;S&gt;P$</span> </div>

answer:
<div class="solution">The standard potential of $\mathrm{Zn} \mid \mathrm{Zn}^{2+}$ half-cell $=-0.76 \mathrm{~V}$<br/>Half-cell reaction,<br/>$\overline{\mathrm{Z}} \mathrm{n}^{2 *}(a q)+\overline{\mathcal{2}} e^{=} \longrightarrow \overline{\mathrm{Z}}(s)$<br/>$E_{\text {red }}=E_{\text {red }}^{\circ}-\frac{0.059}{n} \log \left[\frac{1}{\mathrm{Zn}^{2+}}\right]$<br/>$=-0.76+\frac{0.059}{n} \log \left[\mathrm{Zn}^{2+}\right]$<br/>Lower the concentration of $\mathrm{Zn}^{2+}$, lower is the $E_{\text {red }}$ or vice-versa.<br/>Hence, the correct order is $Q&gt;R&gt;S&gt;P$.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-expressions-1-x-x-2-x-1-2-ax-2-bx-x-c-c-d-x-2-e-3-3-x-2-5-x-ab-4-x-2-a-x-b-3-2-5-x-1-x-5-2-which-of-the-above-are-rational-expr

slug:
consider-the-following-expressions-1-x-x-2-x-1-2-ax-2-bx-x-c-c-d-x-2-e-3-3-x-2-5-x-ab-4-x-2-a-x-b-3-2-5-x-1-x-5-2-which-of-the-above-are-rational-expr-14366

question:
<div class="question">Consider the following expressions:<br/>1.$x+x^{2}-\frac{1}{x}$<br/>2.$\sqrt{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{x}-\mathrm{c}+\frac{\mathrm{d}}{\mathrm{c}}-\frac{\mathrm{e}}{\mathrm{x}^{2}}}$<br/>3.$3 x^{2}-5 x+a b$<br/>4.$\frac{2}{x^{2}-a x+b^{3}}$<br/>5.$\frac{1}{\mathrm{x}}-\frac{2}{\mathrm{x}+5}$<br/>Which of the above are rational expressions?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,4 and 5 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1,3,4$ and 5 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">C</span> <span class="option-data">2,4 and 5 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 2 only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1,3,4$ and 5 only</span> </div>

answer:
<div class="solution">A rational expression isnothing more than a fraction in which the numerator and denominator are polynomials. Here are some example of rational expressions are $\left(x+x^{2}+\frac{1}{x}\right),\left(3 x^{2}-5 x+a b\right), \frac{2}{x^{2}-a x+a b}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-for-any-three-nonempty-sets-a-b-and-c-1-a-b-c-a-b-a-c-2-2-a-b-a-a-b-3-a-a-b-a-b-which-of-the-above-isare-correct

slug:
consider-the-following-for-any-three-nonempty-sets-a-b-and-c-1-a-b-c-a-b-a-c-2-2-a-b-a-a-b-3-a-a-b-a-b-which-of-the-above-isare-correct-78862

question:
<div class="question">Consider the following for any three non-empty sets $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$. <br/>1.$A-(B \cup C)=(A-B) \cup(A-C)$<br/>2.$2 \quad A-B=A-(A \cap B)$<br/>3.$\mathrm{A}=(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A}-\mathrm{B})$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Only 1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 and 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">C</span> <span class="option-data">1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 3</span> </div>

answer:
<div class="solution">A Venndiagram ofthe three non-empty andintersecting sets is drawn, dividing into 7 regions, a to gas showr below and consider statements one by one Statement $(1): \mathrm{A}-(\mathrm{B} \cup \mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup(\mathrm{A}-\mathrm{C})$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/j1IFiKGsGZSRBzbaQyb6J2aX2XMo_jg1pQGWI1e9OyY.original.fullsize.png"/><br/><br/>LHS = regions $a, b, c, d-b, c, d, e, f, g=a$.<br/>$\mathrm{RHS}=\mathrm{a}, \mathrm{d}, \cup \mathrm{a}, \mathrm{b}=\mathrm{a}, \mathrm{b}, \mathrm{d}$<br/>So, statement $(1)$ is not correct. Statement $(2): A-B=A-(A \cap B)$<br/>LHS=region a, $\mathrm{d}$. $\mathrm{RHS}=$ region $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}-($ region $\mathrm{b}, \mathrm{c},=\mathrm{a}, \mathrm{d}$<br/>So, statement $(2)$ iscorrect. Statement $(3): A=(A \cap B) \cup(A-B)$<br/>LHS = regions $a, b, c, d$. $\mathrm{RHS}=$ regions $\mathrm{b}, \mathrm{c}, \cup$ region $\mathrm{a}, \mathrm{d}=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ so,<br/>statement (3) is correct.</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-for-triangle-abc-2017-i-1-sin-2-b-c-cos-2-a-2-tan-2-b-c-cot-2-a-3-sin-b-c-cos-a-4-tan-b-c-cot-a-which-of-the-above-are-correct

slug:
consider-the-following-for-triangle-abc-2017-i-1-sin-2-b-c-cos-2-a-2-tan-2-b-c-cot-2-a-3-sin-b-c-cos-a-4-tan-b-c-cot-a-which-of-the-above-are-correct-24325

question:
<div class="question">Consider the following for triangle ABC: $\quad[2017-I]$<br/>1) $\quad \sin \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)=\cos \left(\frac{\mathrm{A}}{2}\right)$<br/>2) $\tan \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)=\cot \left(\frac{\mathrm{A}}{2}\right)$<br/>3) $\sin (B+C)=\cos A$<br/>4) $\tan (\mathrm{B}+\mathrm{C})=-\cot \mathrm{A}$<br/>Which of the above are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 3</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1 and 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 and 4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 2</span> </div>

answer:
<div class="solution">In triangle $\mathrm{ABC}, \mathrm{A}+\mathrm{B}+\mathrm{C}=\pi$<br/>$\Rightarrow \frac{\mathrm{A}+\mathrm{B}+\mathrm{C}}{2}=\frac{\pi}{2} \Rightarrow \frac{\mathrm{A}}{2}+\frac{\mathrm{B}}{2}+\frac{\mathrm{C}}{2}=\frac{\pi}{2}$<br/>$\Rightarrow \frac{\mathrm{B}}{2}+\frac{\mathrm{C}}{2}=\frac{\pi}{2}-\frac{\mathrm{A}}{2}$<br/>$\Rightarrow \sin \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)=\sin \left(\frac{\pi}{2}-\frac{\mathrm{A}}{2}\right)=\cos \frac{\mathrm{A}}{2}$<br/>Also, from (i), $\tan \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)=\tan \left(\frac{\pi}{2}-\frac{\mathrm{A}}{2}\right)=\cot \frac{\mathrm{A}}{2}$</div>


tags: ['Biology', 'Morphology of Flowering Plants', 'NEET', 'NEET 2010 (Mains)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-four-statements-a-b-c-and-d-select-the-right-option-for-two-correct-statements-statements-a-in-vexillary-aestivation-the-large-

slug:
consider-the-following-four-statements-a-b-c-and-d-select-the-right-option-for-two-correct-statements-statements-a-in-vexillary-aestivation-the-large-69974

question:
<div class="question">Consider the following four statements A, B, C and D select the right option for two correct statements. Statements :
<br/>(A) In vexillary aestivation, the large posteior petal is called-standard, two lateral ones are wings and two small anteior petals are termed keel.
<br/>(B) The floral formula for Litiaceae is <img src="https://cdn-question-pool.getmarks.app/pyq/neet/eury8y887guiu.png"/> <br/>(C) In pea flower the stamens are monadelphous
<br/>(D) The floral formula for Solanaceae is <img src="https://cdn-question-pool.getmarks.app/pyq/neet/gy97887556dtr.png"/> <br/>The Correct statements are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">A and C</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">C</span> <span class="option-data">B and C</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">C and D</span> </li> </ul>

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

answer:
<div class="solution">In vexillary aestivation (eg, papilionaceous plants), the large posterior petal is called standard, two lateral ones are wings and two small anterior petals are termed keel.<br/>General floral formula for family-Liliaceae is $\mathrm{Br} \oplus \overbrace{}^2 \mathrm{P}_{3+3} \mathrm{~A}_{3+3} \mathrm{G}_{(3)}$<br/>In pea flower, the stamens are diadelphous, ie, anthers remain free, while filaments unite to form two separate bundles.<br/>The general floral formula for family-Solanaceae is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/neet/Jf-FH_2z1c2r44l_2m-3C_DBorvlZDznZGk9ao35eOw.original.fullsize.png"/><br/></div>


tags: ['Biology', 'Morphology of Flowering Plants', 'NEET']
source_db: MarksBatch1_P2.db
title:
consider-the-following-four-statements-i-ii-iii-and-iv-and-select-the-right-option-i-in-vexillary-aestivation-the-large-posterior-petal-is-called-stan

slug:
consider-the-following-four-statements-i-ii-iii-and-iv-and-select-the-right-option-i-in-vexillary-aestivation-the-large-posterior-petal-is-called-stan-79788

question:
<div class="question">Consider the following four statements (i) ,(ii) ,(iii) and (iv) and select the right option.<br/> <br/>
(i) In vexillary aestivation, the large posterior petal is called - standard, two lateral ones are wings and two small anterior petals are termed keel.<br/>
(ii) The floral formula for liliaceae is <math> <mrow> <mo>â</mo> <munderover> <mtext>o</mtext> <mo>+</mo> <mo>â</mo> </munderover> <msub> <mtext>P</mtext> <mn>3</mn> </msub> <msub> <mo>+</mo> <mn>3</mn> </msub> <msub> <mtext>A</mtext> <mrow> <mn>3</mn> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <munder accentunder="true"> <mrow> <msub> <mtext>G</mtext> <mn>3</mn> </msub> </mrow> <mo>Â¯</mo> </munder></mrow></math> <br/>
(III) In pea flower the stamens are monadelphous<br/>
(iv) The floral formula for solanaceae is <math> <mrow> <mo>â</mo> <munderover> <mtext>o</mtext> <mo>+</mo> <mo>â</mo> </munderover> <msub> <mtext>P</mtext> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </msub> <msub> <mtext>C</mtext> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </msub> <msub> <mtext>A</mtext> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </msub> <mo>+</mo> <mrow> <mrow> <msub> <mrow> <munder accentunder="true"> <mtext>G</mtext> <mo>Â¯</mo> </munder> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo>  </mrow> </msub> </mrow> </mrow> </mrow> </math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(i) and (iii)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">(i) and (ii)</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">(ii) and (iii) </span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(iii) and (iv)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(i) and (ii)</span> </div>

answer:
<div class="solution">Vexillary or Descending imbricate: The posterior petal is outermost and largest. In it standard or vexillum covers two lateral petals. These two laterals covers two anterior ones. Vexillary arrangement is present in pea family. E.g. Pea, Bean.<br/><br/>The general characters of Liliaceae family is the flowers are bracteate, bisexual, complete, actinomorphic symmetry, hypogynous and trimerous, epiphyllous and in some plants may be epipetalous. They have polyandrous andoecium and superior ovary. </div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=y$</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-frequency-distribution-beginarraycccccc-hline-class-interval-010-1020-2030-3040-4050-hline-frequency-14-mathrmx-27-mathrmy-15-h

slug:
consider-the-following-frequency-distribution-beginarraycccccc-hline-class-interval-010-1020-2030-3040-4050-hline-frequency-14-mathrmx-27-mathrmy-15-h-19713

question:
<div class="question">Consider the following frequency distribution:<br/>\begin{array}{|c|c|c|c|c|c|}<br/>\hline Class interval &amp; 0-10 &amp; 10-20 &amp; 20-30 &amp; 30-40 &amp; 40-50 \\<br/>\hline Frequency &amp; 14 &amp; \mathrm{x} &amp; 27 &amp; \mathrm{y} &amp; 15 \\<br/>\hline<br/>\end{array} If the total of the frequencies is 100 and mode is 25, then which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=2 y$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x=y$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x=y$</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">$x=3 y$</span> </li> </ul>

answer:
<div class="solution">Since the total of the frequencies $=100$ $14+x+27+y+15=100$<br/>$\Rightarrow x+y=44$<br/>Now, to exist the mode $x$ and $y$ should be equal $\mathrm{x}=\mathrm{y}$ is only possibility</div>


title: consider-the-following-function-f-r-r-such-that-following-is-correct
slug: consider-the-following-function-f-r-r-such-that-following-is-correct-35511
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the following function $f: R \rightarrow R$ such that<br/>following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(x)$ is continuous at every $x \in R$</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">$f(x)$ is continuous at $x=0$ only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)$ is discontinous at $x=0$ only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$f(x)$ is discontinuous at every $x \in R$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)$ is continuous at every $x \in R$</span> </div>

answer:
<div class="solution">Given, $f(x)=\left\{\begin{aligned} x, &amp; x \geq 0 \\-x^{2}, &amp; x   &lt; 0 \end{aligned}\right.$<br/>$\mathrm{LHL}=\lim _{x \rightarrow 0^{-}} f(x)=-\lim _{x \rightarrow 0} x^{2}=0$<br/>$\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} x=0$<br/>and $f(0)=0$ Since, $\mathrm{LHL}=\mathrm{RHL}=f(0)$<br/>$\therefore f(x)$ is continuous at $x=0$. Also, $f(x)$ is continuous in the given interval ie. $\mathrm{R}$ Hence, $f(x)$ is continuous in every $x \in R$</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-functions-1-f-x-e-x-where-x-0-2-g-x-x-3-which-of-the-above-functions-isare-continuous

slug:
consider-the-following-functions-1-f-x-e-x-where-x-0-2-g-x-x-3-which-of-the-above-functions-isare-continuous-71292

question:
<div class="question">Consider the following functions:<br/>1- $f(x)=e^{x}$, where $x&gt;0$<br/>2- $g(x)=|x-3|$<br/>Which of the above functions is/are continuous?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$, where $\mathrm{x}&gt;0$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/V4yrTD1Jzni9SofppmoaA56x-eqIHroyNQDOkrDWj9I.original.fullsize.png"/><br/><br/>According to graph, Graph of $\mathrm{e}^{\mathrm{x}}$ is not breaking when $\mathrm{x}&gt;0$ Therefore, graph is continuous at $\mathrm{x}&gt;0$ Statement II $g(x)=|x-3|$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/HcwR7NjOC_gnnIj-hoCYyHjkc5OGybOxduOrqF8Zmt8.original.fullsize.png"/><br/><br/>Graph of $|x-3|$ is not breaking but have sharp turn at $x$ $=3 .$<br/>So, it is continuous</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-functions-1-f-x-x-3-x-r-2-f-x-sin-x-0-x-2-3-f-x-e-x-x-r-which-of-the-above-functions-have-inverse-defined-on-thein-ranges

slug:
consider-the-following-functions-1-f-x-x-3-x-r-2-f-x-sin-x-0-x-2-3-f-x-e-x-x-r-which-of-the-above-functions-have-inverse-defined-on-thein-ranges-56091

question:
<div class="question">Consider the following functions:<br/>1- $\quad \mathrm{f}(\mathrm{x})=\mathrm{x}^{3}, x \in \mathbb{R}$<br/>2- $\quad f(x)=\sin x, 0   &lt; x &lt; 2 \pi$<br/>3- $\quad f(x)=e^{x}, x \in \mathbb{R}$<br/>Which of the above functions have inverse defined on thein ranges?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 and 3 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1 and 3 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">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 3 only</span> </div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-functions-2015-ii-1-f-x-x-1-0-if-if-x-0-x-0-2-f-x-2-x-5-x-2-2-x-5-if-if-x-0-x-0-which-of-the-above-functions-isare-derivable-at

slug:
consider-the-following-functions-2015-ii-1-f-x-x-1-0-if-if-x-0-x-0-2-f-x-2-x-5-x-2-2-x-5-if-if-x-0-x-0-which-of-the-above-functions-isare-derivable-at-12737

question:
<div class="question">Consider the following functions: $\quad[2015-I I]$<br/>1- $\quad f(x)=\left\{\begin{array}{lll}\frac{1}{x} &amp; \text { if } &amp; x \neq 0 \\ 0 &amp; \text { if } &amp; x=0\end{array}\right.$<br/>2- $\quad f(x)=\left\{\begin{array}{cll}2 x+5 &amp; \text { if } &amp; x&gt;0 \\ x^{2}+2 x+5 &amp; \text { if } &amp; x \leq 0\end{array}\right.$<br/>Which of the above functions is/are derivable at $\mathrm{x}=0$ ?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\frac{1}{\mathrm{x}} &amp; \mathrm{x} \neq 0 \\ 0 &amp; \mathrm{x}=0\end{array}\right.$<br/>as there is a discontinuity at $\mathrm{x}=0$, so function is not differentiable at $\mathrm{x}=0$<br/>$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}2 \mathrm{x}+5 &amp; \mathrm{x}&gt;0 \\ \mathrm{x}^{2}+2 \mathrm{x}+5 \quad \mathrm{x} \leq 0\end{array}\right.$<br/>$\mathrm{f}(0)=5$<br/>$\mathrm{LHD}=\lim _{\mathrm{x} \rightarrow 0^{-}} \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(0)}{\mathrm{x}-0}$<br/>$=\lim _{\mathrm{x} \rightarrow 0} \frac{\mathrm{x}^{2}+2 \mathrm{x}+5-5}{\mathrm{x}}=\lim _{\mathrm{x} \rightarrow 0} \mathrm{x}+2=2$<br/>$\mathrm{RHD}=\lim _{\mathrm{x} \rightarrow 0^{+}} \frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(0)}{\mathrm{x}-0}$<br/>$=\lim _{\mathrm{x} \rightarrow 0} \frac{2 \mathrm{x}+5-5}{\mathrm{x}}=2$<br/>$\therefore$ It is differentiable at $\mathrm{x}=0$ <br/>$\therefore$ Only $(2)$ is differentiable at $\mathrm{x}=0$</div>


title: consider-the-following-i-cosec-1-3-2-3-ii-sec-1-3-2-6-which-of-the-above-isare-correct
slug: consider-the-following-i-cosec-1-3-2-3-ii-sec-1-3-2-6-which-of-the-above-isare-correct-27778
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch1_P2.db
question:
<div class="question">$\quad$ Consider the following <br/>I. $\quad \operatorname{cosec}^{-1}\left(-\frac{2}{\sqrt{3}}\right)=-\frac{\pi}{3}$<br/>II. $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi}{6}$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Only I</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Only II</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both I and II</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">Neither Inor II</span> </li> </ul>

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

answer:
<div class="solution">Consider (I):<br/>$$<br/>\operatorname{cosec}^{-1}\left(\frac{-2}{\sqrt{3}}\right)=\frac{-\pi}{3}<br/>$$<br/>Let $\operatorname{cosec}^{-1}\left(\frac{-2}{\sqrt{3}}\right)=\theta$<br/>$$<br/>\begin{array}{l}<br/>\Rightarrow \frac{-2}{\sqrt{3}}=\operatorname{cosec} \theta \Rightarrow \frac{-\sqrt{3}}{2}=\frac{1}{\operatorname{cosec} \theta} \\<br/>\Rightarrow \frac{-\sqrt{3}}{2}=\sin \theta \Rightarrow \theta=\frac{-\pi}{3}<br/>\end{array}<br/>$$<br/>Now, consider (II):<br/>$$<br/>\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi}{6}<br/>$$<br/>Let $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=\theta$<br/>$\Rightarrow \frac{2}{\sqrt{3}}=\sec \theta \Rightarrow \frac{\sqrt{3}}{2}=\frac{1}{\sec \theta}$<br/>$\Rightarrow \frac{\sqrt{3}}{2}=\cos \theta \quad \Rightarrow \theta=\frac{\pi}{6}$<br/>Hence, both statements I and II are correct.</div>


tags: ['Biology', 'Transport in Plants', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">i only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-i-for-a-solution-at-atmospheric-pressure-w-s-ii-w-water-potential-of-a-cell-is-affected-by-solute-potential-only-iii-p-pressure

slug:
consider-the-following-i-for-a-solution-at-atmospheric-pressure-w-s-ii-w-water-potential-of-a-cell-is-affected-by-solute-potential-only-iii-p-pressure-86013

question:
<div class="question">Consider the following:<br/> <br/>
i. For a solution at atmospheric pressure <math><msub><mrow><mi mathvariant="normal">Î¨</mi></mrow><mrow><mi>w</mi></mrow></msub></math> = <math><msub><mrow><mi mathvariant="normal">Î¨</mi></mrow><mrow><mi>s</mi></mrow></msub></math><br/>
ii. <math><msub><mrow><mi mathvariant="normal">Î¨</mi></mrow><mrow><mi>w</mi></mrow></msub><mi> </mi></math> (water potential) of a cell is affected by solute potential only<br/>
iii. <math><msub><mrow><mi mathvariant="normal">Î¨</mi></mrow><mrow><mi>p</mi></mrow></msub><mi> </mi></math> (pressure potential) is usually positive in xylem during the day<br/> <br/>
Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">ii &amp; iii</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">i 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">C</span> <span class="option-data">ii only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">i, ii &amp; iii</span> </li> </ul>

answer:
<div class="solution">Water potential of a solution is the sum total of four components: gravitational potential, osmotic(solute) potential, matric potential and pressure potential. But, since matric potential and gravitational potential are negligible, water potential is generally more affected by solute and pressure potential. Pressure potential of water during the standard stage is 0 MPa. The positive value of water potential within the cell is referred to as turgor pressure. During the day when transpiration is high, pressure potential of the xylem mostly is negative since water is evaporated and its turgor pressure is less.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-if-a-and-b-are-the-vectors-forming-consecutive-sides-of-a-regular-hexagon-abcdef-then-1-ce-b-2-a-2-ae-2-b-a-3-fa-a-b-which-of-t

slug:
consider-the-following-if-a-and-b-are-the-vectors-forming-consecutive-sides-of-a-regular-hexagon-abcdef-then-1-ce-b-2-a-2-ae-2-b-a-3-fa-a-b-which-of-t-49680

question:
<div class="question">Consider the following<br/>If $\vec{a}$ and $\vec{b}$ are the vectors forming consecutive sides of a regular hexagon ABCDEF, then<br/>$1.$ $\overrightarrow{\mathrm{CE}}=\overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}$<br/>$2.$ $\overrightarrow{\mathrm{AE}}=2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>$3.$ $\quad \overrightarrow{\mathrm{FA}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$<br/>Which of the above are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 3 only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1,2 and 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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1,2 and 3</span> </div>

answer:
<div class="solution">Let ABCDEF be the regular hexagen as shown in the figure.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/t7Pdvi6vXLNmJhtGkSlKMs8nbFPpnX4agnebuEW0uDE.original.fullsize.png"/><br/><br/>Let $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{b}}$<br/>Join $\mathrm{AD}, \mathrm{FC}$ and $\mathrm{EB}$. They meet at a common point $\mathrm{O}$, which is the centre of hexagon.<br/>$\mathrm{AO} \| \mathrm{BC}$ so, $\overrightarrow{\mathrm{AO}}=\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{b}}$<br/>$\mathrm{OC} \| \mathrm{AB} \mathrm{so}, \overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{a}}$<br/>OAB forms a triangle, $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BO}}=\overrightarrow{\mathrm{AO}}$<br/>$\Rightarrow \overrightarrow{\mathrm{BO}}=\overrightarrow{\mathrm{AO}}-\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>$\mathrm{BO}=\mathrm{OE}$ and they are on the same line,<br/>So, $\overrightarrow{\mathrm{BO}}=\overrightarrow{\mathrm{OE}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>$\operatorname{In} \Delta \mathrm{OCE}, \overrightarrow{\mathrm{CO}}+\overrightarrow{\mathrm{OE}}=\overrightarrow{\mathrm{CE}}$<br/>$\Rightarrow \overrightarrow{\mathrm{CE}}=-\overrightarrow{\mathrm{OC}}+\mathrm{OE}=-\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}-\overrightarrow{2 \mathrm{a}}$<br/>So, (1) is correct.<br/>$\overrightarrow{\mathrm{BE}}=2 \overrightarrow{\mathrm{OB}}$ In $\Delta \mathrm{AEB}, \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BE}}=\overrightarrow{\mathrm{AE}}$<br/>$\Rightarrow \overrightarrow{\mathrm{AE}}=\overrightarrow{\mathrm{AB}}+2 \overrightarrow{\mathrm{BO}}=\overrightarrow{\mathrm{A}}+2(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})$<br/>$\Rightarrow \overrightarrow{\mathrm{AE}}=\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}-2 \overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}$<br/>So, $(2)$ is also correct.<br/>$\mathrm{FA} \| \mathrm{OB} \Rightarrow \overrightarrow{\mathrm{FA}}=-\overrightarrow{\mathrm{BO}}=-(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}})=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$<br/>So, (3) is also correct.<br/>So, $(1),(2) \&amp;(3)$ are correct.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-matrices-a-and-b-of-same-order-2018-ii-1-a-2-b-2-a-b-a-b-2-a-i-i-a-0-a-2-i-where-i-is-the-identity-matrix-and-o-i

slug:
consider-the-following-in-respect-of-matrices-a-and-b-of-same-order-2018-ii-1-a-2-b-2-a-b-a-b-2-a-i-i-a-0-a-2-i-where-i-is-the-identity-matrix-and-o-i-89814

question:
<div class="question">Consider the following in respect of matrices $\mathrm{A}$ and $\mathrm{B}$ of same order: $\quad[2018-I I]$<br/>$1.$ $\mathrm{A}^{2}-\mathrm{B}^{2}=(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B})$<br/>$2.$ $(A-I)(I+A)=0 \Leftrightarrow A^{2}=I$<br/>Where I is the identity matrix and $\mathrm{O}$ is the null matrix.<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">Matrix product is commutative if both are diagonal matrices of same order.<br/>$\Rightarrow \quad A^{2}-B^{2}=(A+B)(A-B)$ is not true.<br/>Next, $(\mathrm{A}-\mathrm{I})(\mathrm{A}+\mathrm{I})=0$<br/>$\Rightarrow \quad \mathrm{A}^{2}+\mathrm{AI}-\mathrm{IA}-\mathrm{I}^{2}=0(\because \mathrm{AI}=\mathrm{IA})$<br/>$\Rightarrow \mathrm{A}^{2}=\mathrm{I}$ is correct.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">By property, statement 1 and 3 are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-matrices-a-b-and-c-of-same-order-1-a-b-c-a-b-c-2-ab-a-b-3-abc-c-b-a-where-a-is-the-transpose-of-the-matrix-a-whic

slug:
consider-the-following-in-respect-of-matrices-a-b-and-c-of-same-order-1-a-b-c-a-b-c-2-ab-a-b-3-abc-c-b-a-where-a-is-the-transpose-of-the-matrix-a-whic-28763

question:
<div class="question">Consider the following in respect of matrices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of same order: <br/>$1.$ $(\mathrm{A}+\mathrm{B}+\mathrm{C})^{\prime}=\mathrm{A}^{\prime}+\mathrm{B}^{\prime}+\mathrm{C}^{\prime}$<br/>$2.$ $(\mathrm{AB})^{\prime}=\mathrm{A}^{\prime} \mathrm{B}^{\prime}$<br/>$3.$ $(\mathrm{ABC})^{\prime}=\mathrm{C}^{\prime} \mathrm{B}^{\prime} \mathrm{A}^{\prime}$<br/>Where $\mathrm{A}^{\prime}$ is the transpose of the matrix $\mathrm{A}$. Which of the above are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 and 3 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1 and 3 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">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 3 only</span> </div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-sets-a-and-b-1-a-b-b-a-2-a-b-a-a-3-a-b-b-4-a-b-a-b-b-which-of-the-above-are-correct

slug:
consider-the-following-in-respect-of-sets-a-and-b-1-a-b-b-a-2-a-b-a-a-3-a-b-b-4-a-b-a-b-b-which-of-the-above-are-correct-24256

question:
<div class="question">Consider the following in respect of sets $\mathrm{A}$ and $\mathrm{B}$ <br/>1.$(\mathrm{A}-\mathrm{B}) \cup \mathrm{B}=\mathrm{A}$<br/>2.$\quad(\mathrm{A}-\mathrm{B}) \cup \mathrm{A}=\mathrm{A}$<br/>3.$(\mathrm{A}-\mathrm{B}) \cap \mathrm{B}=\phi$<br/>4.$\quad \mathrm{A} \subset \mathrm{B} \Rightarrow \mathrm{A} \cup \mathrm{B}=\mathrm{B}$<br/>Which of the above are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,2 and 3</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2,3 and 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">1,3 and 4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2,3 and 4</span> </div>

answer:
<div class="solution">Venn diagram<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/h6rTXg4WpYwF1Y0iX4KpAHXoXT1I6HPgWthjePBEDRc.original.fullsize.png"/><br/><br/>from, Venn diagram we can observe that $\mathrm{A}-\mathrm{B}$ is the shaded part.<br/>$(\mathrm{A}-\mathrm{B}) \cup \mathrm{A}=\mathrm{A}$<br/>$(\mathrm{A}-\mathrm{B}) \cap \mathrm{B}=\phi$<br/>$A \subseteq B \Rightarrow A \cup B=B$</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-the-differential-equation-d-x-2-d-2-y-2-d-x-d-y-2-9-y-x-1the-degree-of-the-differential-equation-is-1-2the-order-

slug:
consider-the-following-in-respect-of-the-differential-equation-d-x-2-d-2-y-2-d-x-d-y-2-9-y-x-1the-degree-of-the-differential-equation-is-1-2the-order-96613

question:
<div class="question">Consider the following in respect of the differential equation:<br/>$\frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+9 y=x \quad$<br/>1.The degree of the differential equation is $1 .$<br/>2.The order of the differential equation is 2 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+9 y=x$<br/>Order $=2$<br/>Degree $=1$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-the-function-2016ii-f-x-2-x-x-0-2-x-x-0-1-limf-x-1-x-does-not-exist-2-f-x-is-differentiable-at-x-0-3-f-x-is-conti

slug:
consider-the-following-in-respect-of-the-function-2016ii-f-x-2-x-x-0-2-x-x-0-1-limf-x-1-x-does-not-exist-2-f-x-is-differentiable-at-x-0-3-f-x-is-conti-18155

question:
<div class="question">Consider the following in respect of the function [2016-II]<br/>$f(x)=\left\{\begin{array}{l}2+x, \quad x \geq 0 \\ 2-x, x   &lt; 0\end{array}\right.$<br/>1- $\operatorname{limf}_{x \rightarrow 1}(x)$ does not exist.<br/>2- $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$<br/>3- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$ Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3 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">C</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 3 only</span> </li> </ul>

answer:
<div class="solution">For $x \geq 1$<br/>$\lim _{x \rightarrow 1} f(x)=\lim _{x \rightarrow 1} 2+x=2+1=3$<br/>For $x   &lt; 1$<br/>$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} 2+x=2+1=3$<br/>So, $\lim _{x \rightarrow 1} f(x)$ exist.<br/>At $x=0$<br/>RHL: $\lim _{h \rightarrow 0^{+}} f(0+h)=\lim _{h \rightarrow 0} 2+h=2$<br/>LHL $\lim _{h \rightarrow 0^{-}} f(0-h)=\lim _{h \rightarrow 0} 2-h=2$<br/>$f(0)=2+0=2 .$<br/>So, $\mathrm{RHL}=\mathrm{LHL}=f(0)$<br/>$\Rightarrow f(x)$ is continuous at $x=0$ Differentiability at $x=0$<br/>LHD: $\lim _{h \rightarrow 0^{-}} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0^{-}} \frac{2+h-2}{-h}$<br/>$=\frac{-h}{h}=-1$<br/>RHD : $\lim _{h \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{h}=\lim _{h \rightarrow 0^{+}} \frac{2+h-2}{h}=1$<br/>Since $\mathrm{LHD} \neq \mathrm{RHD}$<br/>So, $f(x)$ is not differentiable at $x=0$.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-the-function-f-x-x-3-1-f-x-is-continuous-at-x-3-2-f-x-is-differentiable-at-x-0-which-of-the-above-statements-isar

slug:
consider-the-following-in-respect-of-the-function-f-x-x-3-1-f-x-is-continuous-at-x-3-2-f-x-is-differentiable-at-x-0-which-of-the-above-statements-isar-78923

question:
<div class="question">Consider the following in respect of the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-3|:$<br/>1- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=3$<br/>2- $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Given $\mathrm{f}(\mathrm{x})=|\mathrm{x}-3|$ is not continuous at $\mathrm{x}=3$ but it is differentiable at $\mathrm{x}=0$.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-the-matrix-a-1-1-1-1-1-a-2-a-2-a-3-4-a-which-of-the-above-isare-correct

slug:
consider-the-following-in-respect-of-the-matrix-a-1-1-1-1-1-a-2-a-2-a-3-4-a-which-of-the-above-isare-correct-62094

question:
<div class="question">Consider the following in respect of the matrix $\mathrm{A}=\left(\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right):$<br/>$1.$ $\mathrm{A}^{2}=-\mathrm{A}$<br/>$2.$ $A^{3}=4 A$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$A=\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right]$<br/>$\begin{aligned} A \cdot A &amp;=\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right]\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right] \\ &amp;=\left[\begin{array}{cc}2 &amp; -2 \\ -2 &amp; 2\end{array}\right]=-2\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right] \\ A^{2} &amp;=-2 A \end{aligned}$<br/>$A^{2}, A=-2\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right]\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right]$<br/>$\quad=-2\left[\begin{array}{cc}2 &amp; -2 \\ -2 &amp; 2\end{array}\right]=4\left[\begin{array}{cc}-1 &amp; 1 \\ 1 &amp; -1\end{array}\right]$<br/>$A^{3}=4 A$<br/>Hence $A^{2} \neq-A, A^{3}=4 A$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-two-events-a-and-b-1-p-a-occurs-but-not-b-p-a-p-b-if-b-a-2-p-a-alone-or-b-alone-occurs-p-a-p-b-p-a-b-3-p-a-b-p-a-

slug:
consider-the-following-in-respect-of-two-events-a-and-b-1-p-a-occurs-but-not-b-p-a-p-b-if-b-a-2-p-a-alone-or-b-alone-occurs-p-a-p-b-p-a-b-3-p-a-b-p-a-11093

question:
<div class="question">Consider the following in respect of two events $\mathrm{A}$ and $\mathrm{B}$ :<br/>$1.$ $\quad \mathrm{P}($ A occurs but not $\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$ if $\mathrm{B} \subset \mathrm{A}$<br/>$2.$ $\quad \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone occurs $)=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$3.$ $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$ if $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1 and 3 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">C</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 2 only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 3 only</span> </div>

answer:
<div class="solution">Statement $1:$ If $B \subset A$, then $\mathrm{P}(\mathrm{A}-\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$<br/>$\therefore$ It is correct.<br/>Statement $2: \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone $)$ $=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(-\mathrm{A} \cap \mathrm{B})$<br/>$\therefore$ It is false. Statement $3:$ If $A, B$ are mutually exclusive events, then $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$<br/>$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$<br/>It is correct.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-in-respect-of-two-nonsingular-matrices-a-and-b-of-same-order-2015-i-1-det-a-b-det-a-det-b-2-a-b-1-a-1-b-1-which-of-the-above-is

slug:
consider-the-following-in-respect-of-two-nonsingular-matrices-a-and-b-of-same-order-2015-i-1-det-a-b-det-a-det-b-2-a-b-1-a-1-b-1-which-of-the-above-is-82282

question:
<div class="question">Consider the following in respect of two non-singular matrices $A$ and $B$ of same order: $\quad[2015-I]$<br/>$1.$ $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$<br/>$2.$ $(A+B)^{-1}=A^{-1}+B^{-1}$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 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> </ul>

answer:
<div class="solution">Non-singular matrix is a matrix whose determinate Value is non-zero.<br/>Let $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}2 &amp; 1 \\ 1 &amp; 2\end{array}\right]$<br/>Here, $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrix Now from Statement 1 :<br/>$\mathrm{A}+\mathrm{B}=\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 2\end{array}\right]+\left[\begin{array}{ll}2 &amp; 1 \\ 1 &amp; 2\end{array}\right]=\left[\begin{array}{ll}3 &amp; 2 \\ 2 &amp; 4\end{array}\right]$<br/>$\operatorname{det}(\mathrm{A}+\mathrm{B})=12-4=8$<br/>Now det $(\mathrm{A})=\left|\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 2\end{array}\right|=1$<br/>and det $(\mathrm{B})=\left|\begin{array}{ll}2 &amp; 1 \\ 1 &amp; 2\end{array}\right|=3$<br/>Now, $\operatorname{det}(\mathrm{A}+\mathrm{B})=4 \neq \operatorname{det}(\mathrm{A}+\mathrm{B})$<br/>Statement 1 is wrong. Now from Statement 2 :<br/>$(\mathrm{A}+\mathrm{B})^{-1}=\frac{1}{|\mathrm{~A}+\mathrm{B}|} \operatorname{adj}(\mathrm{A}+\mathrm{B})$<br/>$=\frac{1}{8}\left[\begin{array}{cc}4 &amp; -2 \\ -2 &amp; 3\end{array}\right]$<br/>and $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})=\frac{1}{1}\left[\begin{array}{cc}2 &amp; -1 \\ -1 &amp; 1\end{array}\right]$<br/>and $\mathrm{B}^{-1}=\frac{1}{|\mathrm{~B}|} \operatorname{adj}(\mathrm{B})=\frac{1}{3}\left[\begin{array}{cc}2 &amp; -1 \\ -1 &amp; 2\end{array}\right]$<br/>Now, $A^{-1}+B^{-1}=\left[\begin{array}{cc}2 &amp; -1 \\ -1 &amp; 1\end{array}\right]+\left[\begin{array}{cc}\frac{2}{3} &amp; \frac{-1}{3} \\ \frac{-1}{3} &amp; \frac{2}{3}\end{array}\right]$<br/>$=\left[\begin{array}{cc}\frac{8}{3} &amp; \frac{-4}{3} \\ \frac{-4}{3} &amp; \frac{5}{3}\end{array}\right] \neq(\mathrm{A}+\mathrm{B})^{-1}$<br/>$\therefore$ Statement 2 is wrong<br/>$\therefore$ Option (d) is correct.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-inequalities-in-respect-of-vectors-a-and-b-1-a-b-a-a-b-2-a-b-3-a-b-which-of-the-above-isare-correct

slug:
consider-the-following-inequalities-in-respect-of-vectors-a-and-b-1-a-b-a-a-b-2-a-b-3-a-b-which-of-the-above-isare-correct-11355

question:
<div class="question">Consider the following inequalities in respect of vectors<br/>$\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}:$<br/>$1.$ $\quad|\vec{a}+\vec{b}|_{ÃÂ£}|\vec{a}|+|\vec{b}|$<br/>$2.$ $\quad|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{3}|\overrightarrow{\mathrm{a}}|-|\overrightarrow{\mathrm{b}}|$<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$\begin{aligned} \text {  } &amp;|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}| \leq|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}| \\ &amp;|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}| \geq|\overrightarrow{\mathrm{a}}|-|\overrightarrow{\mathrm{b}}| \end{aligned}$<br/>both are correct.</div>


tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-following-perhalate-ion-in-acidic-medium-cl-o-4-i-br-o-4-ii-i-o-4-iii-arrange-these-in-the-decreasing-order-of-oxidizing-power

slug:
consider-the-following-perhalate-ion-in-acidic-medium-cl-o-4-i-br-o-4-ii-i-o-4-iii-arrange-these-in-the-decreasing-order-of-oxidizing-power-72548

question:
<div class="question">Consider the following perhalate ion in acidic medium<br/> <math><mtext>Cl</mtext><msubsup><mrow><mtext>O</mtext></mrow><mrow><mn>4</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mfenced separators="|"><mrow><mi mathvariant="normal">I</mi></mrow></mfenced><mo>,</mo><mtext>Br</mtext><msubsup><mrow><mtext>O</mtext></mrow><mrow><mn>4</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mfenced separators="|"><mrow><mtext>II</mtext></mrow></mfenced><mo>,</mo><mtext>I</mtext><msubsup><mrow><mtext>O</mtext></mrow><mrow><mn>4</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mfenced separators="|"><mrow><mtext>III</mtext></mrow></mfenced></math><br/>
Arrange these in the decreasing order of oxidizing power.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi><mo>&gt;</mo><mi mathvariant="normal">I</mi></math></span> </div>

answer:
<div class="solution">Standard reduction potential (in volts) of the perhalate ions are gives as belowâ<br/><math> <mrow> <msubsup> <mrow> <mtext>ClO</mtext></mrow> <mtext>4</mtext> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2H</mtext></mrow> <mtext>+</mtext> </msup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2e</mtext></mrow> <mo>â</mo> </msup> <mo>â</mo><msubsup> <mrow> <mtext>ClO</mtext></mrow> <mn>3</mn> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msub> <mtext>H</mtext> <mtext>2</mtext> </msub> <mtext>O;</mtext><mtext>â</mtext><msup> <mtext>E</mtext> <mtext>o</mtext> </msup> <mtext>â</mtext><mtext>=</mtext><mtext>â</mtext><mtext>1</mtext><mtext>.19</mtext><mtext>â</mtext><mtext>V</mtext></mrow> </math><br/><math> <mrow> <msubsup> <mrow> <mtext>BrO</mtext></mrow> <mtext>4</mtext> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2H</mtext></mrow> <mtext>+</mtext> </msup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2e</mtext></mrow> <mo>â</mo> </msup> <mo>â</mo><msubsup> <mrow> <mtext>BrO</mtext></mrow> <mn>3</mn> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msub> <mtext>H</mtext> <mtext>2</mtext> </msub> <mtext>O;</mtext><mtext>â</mtext><msup> <mtext>E</mtext> <mtext>o</mtext> </msup> <mtext>â</mtext><mtext>=</mtext><mtext>â</mtext><mtext>1</mtext><mtext>.74</mtext><mtext>â</mtext><mtext>V</mtext></mrow> </math><br/><math> <mrow> <msubsup> <mrow> <mtext>IO</mtext></mrow> <mtext>4</mtext> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2H</mtext></mrow> <mtext>+</mtext> </msup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msup> <mrow> <mtext>2e</mtext></mrow> <mo>â</mo> </msup> <mo>â</mo><msubsup> <mrow> <mtext>IO</mtext></mrow> <mn>3</mn> <mo>â</mo> </msubsup> <mtext>â</mtext><mtext>+</mtext><mtext>â</mtext><msub> <mtext>H</mtext> <mtext>2</mtext> </msub> <msup> <mrow> <mtext>O; E</mtext></mrow> <mtext>o</mtext> </msup> <mtext>â</mtext><mtext>=</mtext><mtext>â</mtext><mtext>+1</mtext><mtext>.65</mtext><mtext>â</mtext><mtext>V</mtext></mrow> </math><br/>More the standard reduction potential, more is the tendency of perhalite ion to undergo reduction and stronger will be an oxidizing agent.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-points-1-0-5-2-2-1-3-3-4-which-of-the-above-lie-on-the-line-3-x-y-5-and-at-a-distance-10-from-1-2

slug:
consider-the-following-points-1-0-5-2-2-1-3-3-4-which-of-the-above-lie-on-the-line-3-x-y-5-and-at-a-distance-10-from-1-2-13044

question:
<div class="question">Consider the following points:<br/>1) $(0,5)$ <br/>2) $(2,-1)$<br/>3) $(3,-4)$<br/>Which of the above lie on the line $3 x+y=5$ and at a distance<br/>$\sqrt{10}$ from $(1,2) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1 and 2 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">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1 and 2 only</span> </div>

answer:
<div class="solution">All three points $(0,5),(2,-1)$ and $(3,-4)$ lie on $3 x+y=5$<br/>$\sqrt{(0-1)^{2}+(5-2)^{2}}=\sqrt{10}$<br/>$\sqrt{(2-1)^{2}+(-1-2)^{2}}=\sqrt{10}$<br/>$\sqrt{(3-1)^{2}+(-4-2)^{2}}=\sqrt{40}=2 \sqrt{10}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Main', 'JEE Main 2024 (31 Jan Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">163</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-reaction-at-298-k-3-2-o-2-g-o-3-g-k-p-2-47-10-29-r-g-0-for-the-reaction-is-kj-given-r-8-314-jk-1-mol-1-round-off-your-answer-to

slug:
consider-the-following-reaction-at-298-k-3-2-o-2-g-o-3-g-k-p-2-47-10-29-r-g-0-for-the-reaction-is-kj-given-r-8-314-jk-1-mol-1-round-off-your-answer-to-98096

question:
<div class="question"><p>Consider the following reaction at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>298</mn><mo>Â </mo><mi mathvariant="normal">K</mi></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mfenced><mi mathvariant="normal">g</mi></mfenced></mrow></msub><mo>â</mo><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mfenced><mi mathvariant="normal">g</mi></mfenced></mrow></msub><mo>.</mo><msub><mi mathvariant="normal">K</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><mn>2</mn><mo>.</mo><mn>47</mn><mo>Ã</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>29</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mo>â</mo><mi mathvariant="normal">r</mi></msub><msup><mi mathvariant="normal">G</mi><mn>0</mn></msup></math>Â for the reaction is _________ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>kJ</mi></math>. (Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo>Â </mo><msup><mi>JK</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo>Â </mo><msup><mi>mol</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math>)</p><p>Round off your answer to the nearest integer.</p></div>

answer:
<div class="solution"><p>GivenÂ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo>Â </mo><msup><mi>JK</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup><mo>Â </mo><msup><mi>mol</mi><mrow><mo>â</mo><mn>1</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mfenced><mi mathvariant="normal">g</mi></mfenced></mrow></msub><mo>â</mo><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mfenced><mi mathvariant="normal">g</mi></mfenced></mrow></msub><mo>.</mo><mo>Â </mo><msub><mi mathvariant="normal">K</mi><mi mathvariant="normal">p</mi></msub><mo>=</mo><mn>2</mn><mo>.</mo><mn>47</mn><mo>Ã</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>29</mn></mrow></msup></math></p><p>Standard Gibbs free energy of the formation of a compound is basically the change of Gibbs free energy that is followed by the formation of one mole of that substance from its component element available at their standard states or the most stable form of the element which is at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>Â°</mo><mi mathvariant="normal">C</mi></math> Â and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> kPa. Its symbol is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Î</mi><mi mathvariant="normal">f</mi></msub><mi mathvariant="normal">G</mi><mo>Ë</mo></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mo>â</mo><mi mathvariant="normal">r</mi></msub><msup><mi mathvariant="normal">G</mi><mn>0</mn></msup><mo>=</mo><mo>-</mo><mi>RT</mi><mo>Â </mo><mi>ln</mi><mo>Â </mo><msub><mi mathvariant="normal">K</mi><mi mathvariant="normal">p</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo>Ã</mo><msup><mn>10</mn><mrow><mo>â</mo><mn>3</mn></mrow></msup><mo>Ã</mo><mn>298</mn><mo>Ã</mo><mi>ln</mi><mo>Â </mo><mo>(</mo><mn>2</mn><mo>.</mo><mn>47</mn><mo>Ã</mo><msup><mn>10</mn><mrow><mo>â</mo><mn>29</mn></mrow></msup><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>â</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo>Ã</mo><msup><mn>10</mn><mrow><mo>â</mo><mn>3</mn></mrow></msup><mo>Ã</mo><mn>298</mn><mo>Ã</mo><mo>(</mo><mo>â</mo><mn>65</mn><mo>.</mo><mn>87</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>163</mn><mo>.</mo><mn>19</mn><mo>Â </mo><mi>kJ</mi></math></p></div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Only I</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-relations-among-the-angles-and-made-by-a-vector-with-the-coordinate-axes-i-cos-2-cos-2-cos-2-1-ii-sin-2-sin-2-sin-2-1-which-of-

slug:
consider-the-following-relations-among-the-angles-and-made-by-a-vector-with-the-coordinate-axes-i-cos-2-cos-2-cos-2-1-ii-sin-2-sin-2-sin-2-1-which-of-46311

question:
<div class="question">Consider the following relations among the angles $\alpha, \beta$ and $\gamma$ made by a vector with the coordinate axes<br/>I. $\quad \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$<br/>II. $\quad \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Only I</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">Only II</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both I and II</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither I nor II</span> </li> </ul>

answer:
<div class="solution">We have, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$<br/>$\Rightarrow 2 \cos ^{2} \alpha+2 \cos ^{2} \beta+2 \cos ^{2} \gamma=2$<br/>$\Rightarrow 2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1=2-3$<br/>$\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$<br/>Hence statement $-\mathrm{I}$ is correct.<br/>and now from (i), $1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$<br/>$\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$<br/>Hence, only statement I is correct.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-relations-from-a-to-b-where-a-u-v-w-x-y-z-and-b-p-q-r-s-1-u-p-v-p-w-p-x-q-y-q-z-q-2u-pv-qw-rz-s-3u-sv-rw-qu-pv-qz-q-4-u-q-v-p-w

slug:
consider-the-following-relations-from-a-to-b-where-a-u-v-w-x-y-z-and-b-p-q-r-s-1-u-p-v-p-w-p-x-q-y-q-z-q-2u-pv-qw-rz-s-3u-sv-rw-qu-pv-qz-q-4-u-q-v-p-w-71067

question:
<div class="question">Consider the following relations from $A$ to $B$ where $A=\{u, v, w, x, y, z\}$ and $B=\{p, q, r, s\}$<br/>1.$\{(u, p),(v, p),(w, p),(x, q),(y, q),(z, q)\}$<br/>2.\{(u, p),(v, q),(w, r),(z, s)\} <br/>3.\{(u, s),(v, r),(w, q),(u, p),(v, q),(z, q)<br/>4.$\{(u, q),(v, p),(w, s),(x, r),(y, q),(z, s)$<br/>Which of the above relations are not functions ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 and 4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2 and 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">3 and 4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 3</span> </div>

answer:
<div class="solution">Given that, $A=\{u, v, x, y, z\} ; B=\{p, q, r, s\}$ As we know, a mapping $\mathrm{f}: \mathrm{x} \rightarrow \mathrm{y}$ is said to be a function if each element in the set $\mathrm{x}$ has its image in set $\mathrm{y} .$ It $\mathrm{k}$<br/>also possible that these are few elements in set y which are not the image of any element in set $\mathrm{x}$. Every element<br/>in set $x$ should have one and only one image.<br/>(i)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/zBzdjTmDGjYoeqZxXDUF9a8QP1TYREnDLozV2zSGrV0.original.fullsize.png"/><br/><br/>(ii)<img src="https://cdn-question-pool.getmarks.app/pyq/nda/pxVsDNm70kIdfJ94tjJyRgk8arjnQSQ5ZHv75HcW_Mk.original.fullsize.png"/><br/><br/>(iii)<img src="https://cdn-question-pool.getmarks.app/pyq/nda/Pu9BbiDfaAZGqE8lyqxH7m9dLRAsXAi9_DT8u3oj4CI.original.fullsize.png"/><br/><br/>(iv)<img src="https://cdn-question-pool.getmarks.app/pyq/nda/rJrYYseFZq869M2Y6pdipVnLu5D_ayEm8BGh9GJUdss.original.fullsize.png"/><br/><br/>(ii) and (iii) are not function</div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2017 (08 Apr Online)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d-1

slug:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d-1-19843

question:
<div class="question">Consider the following standard electrode potentials (<math><msup><mrow><mtext>E</mtext></mrow><mrow><mtext>o</mtext></mrow></msup></math> in volts) in aqueous solution:<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><munder><mrow><mtext>E</mtext><mtext>l</mtext><mtext>e</mtext><mtext>m</mtext><mtext>e</mtext><mtext>n</mtext><mtext>t</mtext></mrow><mrow></mrow></munder></mtd><mtd><munder><mrow><msup><mtext>M</mtext><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mtext>M</mtext></mrow><mrow></mrow></munder></mtd><mtd><munder><mrow><msup><mtext>M</mtext><mo>+</mo></msup><mo>/</mo><mtext>M</mtext></mrow><mrow></mrow></munder></mtd></mtr><mtr><mtd><mtext>A</mtext><mtext>l</mtext></mtd><mtd><mo>-</mo><mn>1.66</mn></mtd><mtd><mo>+</mo><mn>0.55</mn></mtd></mtr><mtr><mtd><mtext>T</mtext><mtext>l</mtext></mtd><mtd><mo>+</mo><mn>1.26</mn></mtd><mtd><mo>-</mo><mn>0.34</mn></mtd></mtr></mtable></math><br/>Based on these data, which of the following statements is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math>  is more stable than  <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></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><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math></span> </div>

answer:
<div class="solution"><math><mo>∆</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mo>-</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">F</mtext><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup></math><br/><br/>if <math><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">p</mtext><mtext mathvariant="normal">o</mtext><mtext mathvariant="normal">s</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext><mo>;</mo><mtext mathvariant="normal"> </mtext></math>then <math><mo>∆</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">g</mtext><mtext mathvariant="normal">a</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext></math>,<br/><br/>For those, conversions whose <math> <mrow> <msup> <mtext>E</mtext> <mtext>o</mtext> </msup> </mrow> </math> values are more positive are more favourable.<br/>As we can see from then electrode potential data, that reduction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Al</mi><mo>+</mo></msup></math> is spontaneous, but reduction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Tl</mi><mo>+</mo></msup></math> is not. So, we can conclude from this that <math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> </div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main']
source_db: MarksBatch1_P2.db
title:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d-2

slug:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d-2-92456

question:
<div class="question">Consider the following standard electrode potentials (<math><msup><mrow><mtext>E</mtext></mrow><mrow><mtext>o</mtext></mrow></msup></math> in volts) in aqueous solution:<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><munder><mrow><mtext>E</mtext><mtext>l</mtext><mtext>e</mtext><mtext>m</mtext><mtext>e</mtext><mtext>n</mtext><mtext>t</mtext></mrow><mrow></mrow></munder></mtd><mtd><munder><mrow><msup><mtext>M</mtext><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mtext>M</mtext></mrow><mrow></mrow></munder></mtd><mtd><munder><mrow><msup><mtext>M</mtext><mo>+</mo></msup><mo>/</mo><mtext>M</mtext></mrow><mrow></mrow></munder></mtd></mtr><mtr><mtd><mtext>A</mtext><mtext>l</mtext></mtd><mtd><mo>-</mo><mn>1.66</mn></mtd><mtd><mo>+</mo><mn>0.55</mn></mtd></mtr><mtr><mtd><mtext>T</mtext><mtext>l</mtext></mtd><mtd><mo>+</mo><mn>1.26</mn></mtd><mtd><mo>-</mo><mn>0.34</mn></mtd></mtr></mtable></math><br/>Based on these data, which of the following statements is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math>  is more stable than  <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></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><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math></span> </div>

answer:
<div class="solution"><math><mo>∆</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mo>-</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">F</mtext><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup></math><br/><br/>if <math><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">p</mtext><mtext mathvariant="normal">o</mtext><mtext mathvariant="normal">s</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext><mo>;</mo><mtext mathvariant="normal"> </mtext></math>then <math><mo>∆</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">g</mtext><mtext mathvariant="normal">a</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext></math>,<br/><br/>For those, conversions whose <math> <mrow> <msup> <mtext>E</mtext> <mtext>o</mtext> </msup> </mrow> </math> values are more positive are more favourable.<br/>As we can see from then electrode potential data, that reduction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Al</mi><mo>+</mo></msup></math> is spontaneous, but reduction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Tl</mi><mo>+</mo></msup></math> is not. So, we can conclude from this that <math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> </div>


tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch1_P2.db
title:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d

slug:
consider-the-following-standard-electrode-potentials-e-o-in-volts-in-aqueous-solution-e-l-e-m-e-n-t-m-3-m-m-m-a-l-166-055-t-l-126-034-based-on-these-d-50631

question:
<div class="question">Consider the following standard electrode potentials (<math><msup><mrow><mtext>E</mtext></mrow><mrow><mtext>o</mtext></mrow></msup></math> in volts) in aqueous solution:<br/> <br/> <math><mtable><mtr><mtd><munder underaccent="false"><mrow><mtext mathvariant="normal">E</mtext><mtext mathvariant="normal">l</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">m</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">t</mtext></mrow><mo>_</mo></munder></mtd><mtd><munder underaccent="false"><mrow><mrow><mrow><msup><mrow><mtext>M</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></mrow><mo>/</mo><mrow><mtext>M</mtext></mrow></mrow></mrow><mo>_</mo></munder></mtd><mtd><munder underaccent="false"><mrow><mrow><mrow><msup><mrow><mtext>M</mtext></mrow><mrow><mo>+</mo></mrow></msup></mrow><mo>/</mo><mrow><mtext>M</mtext></mrow></mrow></mrow><mo>_</mo></munder></mtd></mtr><mtr><mtd><mtext>A</mtext><mtext>l</mtext></mtd><mtd><mo>-</mo><mn>1.66</mn></mtd><mtd><mo>+</mo><mn>0.55</mn></mtd></mtr><mtr><mtd><mtext>T</mtext><mtext>l</mtext></mtd><mtd><mo>+</mo><mn>1.26</mn></mtd><mtd><mo>-</mo><mn>0.34</mn></mtd></mtr></mtable></math><br/> <br/>
Based on these data, which of the following statements is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math>  is more stable than  <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></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><mtext>T</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math> is more stable than <math><mtext>A</mtext><msup><mrow><mtext>l</mtext></mrow><mrow><mo>+</mo></mrow></msup></math></span> </div>

answer:
<div class="solution"><math><mo>â</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mo>-</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">F</mtext><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup></math><br/><br/>if <math><msup><mrow><mtext mathvariant="normal">E</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal"> </mtext><mtext mathvariant="normal">p</mtext><mtext mathvariant="normal">o</mtext><mtext mathvariant="normal">s</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext><mo>;</mo><mtext mathvariant="normal"> </mtext></math>then <math><mo>â</mo><msup><mrow><mtext mathvariant="normal">G</mtext></mrow><mrow><mtext mathvariant="normal">o</mtext></mrow></msup><mo>=</mo><mtext mathvariant="normal">n</mtext><mtext mathvariant="normal">e</mtext><mtext mathvariant="normal">g</mtext><mtext mathvariant="normal">a</mtext><mtext mathvariant="normal">t</mtext><mtext mathvariant="normal">i</mtext><mtext mathvariant="normal">v</mtext><mtext mathvariant="normal">e</mtext></math>,<br/><br/>For those conversions whose <math> <mrow> <msup> <mtext>E</mtext> <mtext>o</mtext> </msup> </mrow> </math> values are more positive are more favourable.<br/> </div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statement-the-mean-of-a-binomial-distribution-is-3-and-variance-is-4-which-of-the-following-is-correct-regarding-this-statement

slug:
consider-the-following-statement-the-mean-of-a-binomial-distribution-is-3-and-variance-is-4-which-of-the-following-is-correct-regarding-this-statement-60005

question:
<div class="question">Consider the following statement:<br/>"The mean of a binomial distribution is 3 and variance is $4 . "$ Which of the following is correct regarding this statement?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">It is always true</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">It is sometimes true</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">It is never true</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">No conclusion can be drawn</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It is never true</span> </div>

answer:
<div class="solution">Given, that $\mathrm{np}=3$ and $\mathrm{npq}=4$ where $\mathrm{p}$ is the probability of success in one trial and $\mathrm{q}$ is the probability of failure and $\mathrm{n}$ is the number of trials<br/>$\Rightarrow q=\frac{4}{3}$<br/>and this is not possible. Thus, the given statement is never true.</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-1-in-radian-measure-is-less-than-002-radians-2-1-radian-in-degree-measure-is-greater-than-4-5-which-of-the-above-s

slug:
consider-the-following-statements-1-1-in-radian-measure-is-less-than-002-radians-2-1-radian-in-degree-measure-is-greater-than-4-5-which-of-the-above-s-10768

question:
<div class="question">Consider the following statements: <br/>1. $1^{\circ}$ in radian measure is less than $0.02$ radians.<br/>2. 1 radian in degree measure is greater than $45^{\circ}$. Which of the above statements is/are correct?</div>

answer:
<div class="solution">(1) $1^{\circ}=\left(\frac{\pi}{180}\right)$ radian $=0.01746$ radian which is less than $0.02$ radian.<br/>(2) 1 radian $=\left(\frac{180}{\pi}\right)^{\circ}=57^{\circ} 16^{\prime} 22^{\prime \prime}$ approx. which is<br/>greater than $45^{\circ}$. Hence, both statements are true.</div>


title: consider-the-following-statements-1-2-which-of-the-statements-given-above-isare-correct
slug: consider-the-following-statements-1-2-which-of-the-statements-given-above-isare-correct-68135
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 1)']
answer: <div class="solution">Both statements are incorrect</div>
source_db: MarksBatch1_P2.db
question:
<div class="question">Consider the following statements<br/>1.$\phi \in\{\phi\}$<br/>2.$\quad\{\phi\} \subseteq \phi$<br/>Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor $2$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Neither 1 nor $2$</span> </div>


tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-10-1-7-0-2-105-1-10-p-10-for-some-prime-number-p-where-1-is-a-cube-root-of-unity-which-of-the-above-statements-isa

slug:
consider-the-following-statements-1-10-1-7-0-2-105-1-10-p-10-for-some-prime-number-p-where-1-is-a-cube-root-of-unity-which-of-the-above-statements-isa-35036

question:
<div class="question">Consider the following statements :<br/>1- $\left(\omega^{10}+1\right)^{7}+\omega=0$<br/>2- $\left(\omega^{105}+1\right)^{10}=\mathrm{p}^{10}$ for some prime number p where $\omega \neq 1$ is a cube root of unity. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement-1 $\left(\omega^{10}+1\right)^{7}+\omega=0$<br/>L.H.S $=\left[\left(\omega^{3}\right)^{3} \omega+1\right]^{7}+\omega$<br/>$\Rightarrow(\omega+1)^{7}+\omega\left(\because \omega^{3}=1\right)$<br/>$\Rightarrow-\left[\omega^{2}\right]^{7}+\omega=-\left(\omega^{3}\right)^{4} \cdot \omega^{2} \cdot+\omega_{.}=-\omega^{2}+\omega \neq 0$<br/>Hence, LHS $\neq$ RHS<br/>Statement $-2$<br/>L.H.S $=\left(\omega^{105}+1\right)^{10}=\left[\left(\omega^{3}\right)^{35}+1\right]^{10}=2^{10}$<br/>where 2 is a prime number. Hence, statement $-2$ is correct.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 1)']
answer: <div class="solution">Both the statements are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-a-matrix-is-not-a-number-2-two-determinants-of-different-order-may-have-the-same-value-which-of-the-above-statemen

slug:
consider-the-following-statements-1-a-matrix-is-not-a-number-2-two-determinants-of-different-order-may-have-the-same-value-which-of-the-above-statemen-27560

question:
<div class="question">Consider the following statements:<br/>$1.$ A matrix is not a number<br/>$2.$ Two determinants of different order may have the same value.<br/>Which of the above statements is/are correct?</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-both-variance-and-standard-deviation-are-measures-of-variability-in-the-population-2-standard-deviation-is-the-squ

slug:
consider-the-following-statements-1-both-variance-and-standard-deviation-are-measures-of-variability-in-the-population-2-standard-deviation-is-the-squ-50187

question:
<div class="question">Consider the following statements:<br/> 1 . Both variance and standard deviation are measures of variability in the population.<br/> 2 . Standard deviation is the square of the variance. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Both statements are wrong. Also, $\mathrm{S} . \mathrm{D}=\sqrt{\text { variance }}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 2)']
answer: <div class="solution">Only statements 2 and 3 are correct</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-coefficient-of-variation-depends-on-the-unit-of-measurement-of-the-variable-2-range-is-a-measure-of-dispersion-3-m

slug:
consider-the-following-statements-1-coefficient-of-variation-depends-on-the-unit-of-measurement-of-the-variable-2-range-is-a-measure-of-dispersion-3-m-87676

question:
<div class="question">Consider the following statements:<br/> 1 . Coefficient of variation depends on the unit of measurement of the variable.<br/> 2 . Range is a measure of dispersion.<br/> 3 . Mean deviation is least when measured about median. Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 and 3 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">C</span> <span class="option-data">1 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 3 only</span> </div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
answer: <div class="solution">All statements are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-derivative-of-f-x-may-not-exist-at-some-point-2-derivative-of-f-x-may-exist-finitely-at-some-point-3-derivative-of

slug:
consider-the-following-statements-1-derivative-of-f-x-may-not-exist-at-some-point-2-derivative-of-f-x-may-exist-finitely-at-some-point-3-derivative-of-73671

question:
<div class="question">Consider the following statements:<br/>1- Derivative of $f(\mathrm{x})$ may not exist at some point.<br/>2- Derivative of $f(\mathrm{x})$ may exist finitely at some point.<br/>3- Derivative of $f(\mathrm{x})$ may be infinite (geometrically) at some point. Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 3 only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1,2 and 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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1,2 and 3</span> </div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-determinant-is-a-square-matrix-2-determinant-is-a-number-associated-with-a-square-matrix-which-of-the-above-statem

slug:
consider-the-following-statements-1-determinant-is-a-square-matrix-2-determinant-is-a-number-associated-with-a-square-matrix-which-of-the-above-statem-77531

question:
<div class="question">Consider the following statements: <br/>$1.$ Determinant is a square matrix.<br/>$2.$ Determinant is a number associated with a square matrix. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 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">C</span> <span class="option-data">Both 1 and 2 .</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">$1.$ We know that, determinant is not a square matrix, so it is not a true statement.<br/>$2.$ It is true that, determinant is a number associated with a square matrix. <br/>Hence, Statement 2 is correct</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
answer: <div class="solution">Statements (1), (2) are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-dx-dy-at-a-point-on-the-curve-given-slope-of-the-tangent-at-that-point-2if-at-denotes-acceleration-of-a-particle-t

slug:
consider-the-following-statements-1-dx-dy-at-a-point-on-the-curve-given-slope-of-the-tangent-at-that-point-2if-at-denotes-acceleration-of-a-particle-t-49154

question:
<div class="question">Consider the following statements: <br/>1.$\frac{\mathrm{dy}}{\mathrm{dx}}$ at a point on the curve given slope of the tangent<br/>at that point.<br/>2.If a(t) denotes acceleration of a particle, then $\int \mathrm{a}(\mathrm{t}) \mathrm{dt}+\mathrm{c}$ gives velocity of the particle.<br/>3.If $\mathrm{s}(\mathrm{t})$ gives displacement of a particle at time $\mathrm{t}$, then<br/>$\frac{\mathrm{ds}}{\mathrm{dt}}$ gives its acceleration at that instant. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 and 2 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">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-equations-ax-b-y-cz-d-0-a-x-b-y-c-z-d-0-represent-a-straight-line-2-equation-of-the-form-x-m-y-n-z-represent-a-str

slug:
consider-the-following-statements-1-equations-ax-b-y-cz-d-0-a-x-b-y-c-z-d-0-represent-a-straight-line-2-equation-of-the-form-x-m-y-n-z-represent-a-str-52160

question:
<div class="question">Consider the following statements:<br/> 1 . Equations ax $+b y+c z+d=0, a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$<br/>represent a straight line.<br/> 2 . Equation of the form<br/>$\frac{\mathrm{x}-\alpha}{\ell}=\frac{\mathrm{y}-\beta}{\mathrm{m}}=\frac{\mathrm{z}-\gamma}{\mathrm{n}}$<br/>represent a straight line passing through the point $(\alpha, \beta, \gamma)$ and having direction ratio proportional to $l, \mathrm{~m}, \mathrm{n}$. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Equations $a x+b y+c z+d=0, a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$<br/>represent a straight line and equation of the form $\frac{x-\alpha}{\ell}=\frac{y-\beta}{m}=\frac{z-y}{n}$<br/>represent a straight line passing through the point $(\alpha, \beta, \gamma)$ and having direction ratios proportional to $l, \mathrm{~m}$<br/>n. Thus, both statements are correct.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-every-function-has-a-primitive-2-a-primitive-of-a-function-is-unique-which-of-the-statements-given-above-isare-cor

slug:
consider-the-following-statements-1-every-function-has-a-primitive-2-a-primitive-of-a-function-is-unique-which-of-the-statements-given-above-isare-cor-68113

question:
<div class="question">Consider the following statements<br/>1- Every function has a primitive.<br/>2- A primitive of a function is unique. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">We know every function does not has a primitive but a primitive of a function is unique.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-every-zero-matrix-is-a-square-matrix-2-a-matrix-has-a-numerical-value-3-a-unit-matrix-is-a-diagonal-matrix-which-o

slug:
consider-the-following-statements-1-every-zero-matrix-is-a-square-matrix-2-a-matrix-has-a-numerical-value-3-a-unit-matrix-is-a-diagonal-matrix-which-o-78418

question:
<div class="question">Consider the following statements: <br/>$1.$ every zero matrix is a square matrix.<br/>$2.$ A matrix has a numerical value.<br/>$3.$ A unit matrix is a diagonal matrix.<br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3 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">C</span> <span class="option-data">2 and 3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1 and 3</span> </li> </ul>

answer:
<div class="solution">Only statement $-3$ is correct<br/>$\text { Unit matrix }=\left[\begin{array}{ccc}<br/>1 &amp; 0 &amp; 0 \\<br/>0 &amp; 1 &amp; 0 \\<br/>0 &amp; 0 &amp; 1<br/>\end{array}\right]$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-f-2-0-does-not-exist-2-f-2-0-does-not-exist-let-f-x-be-a-function-defined-in-1-x-by-2014-i-f-x-2-x-3-x-x-2-for-1-x

slug:
consider-the-following-statements-1-f-2-0-does-not-exist-2-f-2-0-does-not-exist-let-f-x-be-a-function-defined-in-1-x-by-2014-i-f-x-2-x-3-x-x-2-for-1-x-34392

question:
<div class="question">Consider the following statements :<br/>1- $f^{\prime}(2+0)$ does not exist.<br/>2- $f^{\prime}(2-0)$ does not exist.<br/>Let $f(x)$ be a function defined in $1 \leq x   &lt; \infty$ by $\quad[2014-I]$<br/>$f(x)=\left\{\begin{array}{cc}2-x &amp; \text { for } 1 \leq x \leq 2 \\ 3 x-x^{2} &amp; \text { for } x&gt;2\end{array}\right.$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\mathrm{f}^{\prime}(2+0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}^{\prime}(2+\mathrm{h})$<br/>$=\lim _{\mathrm{h} \rightarrow 0} 3-4-2 \mathrm{~h}=-1$<br/>$\mathrm{f}^{\prime}(2-0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}^{\prime}(2-\mathrm{h})=(3-4-2 \mathrm{~h})=-1$<br/>So, $\mathrm{f}^{\prime}(\mathrm{x})$ exist at $\mathrm{x}=2$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-f-x-x-where-is-the-greatest-integer-function-is-discontinuous-at-x-n-where-n-z-2-f-x-cot-x-is-discontinuous-at-x-n

slug:
consider-the-following-statements-1-f-x-x-where-is-the-greatest-integer-function-is-discontinuous-at-x-n-where-n-z-2-f-x-cot-x-is-discontinuous-at-x-n-42302

question:
<div class="question">Consider the following statements:<br/>1- $f(x)=[x]$, where $[.]$ is the greatest integer function, is discontinuous at $x=n$, where $n \in Z$.<br/>2- $f(x)=\cot x$ is discontinuous at $x=n \pi$, where $n \in Z$.<br/>which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither 1 or 2</span> </li> </ul>

answer:
<div class="solution">From statement-1 From the definition of greatest integer function $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$ is discontinuous at $\mathrm{x}=\mathrm{n}$ for any value of $\mathrm{n} \in \mathrm{Z}$<br/>$\therefore \quad$ Statement 1 is correct From statement- 2 $f(x)=\cot x$<br/>for $x=\pi=\cot \pi=-\infty$<br/>for $\mathrm{x}=2 \pi$<br/>$f(2 \pi)=\cot 2 \pi=-\infty$<br/>Hence the function $\mathrm{f}(\mathrm{x})=\cot \mathrm{x}$ is discontinuous at $\mathrm{x}=\mathrm{n} \pi$ where $\mathrm{n} \in \mathrm{Z}$.<br/>$\therefore$ Option (c) is correct.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-for-any-three-vectors-a-b-c-a-b-c-a-b-c-0-2-for-any-three-coplanar-unit-vectors-d-e-f-d-e-f-1-which-of-the-stateme

slug:
consider-the-following-statements-1-for-any-three-vectors-a-b-c-a-b-c-a-b-c-0-2-for-any-three-coplanar-unit-vectors-d-e-f-d-e-f-1-which-of-the-stateme-45960

question:
<div class="question">Consider the following statements<br/>$1.$ For any three vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$;<br/>$\overrightarrow{\mathbf{a}} \cdot\{(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})\}=0$<br/>$2.$ For any three coplanar unit vectors<br/>$\overrightarrow{\mathbf{d}}, \overrightarrow{\mathbf{e}}, \overrightarrow{\mathbf{f}} ;(\overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{e}}) \cdot \overrightarrow{\mathbf{f}}=1$<br/>Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Consider statement 1<br/>$\vec{a} \cdot\{(\vec{b}+\vec{c}) \times(\vec{a}+\vec{b}+\vec{c})\}=0$<br/>$=\vec{a} \cdot\{\vec{b} \times \vec{a}+\vec{b} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}+\vec{c} \times \vec{b}+\vec{c} \times \vec{c}\}$<br/>$=0+0+\vec{a} \cdot(\vec{b} \times \vec{c})+0+\vec{a} \cdot(\vec{c} \times \vec{b})+0 \quad(\because \vec{a} \times \vec{a}=0)$<br/>$=\vec{a} \cdot(\vec{b} \times \vec{c})-\vec{a} \cdot(\vec{b} \times \vec{c})=0$<br/>and for any three coplanar vectors $\overrightarrow{\mathrm{d}}, \overrightarrow{\mathrm{e}}, \overrightarrow{\mathrm{f}}$<br/>$(\vec{d} \times \vec{e}) \cdot \vec{f}=0$<br/>Hence, statement (1) is correct and statement-2 is incorrect.</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-3-is-an-acute-angle-such-that-sin-3-cos-2-then-the-mesurement-of-in-radian-equals-to-10-2-one-radian-is-the-ang

slug:
consider-the-following-statements-1-if-3-is-an-acute-angle-such-that-sin-3-cos-2-then-the-mesurement-of-in-radian-equals-to-10-2-one-radian-is-the-ang-70798

question:
<div class="question">Consider the following statements:<br/>1. If $3 \theta$ is an acute angle such that $\sin 3 \theta=\cos 2 \theta$, then<br/>the mesurement of $\theta$ in radian equals to $\frac{\pi}{10}$.<br/>2 One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement: 1 $\sin 3 \theta=\cos 2 \theta$<br/>$\sin 3 \theta=\sin \left(\frac{\pi}{2}-2 \theta\right)$<br/>$3 \theta=\frac{\pi}{2}-2 \theta$<br/>$5 \theta=\frac{\pi}{2} \Rightarrow \theta=\frac{\pi}{10}$<br/>Statement : 2 One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to radius of that circle. Hence, statement 1 is correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-10-is-added-to-each-entry-on-a-list-then-the-average-increase-by-10-2-if-10-is-added-to-each-entry-on-a-list-th

slug:
consider-the-following-statements-1-if-10-is-added-to-each-entry-on-a-list-then-the-average-increase-by-10-2-if-10-is-added-to-each-entry-on-a-list-th-41481

question:
<div class="question">Consider the following statements: $\quad$<br/> 1 . If 10 is added to each entry on a list then the average increase by $10 .$<br/> 2 . If 10 is added to each entry on a list, then the standard deviation increase by 10 .<br/> 3 . If each entry on a list is doubled, then the average doubles<br/>Which of the above statement are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,2 and 3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1 and 3 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">D</span> <span class="option-data">2 and 3 only</span> </li> </ul>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-a-a-then-a-is-a-singular-matrix-where-a-is-the-transpose-of-a-2-if-a-is-a-square-matrix-such-that-a-3-i-then-a-

slug:
consider-the-following-statements-1-if-a-a-then-a-is-a-singular-matrix-where-a-is-the-transpose-of-a-2-if-a-is-a-square-matrix-such-that-a-3-i-then-a-57236

question:
<div class="question">Consider the following statements<br/>$1.$ If $\mathrm{A}^{\prime}=\mathrm{A}$; then $A$ is a singular matrix, where $\mathrm{A}^{\prime}$ is the transpose of $A$.<br/>$2.$ If $A$ is a square matrix such that $A^{3}=I$, then $A$ is non-singular. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">If $A^{\prime}=\mathrm{A}$ where $A$ ' is transpose of matrix then $|A|=\left|A^{\prime}\right|$<br/>But it is not neccessary that $|A|=0$<br/>i.e. $A$ is singular matrix Hence, statement 1 is wrong. Given $A^{3}=I$<br/>$\left|A^{3}\right|=|I|=1$<br/>$\Rightarrow|A|=1$<br/>Thus, $A$ is non-singular matrix. Hence, only statement 2 is correct.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-a-and-b-are-exhaustive-events-then-their-union-is-the-sample-space-2-if-a-and-b-are-exhaustive-events-then-thei

slug:
consider-the-following-statements-1-if-a-and-b-are-exhaustive-events-then-their-union-is-the-sample-space-2-if-a-and-b-are-exhaustive-events-then-thei-70149

question:
<div class="question">Consider the following statements :<br/>$1$ If $A$ and $B$ are exhaustive events, then their union is the sample space.<br/>$2$ If $A$ and $B$ are exhaustive events, then their intersection must be an empty event. Which of the above statements is/are correct ?</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-det-a-0-then-det-adj-a-0-2-if-a-is-non-singular-then-det-a-1-det-a-1

slug:
consider-the-following-statements-1-if-det-a-0-then-det-adj-a-0-2-if-a-is-non-singular-then-det-a-1-det-a-1-25632

question:
<div class="question">Consider the following statements:<br/>$1.$ If det $\mathrm{A}=0$, then $\operatorname{det}(\operatorname{adj} \mathrm{A})=0$<br/>$2 .$ If A is non- singular, then $\operatorname{det}\left(\mathrm{A}^{-1}\right)=(\operatorname{det} \mathrm{A})^{-1}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither l nor 2</span> </li> </ul>

answer:
<div class="solution">We know that, adj A and A has same value of determinant, if det $\mathrm{A}=0$, then $\operatorname{det}(\operatorname{adj} \mathrm{A})=0$<br/>so, statement (1) is correct. Also If A is a matrix the determinant of $\mathrm{A}^{-1}$ equals inverse of determinant $\mathrm{A}$, so, $\operatorname{det}\left(\mathrm{A}^{-1}\right)=(\operatorname{det} \mathrm{A})^{-1}$, if $\mathrm{A}$<br/>is non singular; Statement 2 is correct. <br/>Thus both (1) and (2) are correct.</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-f-x-x-3-and-g-y-y-3-then-f-g-2-identity-function-is-not-always-a-bijection-which-of-the-above-statements-isare-

slug:
consider-the-following-statements-1-if-f-x-x-3-and-g-y-y-3-then-f-g-2-identity-function-is-not-always-a-bijection-which-of-the-above-statements-isare-71714

question:
<div class="question">Consider the following statements:<br/>1- If $f(x)=x^{3}$ and $g(y)=y^{3}$ then $f=g$.<br/>2- Identity function is not always a bijection. Which of the above statements is/are correct?</div>

answer:
<div class="solution">(1) $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}$ and $\mathrm{g}(\mathrm{y})=\mathrm{y}^{3}$.<br/>$\Rightarrow \mathrm{f}=\mathrm{g}$ is a correct statement.<br/>(2) Identify function is always a bijection.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-the-correlation-coefficient-r-xy-0-then-the-two-line-of-regression-are-parallel-to-each-oth-2-if-the-correlatio

slug:
consider-the-following-statements-1-if-the-correlation-coefficient-r-xy-0-then-the-two-line-of-regression-are-parallel-to-each-oth-2-if-the-correlatio-83305

question:
<div class="question">Consider the following statements:<br/> 1 . If the correlation coefficient $\mathrm{r}_{\mathrm{xy}}=0$, then the two line:<br/>of regression are parallel to each oth<br/> 2 . If the correlation coefficien A then the two lines of regression are perpendicular to each other Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 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> </ul>

answer:
<div class="solution">We know, $\tan \theta=\left|\frac{1-\mathrm{r}_{\mathrm{xy}}^{22}}{\mathrm{r}_{\mathrm{xy}}}\right|\left(\frac{\sigma_{\mathrm{x}} \cdot \sigma_{\mathrm{y}}}{\sigma_{\mathrm{x}}^{2} \cdot \sigma_{\mathrm{y}}^{2}}\right)$<br/>then $\tan \theta=\infty$ and lines are perpendicular.<br/>If $\mathrm{r}_{\mathrm{xy}}$ is 0, then $\tan \theta=\infty$ and lines are<br/>If $\mathrm{r}_{\mathrm{xy}}$ is 1 , then the lines are parallel.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-if-y-ln-sec-x-tan-x-then-d-x-d-y-sec-x-2-if-y-ln-cosec-x-cot-x-then-d-x-d-y-cosec-x-which-of-the-above-isare-corre

slug:
consider-the-following-statements-1-if-y-ln-sec-x-tan-x-then-d-x-d-y-sec-x-2-if-y-ln-cosec-x-cot-x-then-d-x-d-y-cosec-x-which-of-the-above-isare-corre-20634

question:
<div class="question">Consider the following statements:<br/>1 If $y=\ln (\sec x+\tan x)$, then $\frac{d y}{d x}=\sec x$.<br/>2 If $y=\ln (\operatorname{cosec} x-\cot x)$, then $\frac{d y}{d x}=\operatorname{cosec} x$.<br/>Which of the above is/are correct?</div>

answer:
<div class="solution">$(1) y=\ell n(\sec x+\tan x)$<br/>$\frac{d y}{d x}=\frac{\sec x \tan x+\sec ^{2} x}{\sec x+\tan x}$<br/>$\quad=\frac{\sec x(\sec x+\tan x)}{\sec x+\tan x}=\sec x$<br/>$y=\ell n(\operatorname{cosec} x-\cot x)$<br/>$\frac{d y}{d x}=\frac{\operatorname{cosec} x(\operatorname{cosec} x-\cot x)}{\operatorname{cosec} x-\cot x}=\operatorname{cosec} x$<br/>Hence, both statements are correct.</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-lim-x-0-sin-x-1-does-not-exist-2-lim-x-0-sin-x-1-exists-which-of-the-above-statements-correct

slug:
consider-the-following-statements-1-lim-x-0-sin-x-1-does-not-exist-2-lim-x-0-sin-x-1-exists-which-of-the-above-statements-correct-76871

question:
<div class="question">Consider the following statements:<br/>1- $\lim _{x \rightarrow 0} \sin \frac{1}{x}$ does not exist.<br/>2- $\lim _{x \rightarrow 0} \sin \frac{1}{x}$ exists<br/>Which of the above statements correct?</div>

answer:
<div class="solution">Since $\sin \frac{1}{x}$ is an oscillatory function<br/>$\therefore \quad \lim _{x \rightarrow 0} \sin \frac{1}{x}$ has a finite value between $-1$ and 1 .<br/>Now, At $x=0$<br/>L. H. $\mathrm{L}=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(0-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0}(-\mathrm{h}) \sin \left(-\frac{1}{\mathrm{~h}}\right)$<br/>$=0 \times$ a finite value between $-1$ and 1 $\quad=0$<br/>Similarly $\mathrm{R}$. H. $\mathrm{L}=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(0+\mathrm{h})=0$<br/>Also, $\mathrm{f}(0)=0$<br/>$\therefore \quad \lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \sin \frac{1}{\mathrm{x}}$ exists.<br/>Hence Both statements are correct.</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-lim-x-0-x-x-2-exists-2-x-x-2-is-not-continuous-at-x-0-3-lim-x-0-x-x-does-not-exist-which-of-the-statements-given-a

slug:
consider-the-following-statements-1-lim-x-0-x-x-2-exists-2-x-x-2-is-not-continuous-at-x-0-3-lim-x-0-x-x-does-not-exist-which-of-the-statements-given-a-26988

question:
<div class="question">Consider the following statements:<br/>1- $\lim _{x \rightarrow 0} \frac{x^{2}}{x}$ exists.<br/>2- $\left(\frac{x^{2}}{x}\right)$ is not continuous at $x=0$<br/>3- $\lim _{x \rightarrow 0} \frac{|x|}{x}$ does not exist.<br/>Which of the statements given above are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,2 and 3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2 and 3 only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1 and 3 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> </ul>

answer:
<div class="solution">1- $\lim _{x \rightarrow 0} \frac{x^{2}}{x}=\lim _{x \rightarrow 0}(x)=0$<br/>2- $\quad \frac{x^{2}}{x}=x$,<br/>since a polynomial is continuous everywhere, so it is continuous at $x=0$<br/>3- LHL $=\lim _{h \rightarrow 0} \frac{|0-h|}{(0-h)}=\lim _{h \rightarrow 0} \frac{h}{-h}=-1$<br/>RHL $=\lim _{h \rightarrow 0} \frac{|0+h|}{(0+h)}=\lim _{h \rightarrow 0} \frac{h}{h}=1$<br/>$\therefore L H L \neq \mathrm{RHL}$<br/>So, it does not exist.<br/>Thus, the statements 1 and 3 are correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-mean-is-independent-of-change-in-scale-and-change-in-origin-2-variance-is-independent-of-change-in-scale-but-not-i

slug:
consider-the-following-statements-1-mean-is-independent-of-change-in-scale-and-change-in-origin-2-variance-is-independent-of-change-in-scale-but-not-i-18294

question:
<div class="question">Consider the following statements:<br/> 1 . Mean is independent of change in scale and change in<br/>origin.<br/> 2 . Variance is independent of change in scale but not in origin. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither $1 \mathrm{n}$</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">Neither $1 \mathrm{n}$</span> </div>

answer:
<div class="solution">1 . Mean is dependent with change in origin. 2 . Variance is independent with change in origin.</div>


tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-n-sin-2-67-2-1-sin-2-22-2-1-1-for-all-positive-integers-n-2-2-if-x-is-any-positive-real-number-then-n-x-1-for-all-

slug:
consider-the-following-statements-1-n-sin-2-67-2-1-sin-2-22-2-1-1-for-all-positive-integers-n-2-2-if-x-is-any-positive-real-number-then-n-x-1-for-all-29530

question:
<div class="question">Consider the following statements:<br/>1. $n\left(\sin ^{2} 67 \frac{1^{\circ}}{2}-\sin ^{2} 22 \frac{1^{\circ}}{2}\right)&gt;1$ for all positive integers<br/>$n \geq 2 .$<br/>2 If $x$ is any positive real number, then $n x&gt;1$ for all positive integers $n \geq 2$. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 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">2 only $y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">Statement 1:<br/>Given $\mathrm{n}\left(\sin ^{2} 67 \frac{1^{\circ}}{2}-\sin ^{2} 22 \frac{1^{\circ}}{2}\right)$<br/>or $\mathrm{n}\left(\sin ^{2} \frac{135^{\circ}}{2}-\sin ^{2} \frac{45^{\circ}}{2}\right)$<br/>$=\mathrm{n}\left(\sin \frac{135^{\circ}}{2}+\sin \frac{45^{\circ}}{2}\right)\left(\sin \frac{135^{\circ}}{2}-\sin \frac{45^{\circ}}{2}\right)$<br/>$=n\left[2 \sin \frac{\left(\frac{135^{\circ}}{2}+\frac{45^{\circ}}{2}\right)}{2} \cdot \cos \frac{\left(\frac{135^{\circ}}{2}-\frac{45^{\circ}}{2}\right)}{2}\right]$<br/>$\left[2 \cdot \cos \frac{\left(\frac{135^{\circ}+45^{\circ}}{2}\right)}{2} \cdot \sin \frac{\left(\frac{135^{\circ}-45^{\circ}}{2}\right)}{2}\right]$<br/>$=\mathrm{n}\left[2 \cdot \sin \left(\frac{90^{\circ}}{2}\right) \cdot \cos \left(\frac{45^{\circ}}{2}\right)\right]$<br/>$\left[2 \cdot \cos \left(\frac{90^{\circ}}{2}\right) \cdot \sin \left(\frac{45^{\circ}}{2}\right)\right]$<br/>$=2 n\left(2 \sin \frac{45^{\circ}}{2} \cdot \cos \frac{45^{\circ}}{2}\right)\left(\sin 45^{\circ} \cdot \cos 45^{\circ}\right)$<br/>$=2 \mathrm{n} \cdot \sin \left(2 \times \frac{45^{\circ}}{2}\right)\left(\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}\right)$<br/>$=2 \mathrm{n} \cdot \sin 45 \times \frac{1}{2}=\mathrm{n} \cdot \frac{1}{\sqrt{2}}=\frac{\mathrm{n}}{\sqrt{2}}$<br/>$\therefore \frac{\mathrm{n}}{\sqrt{2}}&gt;1$ for all positive integers $\mathrm{n} \geq 2$. $\therefore$ Statement 1 is true Statement 2 $n x&gt;1, \forall n \geq 2$<br/>$\Rightarrow \mathrm{n}&gt;\frac{1}{\mathrm{x}}, \forall \mathrm{n} \geq 2$<br/>$\mathrm{x} \in(0, \infty)$, then we take $\mathrm{x}=1$<br/>$\mathrm{n}&gt;1$, but $\mathrm{n}$ is always greater or equal to 2 for all $\mathrm{x}$ positive real number. $\therefore$ Statement 2 is false.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
answer: <div class="solution">The statements (1) and (3) are true.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-p-a-b-p-a-p-b-p-a-b-2-p-a-b-p-b-p-a-b-3-p-a-b-p-b-p-a-b-which-of-the-above-statements-are-correct

slug:
consider-the-following-statements-1-p-a-b-p-a-p-b-p-a-b-2-p-a-b-p-b-p-a-b-3-p-a-b-p-b-p-a-b-which-of-the-above-statements-are-correct-82037

question:
<div class="question">Consider the following statements:<br/>$1.$ $\quad \mathrm{P}(\overline{\mathrm{A}} \cup \mathrm{B})=\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})$<br/>$2.$ $\quad \mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$3.$ $\quad \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{B}) \mathrm{P}(\mathrm{A} \mid \mathrm{B})$<br/>Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1 and 3 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">C</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-pie-diagrams-are-suitable-for-categorical-data-2-the-arc-length-of-a-sector-of-a-pie-diagram-proportional-to-the-v

slug:
consider-the-following-statements-1-pie-diagrams-are-suitable-for-categorical-data-2-the-arc-length-of-a-sector-of-a-pie-diagram-proportional-to-the-v-47329

question:
<div class="question">Consider the following statements<br/> 1 . Pie diagrams are suitable for categorical data<br/> 2 . The arc length of a sector of a pie diagram proportional to the value of the component representec by the sector<br/>Which of the statements given above is/are correct?</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-sin-x-cos-x-is-always-positive-2-sin-x-2-cos-x-2-is-always-positive-which-of-the-above-statements-isare-correct

slug:
consider-the-following-statements-1-sin-x-cos-x-is-always-positive-2-sin-x-2-cos-x-2-is-always-positive-which-of-the-above-statements-isare-correct-93957

question:
<div class="question">Consider the following statements:<br/>1. $\sin |x|+\cos |x|$ is always positive.<br/>2. $\sin \left(x^{2}\right)+\cos \left(x^{2}\right)$ is always positive.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement 1 : $\mathrm{f}_{1}(\mathrm{x})=\sin |\mathrm{x}|+\cos |\mathrm{x}|$, the value of $|\sin \mathrm{x}|$<br/>and $|\cos x|$ depends on its angles. $\sin |x|+\cos |x|$ is not always positive. Statement $2: \mathrm{f}_{2}(\mathrm{x})=\sin \left(\mathrm{x}^{2}\right)+\cos \left(\mathrm{x}^{2}\right)$, the value of $\mathrm{x}^{2}$<br/>between any value which lies in the interval $\left(\pi, \frac{3 \pi}{2}\right)$, then value of $\mathrm{f}_{2}(\mathrm{x})=\sin \left(\mathrm{x}^{2}\right)+\cos \left(\mathrm{x}^{2}\right)$ is always<br/>negative.</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-tan-1-1-tan-1-05-2-2-sin-1-13-cos-1-13-2-which-of-the-above-statements-isare-correct

slug:
consider-the-following-statements-1-tan-1-1-tan-1-05-2-2-sin-1-13-cos-1-13-2-which-of-the-above-statements-isare-correct-25360

question:
<div class="question">Consider the following statements: <br/>1) $\tan ^{-1} 1+\tan ^{-1}(0.5)=\pi / 2$<br/>2) $\sin ^{-1}(1 / 3)+\cos ^{-1}(1 / 3)=\pi / 2$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">1) L.H.S. $\begin{aligned} \tan ^{-1}(1)+\tan ^{-1}\left(\frac{1}{2}\right) \\=\tan ^{-1}(1)+\cot ^{-1}\left(\frac{1}{\frac{1}{2}}\right) &amp;=\tan ^{-1}+\cot ^{-1} 2 \neq \frac{\pi}{2} \\ \text { So, L.H.S. } \neq \mathrm{RHS} \text { . } \\ \text { 2) } \sin ^{-1} \frac{1}{3}+\cos ^{-1}\left(\frac{1}{3}\right)=\frac{\pi}{2} \\\left\{\sin ^{-1} \mathrm{x}+\cos ^{-1} \mathrm{x}=\frac{\pi}{2}\right\} \end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
answer: <div class="solution">Only the statement 1 is correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-algebraic-sum-of-deviations-of-a-set-of-values-from-their-arithmetic-mean-is-always-zero-2-arithmetic-mean-med

slug:
consider-the-following-statements-1-the-algebraic-sum-of-deviations-of-a-set-of-values-from-their-arithmetic-mean-is-always-zero-2-arithmetic-mean-med-39603

question:
<div class="question">Consider the following statements:<br/> 1 . The algebraic sum of deviations of a set of values from their arithmetic mean is always zero.<br/> 2 . Arithmetic mean $&gt;$ Median $&gt;$ Mode for a symmetric distribution. Which of the above statements is/are correct?</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-derivative-where-the-function-attains-maxima-or-minima-be-zero-2-if-a-function-is-differentiable-at-a-point-th-1

slug:
consider-the-following-statements-1-the-derivative-where-the-function-attains-maxima-or-minima-be-zero-2-if-a-function-is-differentiable-at-a-point-th-1-46023

question:
<div class="question">Consider the following statements:<br/>1- The derivative where the function attains maxima or minima be zero.<br/>2- If a function is differentiable at a point, then it must be continuous at that point. Which of the above statements is/are correct?</div>

answer:
<div class="solution">$1.$ Not necessarily true<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/hs8jt5_W3QWNlwy4QtvYQ7KiPXuyNhY6IQVC7pcLwDM.original.fullsize.png"/><br/><br/>$2.$ True $(\because$ Differentiability $\Rightarrow$ Continuity)<br/>#</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-derivative-where-the-function-attains-maxima-or-minima-be-zero-2-if-a-function-is-differentiable-at-a-point-th

slug:
consider-the-following-statements-1-the-derivative-where-the-function-attains-maxima-or-minima-be-zero-2-if-a-function-is-differentiable-at-a-point-th-15310

question:
<div class="question">Consider the following statements:<br/>1- The derivative where the function attains maxima or minima be zero.<br/>2- If a function is differentiable at a point, then it must be continuous at that point. Which of the above statements is/are correct?</div>

answer:
<div class="solution">$1.$ Not necessarily true<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/hs8jt5_W3QWNlwy4QtvYQ7KiPXuyNhY6IQVC7pcLwDM.original.fullsize.png"/><br/><br/>$2.$ True $(\because$ Differentiability $\Rightarrow$ Continuity)<br/>#</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">Using distance between two parallel lines formula.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-distance-between-the-lines-y-mx-c-1-and-y-mx-c-2-is-1-m-2-c-1-c-2-2-the-distance-between-the-lines-ax-by-c-1-a

slug:
consider-the-following-statements-1-the-distance-between-the-lines-y-mx-c-1-and-y-mx-c-2-is-1-m-2-c-1-c-2-2-the-distance-between-the-lines-ax-by-c-1-a-96237

question:
<div class="question">Consider the following statements: <br/>1) The distance between the lines<br/>$\mathrm{y}=\mathrm{mx}+\mathrm{c}_{1}$ and $\mathrm{y}=\mathrm{mx}+\mathrm{c}_{2}$ is $\frac{\left|\mathrm{c}_{1}-\mathrm{c}_{2}\right|}{\sqrt{1-\mathrm{m}^{2}}}$<br/>2) The distance between the lines $\mathrm{ax}+\mathrm{by}+\mathrm{c}_{1}$ and $\mathrm{ax}+\mathrm{by}+\mathrm{c}_{2}=0$ is $\frac{\left|\mathrm{c}_{1}-\mathrm{c}_{2}\right|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$<br/>3) The distance between the lines $\mathrm{x}=\mathrm{c}$ and $\mathrm{x}=\mathrm{c}_{2}$ is $\left|\mathrm{c}_{1}-\mathrm{c}_{2}\right|$. Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 and 3 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">C</span> <span class="option-data">1 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 3 only</span> </div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-3-x-is-continuous-at-all-x-except-at-x-0-2-the-function-f-x-x-is-continuous-at-x-299-where-is-the

slug:
consider-the-following-statements-1-the-function-f-x-3-x-is-continuous-at-all-x-except-at-x-0-2-the-function-f-x-x-is-continuous-at-x-299-where-is-the-60383

question:
<div class="question">Consider the following statements:<br/>1- The function $f(x)=\sqrt[3]{x}$ is continuous at all $x$ except at $x=0$.<br/>2- The function $f(x)=[x]$ is continuous at $x=2.99$ where [.]is the bracket function. Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\begin{aligned} &amp; \text { LHLf }(2.99-0)=\lim _{h \rightarrow 0}(2.99-h) \\ &amp; \lim _{h \rightarrow 0}(2.99-h)=\lim _{h \rightarrow 0} 2=2 \\ &amp; \text { RHL } f(2.99-0)=\lim _{h \rightarrow 0} f(2.99+h) \\ &amp;=\lim _{h \rightarrow 0}(2.99+h)=\lim _{h \rightarrow 0} 2=2 \\ &amp; \text { LHL }=\text { RHL } \end{aligned}$<br/>$\therefore \mathrm{f}(\mathrm{x})$ is continous at $\mathrm{x}=2.99$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-greatest-integer-x-x-r-is-a-continuous-function-2-all-trigonometric-functions-are-continuous-on-r

slug:
consider-the-following-statements-1-the-function-f-x-greatest-integer-x-x-r-is-a-continuous-function-2-all-trigonometric-functions-are-continuous-on-r-14485

question:
<div class="question">Consider the following statements:<br/>1- The function $\mathrm{f}(\mathrm{x})=$ greatest integer $\leq \mathrm{x}, \mathrm{x} \in \mathrm{R}$ is a continuous function.<br/>2- All trigonometric functions are continuous on $\mathrm{R}$. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Here, greatest integer function [x] is discontinuous at its integral value of $\mathrm{x}, \cot \mathrm{x}$ and $\operatorname{cosec} \mathrm{x}$ are discontinuous at $0, \pi, 2 \pi$ etc. and $\tan \mathrm{x}$ and sec $\mathrm{x}$ are discontinuous at<br/>$\mathrm{x}=\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}$ etc. Therefore the greatest integer function and all trigonometric functions are not continuous for $\mathrm{x} \in \mathrm{R}$ Therefore, neither (1) nor (2) are true.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-is-continuous-at-x-0-2-the-function-f-x-is-continuous-at-x-2-which-of-the-above-statements-isare-

slug:
consider-the-following-statements-1-the-function-f-x-is-continuous-at-x-0-2-the-function-f-x-is-continuous-at-x-2-which-of-the-above-statements-isare-36067

question:
<div class="question">Consider the following statements:<br/>1- The function $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$.<br/>2- The function $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=\frac{\pi}{2}$. Which of the above statements is/are correct?<br/>A function $\mathrm{f}(\mathrm{x})$ is defined as follows: $\quad[2016-\mathrm{I}]$ $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{lll}\mathrm{x}+\pi &amp; \text { for } &amp; \mathrm{x} \in[-\pi, 0) \\ \pi \cos \mathrm{x} &amp; \text { for } &amp; \mathrm{x} \in\left[0, \frac{\pi}{2}\right] \\ \left(\mathrm{x}-\frac{\pi}{2}\right)^{2} &amp; \text { for } &amp; \mathrm{x} \in\left(\frac{\pi}{2}, \pi\right]\end{array}\right.$</div>

answer:
<div class="solution">Given<br/>$f(x)=\left\{\begin{array}{cc}(x+\pi) &amp; \text { for } x \in[-\pi, 0) \\ \pi \cos x &amp; \text { for } x \in\left[0, \frac{\pi}{2}\right] \\ \left(x-\frac{\pi}{2}\right)^{2} &amp; \text { for } x \in\left(\frac{\pi}{2}, \pi\right]\end{array}\right.$<br/>For continuity, $f(a)=$ L.H.L. $=$ R.H.L.<br/>At $x=0$<br/>$f(0)=\pi \cos 0=\pi$<br/>L.H.L. $=\lim _{x \rightarrow 0^{-}} f(x-h)$<br/>$=\lim _{h \rightarrow 0} f(-h)$<br/>$=\lim _{h \rightarrow 0}(-h+\pi)=\pi$<br/>R.H.L. $=\lim _{x \rightarrow 0^{+}} f(x+h)$<br/>$=\lim _{h \rightarrow 0} f(0+h)$<br/>$=\lim _{h \rightarrow 0} \pi \cos h$<br/>$\quad=\pi \cos 0=\pi$<br/>$f(0)=$ L.H.L. $=$ R.H.L.<br/>Hence function is continuous at $x=0$. <br/>Statement (1) is correct.<br/>At $x=\frac{\pi}{2}$<br/>$\begin{aligned} \text { L.H.L. } &amp;=\lim _{x \rightarrow \frac{\pi^{-}}{2}} f(x-h) \\ &amp;=\lim _{h \rightarrow 0} f\left(\frac{\pi}{2}-h\right) \end{aligned}$<br/>$$<br/>\begin{array}{l}<br/>=\lim _{h \rightarrow 0} \pi \cos \left(\frac{\pi}{2}-h\right) \\<br/>=\pi \cos \frac{\pi}{2}=0 \\<br/>\mathrm{RH} . \mathrm{L} .=\lim _{x \rightarrow \frac{\pi^{+}}{2}} f(x+h) \\<br/>=\lim _{h \rightarrow 0} f\left(\frac{\pi}{2}+h\right) \\<br/>=\lim _{h \rightarrow 0}\left(\frac{\pi}{2}+h-\frac{\pi}{2}\right)^{2}=0 \\<br/>f\left(\frac{\pi}{2}\right)=\pi \cos \frac{\pi}{2}=0 \\<br/>\text { L.H.L. }=\mathrm{R} . \mathrm{H.L.}=f\left(\frac{\pi}{2}\right)<br/>\end{array}<br/>$$<br/>Hence function is continuous at $x=\frac{\pi}{2}$<br/>Statement $(2)$ is correct.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-is-differentiable-at-x-0-2-the-function-f-x-is-differentiable-at-x-2-which-of-the-above-statement

slug:
consider-the-following-statements-1-the-function-f-x-is-differentiable-at-x-0-2-the-function-f-x-is-differentiable-at-x-2-which-of-the-above-statement-91885

question:
<div class="question">Consider the following statements:<br/>1- The function $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$.<br/>2- The function $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=\frac{\pi}{2}$. Which of the above statements is/are correct?<br/>A function $\mathrm{f}(\mathrm{x})$ is defined as follows: $\quad[2016-\mathrm{I}]$ $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{lll}\mathrm{x}+\pi &amp; \text { for } &amp; \mathrm{x} \in[-\pi, 0) \\ \pi \cos \mathrm{x} &amp; \text { for } &amp; \mathrm{x} \in\left[0, \frac{\pi}{2}\right] \\ \left(\mathrm{x}-\frac{\pi}{2}\right)^{2} &amp; \text { for } &amp; \mathrm{x} \in\left(\frac{\pi}{2}, \pi\right]\end{array}\right.$</div>

answer:
<div class="solution">For differentiability,<br/>L.H.D. = R.H.D. Thus at $x=0$<br/>$\begin{aligned} \text { L.H.D. } &amp;=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h} \\ &amp;=\lim _{h \rightarrow 0} \frac{f(-h)-f(0)}{-h} \\ &amp;=\lim _{h \rightarrow 0} \frac{(-h+\pi)-(\pi \cos 0)}{-h}=1 \\ \text { L.H.D. } &amp;=1 \\ \text { R.H.D. } &amp;=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\pi \cos h-\pi \cos 0}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\pi[\cos h-1]}{h} \\ &amp;=\lim _{h \rightarrow 0} \frac{\pi\left[1-\frac{h^{2}}{2 !}+\frac{h^{4}}{4 !} \ldots . .-1\right]}{h} <br/>\\=\lim _{h \rightarrow 0} \frac{\pi\left[1-\frac{1}{2} h^{2}+\frac{1}{24} h^{4} \ldots \ldots-1\right]}{h} \end{aligned}$<br/>$$<br/>=\lim _{h \rightarrow 0} \frac{\pi\left[-\frac{1}{2} h^{2}+\frac{1}{24} h^{4} \ldots \ldots\right]}{h}=0<br/>$$<br/>L.H.D. $\neq$ R.H.D.<br/>So at $x=0$ function is not differentiable.<br/>Statement (1) is not correct.<br/>$$<br/>\begin{array}{l}<br/>\text { At } x=\frac{\pi}{2} \\<br/>f\left(\frac{\pi}{2}\right)=\pi \cos \frac{\pi}{2}=0 \\<br/>\text { R.H.D. }=\lim _{h \rightarrow 0} \frac{f\left(\frac{\pi}{2}+h\right)-f\left(\frac{\pi}{2}\right)}{-h} \\<br/>\text { L.H.D. }=\lim _{h \rightarrow 0} \frac{f\left(\frac{\pi}{2}-h\right)-f\left(\frac{\pi}{2}\right)}{-h}<br/>\end{array}<br/>$$<br/>$$<br/>\begin{array}{l}<br/>=\lim _{h \rightarrow 0} \frac{\pi \cos \left(\frac{\pi}{2}-h\right)-0}{-h} \\<br/>=\lim _{h \rightarrow 0} \pi\left(\frac{\sin h}{-h}\right) \\<br/>=-\pi \lim _{h \rightarrow 0}\left(\frac{\sin h}{h}\right)=-\pi(1)=-\pi \\<br/>\text { L.H.D. } \neq \text { RH.D. }<br/>\end{array}<br/>$$<br/>Hence function is not differentiable at $x=\frac{\pi}{2}$. Statement $(2)$ is not correct.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-x-2-2-cos-x-is-increasing-in-the-interval-0-2-the-function-f-x-ln-1-x-2-x-is-decreasing-in-the-in

slug:
consider-the-following-statements-1-the-function-f-x-x-2-2-cos-x-is-increasing-in-the-interval-0-2-the-function-f-x-ln-1-x-2-x-is-decreasing-in-the-in-49566

question:
<div class="question">Consider the following statements:<br/>1- The function $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+2 \cos \mathrm{x}$ is increasing in the interval $(0, \pi)$<br/>2- The function $\mathrm{f}(\mathrm{x})=\ln \left(\sqrt{1+\mathrm{x}^{2}}-\mathrm{x}\right)$ is decreasing in<br/>the interval $(-\infty, \infty)$ Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+2 \cos \mathrm{x}$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}-2 \sin \mathrm{x}$<br/>$=2[\mathrm{x}-\sin \mathrm{x}]$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/jUT59flOvDEeVcxUd42lFMSR6S1DT6OVfiHkP5xyLF8.original.fullsize.png"/><br/><br/>Between $(0, \pi) ;(\mathrm{x}-\sin \mathrm{x})$ is always $+\mathrm{ve}$, so $\mathrm{f}^{\prime}(\mathrm{x})$ is always $+\mathrm{ve}$.<br/>Hence it is increasing.<br/>$f(x)=\ln \left(\sqrt{1+x^{2}}-x\right)$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\sqrt{1+\mathrm{x}^{2}}-\mathrm{x}} \times\left[\frac{2 \mathrm{x}}{2 \sqrt{\left(1+\mathrm{x}^{2}\right)}}-1\right]$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\sqrt{1+\mathrm{x}^{2}}-\mathrm{x}} \times\left(\frac{\mathrm{x}}{\sqrt{1+\mathrm{x}^{2}}}-1\right)$<br/>As $\mathrm{f}^{\prime}(\mathrm{x})$ is $-\mathrm{ve}$ always, so this function is decreasing alwavs.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-x-is-not-differentiable-at-x-1-2-the-function-f-x-e-x-is-not-differentiable-at-x-0-which-of-the-a-1

slug:
consider-the-following-statements-1-the-function-f-x-x-is-not-differentiable-at-x-1-2-the-function-f-x-e-x-is-not-differentiable-at-x-0-which-of-the-a-1-94793

question:
<div class="question">Consider the following statements:<br/>1- The function $f(x)=|x|$ is not differentiable at $x=1$.<br/>2- The function $f(x)=e^{x}$ is not differentiable at $x=0$. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement $1: \mathrm{f}(\mathrm{x})=\mid \mathrm{x}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6zwX948yuEo8BJt0FIuIH00-ZVUd8BITvruOo8kSmMo.original.fullsize.png"/><br/><br/>From the graph, the curve has sharp turn at $\mathrm{x}=0$. Therefore, the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}|$ is not differentiable only $\mathrm{x}=0$, it is differentiable at $\mathrm{x}=1$ Statement $2: \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$<br/>$\mathrm{Rf}^{\prime}(0)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}}$<br/>$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{(0+\mathrm{h})}-\mathrm{e}^{\circ}}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-1}{\mathrm{~h}}$<br/>Use L'Hospital rule<br/>$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-0}{1}=\mathrm{e}^{\mathrm{o}}=1$<br/>$f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{1-\mathrm{e}^{-\mathrm{h}}}{\mathrm{h}}$<br/>Use L' Hospital rule<br/>$=\lim _{h \rightarrow 0} \frac{\mathrm{e}^{-\mathrm{h}}}{1}=\mathrm{e}^{-0}=1$<br/>Therefore $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$ is differentiable at $\mathrm{x}=1$.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-x-is-not-differentiable-at-x-1-2-the-function-f-x-e-x-is-not-differentiable-at-x-0-which-of-the-a

slug:
consider-the-following-statements-1-the-function-f-x-x-is-not-differentiable-at-x-1-2-the-function-f-x-e-x-is-not-differentiable-at-x-0-which-of-the-a-15760

question:
<div class="question">Consider the following statements:<br/>1- The function $f(x)=|x|$ is not differentiable at $x=1$.<br/>2- The function $f(x)=e^{x}$ is not differentiable at $x=0$. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement $1: \mathrm{f}(\mathrm{x})=\mid \mathrm{x}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6zwX948yuEo8BJt0FIuIH00-ZVUd8BITvruOo8kSmMo.original.fullsize.png"/><br/><br/>From the graph, the curve has sharp turn at $\mathrm{x}=0$. Therefore, the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}|$ is not differentiable only $\mathrm{x}=0$, it is differentiable at $\mathrm{x}=1$ Statement $2: \mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$<br/>$\mathrm{Rf}^{\prime}(0)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}}$<br/>$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{(0+\mathrm{h})}-\mathrm{e}^{\circ}}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-1}{\mathrm{~h}}$<br/>Use L'Hospital rule<br/>$=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{h}}-0}{1}=\mathrm{e}^{\mathrm{o}}=1$<br/>$f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{1-\mathrm{e}^{-\mathrm{h}}}{\mathrm{h}}$<br/>Use L' Hospital rule<br/>$=\lim _{h \rightarrow 0} \frac{\mathrm{e}^{-\mathrm{h}}}{1}=\mathrm{e}^{-0}=1$<br/>Therefore $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}$ is differentiable at $\mathrm{x}=1$.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-f-x-x-where-is-the-greatest-integer-function-defined-on-r-is-continuous-at-all-points-except-at-x-0-2

slug:
consider-the-following-statements-1-the-function-f-x-x-where-is-the-greatest-integer-function-defined-on-r-is-continuous-at-all-points-except-at-x-0-2-12884

question:
<div class="question">Consider the following statements:<br/>1- The function $f(x)=[x]$, where $[.]$ is the greatest integer function defined on $R$, is continuous at all points except at $x=0 . \quad[2014-I I]$<br/>2- The function $f(x)=\sin |x|$ is continuous for all $x \in R$.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">The greatest integer function is continuous at all statement points except integer. Hence, statement 1is incorrect. Statement $2:$ Let $h(a)=\sin x$ and $g(x)=|x|$<br/>$\operatorname{hog}(x)=\sin \mid x$<br/>$\Rightarrow \mathrm{f}(\mathrm{x})=\operatorname{hog}(\mathrm{x})=\sin |\mathrm{x}|$<br/>Therefore, $\mathrm{g}(\mathrm{x})$ is continuous, $\forall \mathrm{x} \in \mathrm{R}$ and $\mathrm{h}(\mathrm{x})$ is continous $\forall \mathrm{x} \in \mathrm{R}$<br/>When both are continuous then $\operatorname{hog}(\mathrm{x})$ is also continuous. Thus, statement 2 is correct.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-is-continuous-at-every-point-in-the-interval-1-2-the-function-is-differentiable-at-x-15-which-of-the-

slug:
consider-the-following-statements-1-the-function-is-continuous-at-every-point-in-the-interval-1-2-the-function-is-differentiable-at-x-15-which-of-the-50431

question:
<div class="question">Consider the following statements:<br/>1- The function is continuous at every point in the interval $(1, \infty)$<br/>2- The function is differentiable at $x=1.5$.<br/>Which of the above statements is/are correct?<br/>Let $f(x)$ be a function defined in $1 \leq x   &lt; \infty$ by $\quad[2014-I]$<br/>$f(x)=\left\{\begin{array}{cc}2-x &amp; \text { for } 1 \leq x \leq 2 \\ 3 x-x^{2} &amp; \text { for } x&gt;2\end{array}\right.$</div>

answer:
<div class="solution">Statement $1:$ Given $f(x)=\left\{\begin{array}{ccc}2-x &amp; \text { for } &amp; 1 \leq x \leq 2 \\ 3 x-x^{2} &amp; \text { for } &amp; x&gt;2\end{array}\right.$<br/>function defined in $1 \leq \mathrm{x}   &lt; \infty$ the function is polynomial, so it is continous and differentiable in its domain $[1, \infty]-\{2\}$<br/>LHLf $(2-0)=\lim _{h \rightarrow 0} f(2-h)$<br/>$=\lim _{h \rightarrow 0} \mathrm{~h}=0$<br/>$\operatorname{RHLf}(2+0)=\lim _{h \rightarrow 0}(2+h)$<br/>$=6-4=2$<br/>$\mathrm{f}(2)=2-2=0 \therefore \mathrm{LHL} \neq \mathrm{RHL}$<br/>Statement 2 :<br/>$\mathrm{Rf}^{\prime \prime}(1.5)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(1.5+\mathrm{h})-\mathrm{f}(1.5)}{\mathrm{h}}$<br/>$=\lim _{h \rightarrow 0} \frac{(2-1.5+h)-(2-1.5)}{h}=\lim _{h \rightarrow 0} \frac{-h}{h}=-1$<br/>$\mathrm{Lf}^{\prime}(1.5)=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(1.5-\mathrm{h})-\mathrm{f}(1.5)}{-\mathrm{h}}$<br/>$=\lim _{h \rightarrow 0} \frac{(2-1.5-h)-(2-1.5)}{-h}$<br/>$=\lim _{h \rightarrow 0} \frac{h}{-h}=-1$<br/>Therefore, the function is differentiable at $\mathrm{x}=1.5$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-function-is-discontinuous-at-x-3-2-the-function-is-not-differentiable-at-x-0-what-of-the-above-statements-isar

slug:
consider-the-following-statements-1-the-function-is-discontinuous-at-x-3-2-the-function-is-not-differentiable-at-x-0-what-of-the-above-statements-isar-51470

question:
<div class="question">Consider the following statements :<br/>1- The function is discontinuous at $x=3$.<br/>2- The function is not differentiable at $x=0$. What of the above statements is/are correct ?<br/>Consider the function $f(x)=\left\{\begin{array}{ll}x^{2}-5 &amp; x \leq 3 \\ \sqrt{x+13} &amp; x&gt;3\end{array} \quad[2014-I I]\right.$</div>

answer:
<div class="solution">$1.$ $\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)=f(3)$<br/>$\therefore \lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)=4$<br/>Therefore $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=4$<br/>$2.$ Given $f(x)=x^{2}-5 \forall x \leq 3$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}$<br/>$\mathrm{f}^{\prime}(0)=0$<br/>So, $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$ Therefore, neither statement 1 nor 2 is correct.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-length-p-of-the-perpendicular-from-the-origin-to-the-line-a-x-b-y-c-satisfies-the-relation-p-2-a-2-b-2-c-2-2-t

slug:
consider-the-following-statements-1-the-length-p-of-the-perpendicular-from-the-origin-to-the-line-a-x-b-y-c-satisfies-the-relation-p-2-a-2-b-2-c-2-2-t-87629

question:
<div class="question">Consider the following statements:<br/>1) The length $\mathrm{p}$ of the perpendicular from the origin to<br/>the line $a x+b y=c$ satisfies the relation $p^{2}=\frac{c^{2}}{a^{2}+b^{2}}$.<br/>2) The length $\mathrm{p}$ of the perpendicular from the origin to<br/>the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$.<br/>3) The length $\mathrm{p}$ of the perpendicular from the origin to the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1+\mathrm{m}^{2}+\mathrm{c}^{2}}{\mathrm{c}^{2}}$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,2 and 3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1 and 2 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">D</span> <span class="option-data">2 only</span> </li> </ul>

answer:
<div class="solution">1: We know, the perpendicular distance from<br/>$\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is $\frac{\mid \mathrm{ax}_{1}+\mathrm{by}_{1}+\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$<br/>Here, $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=(0,0)$ and distance $=\mathrm{P}$.<br/>$P=\frac{|a(0)+b(0)+(c)|}{\sqrt{a^{2}+b^{2}}}$<br/>$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{a}^{2}+\mathrm{b}^{2}}$<br/>1 is correct.<br/>2: Line is $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1 \Rightarrow\left(\frac{1}{\mathrm{a}}\right) \mathrm{x}+\left(\frac{1}{\mathrm{~b}}\right) \mathrm{y}+(-1)=0$<br/>$P=\frac{\left|\frac{1}{a}(0)+\frac{1}{b}(0)-1\right|}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}$<br/>$=P^{2}=\frac{1}{\frac{1}{a^{2}}+\frac{1}{b^{2}}}=\frac{a^{2} b^{2}}{b^{2}+a^{2}}$<br/>$\Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{a}^{2} \mathrm{~b}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$<br/>2 is correct.<br/>3 : Line is $\mathrm{y}=\mathrm{mx}+\mathrm{c} \Rightarrow \mathrm{mx}-\mathrm{y}+\mathrm{c}=0$<br/>$P=\frac{|\mathrm{m}(0)-0+\mathrm{c}|}{\sqrt{\mathrm{m}^{2}+1}}$<br/>$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{~m}^{2}+1} \Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{m}^{2}+1}{\mathrm{c}^{2}}$<br/>3 is wrong. Only 1 and 2 are correct.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-matrix-1-a-b-2-2-a-2-b-1-1-1-is-singular-2-the-matrix-c-a-b-2-c-2-a-2-b-1-1-1-is-nonsingular-which-of-the-abov

slug:
consider-the-following-statements-1-the-matrix-1-a-b-2-2-a-2-b-1-1-1-is-singular-2-the-matrix-c-a-b-2-c-2-a-2-b-1-1-1-is-nonsingular-which-of-the-abov-56095

question:
<div class="question">Consider the following statements:<br/>$1.$ The matrix $\left(\begin{array}{lll}1 &amp; 2 &amp; 1 \\ a &amp; 2 a &amp; 1 \\ b &amp; 2 b &amp; 1\end{array}\right)$ is singular.<br/>$2.$ The matrix $\left(\begin{array}{lll}c &amp; 2 c &amp; 1 \\ a &amp; 2 a &amp; 1 \\ b &amp; 2 b &amp; 1\end{array}\right)$ is non-singular.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">$1.$ $\left(\begin{array}{ccc}1 &amp; 2 &amp; 1 \\ a &amp; 2 a &amp; 1 \\ b &amp; 2 b &amp; 1\end{array}\right)=2\left(\begin{array}{lll}1 &amp; 1 &amp; 1 \\ a &amp; a &amp; 1 \\ b &amp; b &amp; 1\end{array}\right)=2\left|\begin{array}{lll}1 &amp; 1 &amp; 1 \\ a &amp; a &amp; 1 \\ b &amp; b &amp; 1\end{array}\right|=0$<br/>Hence matrix is singular.<br/>$2.$ $\left|\begin{array}{lll}\mathrm{c} &amp; 2 \mathrm{c} &amp; 1 \\ \mathrm{a} &amp; 2 \mathrm{a} &amp; 1 \\ \mathrm{~b} &amp; 2 \mathrm{~b} &amp; 1\end{array}\right|=2\left|\begin{array}{lll}\mathrm{c} &amp; \mathrm{c} &amp; 1 \\ \mathrm{a} &amp; \mathrm{a} &amp; 1 \\ \mathrm{~b} &amp; \mathrm{~b} &amp; 1\end{array}\right|=2\left|\begin{array}{lll}\mathrm{c} &amp; \mathrm{c} &amp; 1 \\ \mathrm{a} &amp; \mathrm{a} &amp; 1 \\ \mathrm{~b} &amp; \mathrm{~b} &amp; 1\end{array}\right|=0$<br/>Hence matrix is singular.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-mean-and-median-are-equal-in-symmetric-distribution-2-the-range-is-the-difference-between-the-maximum-value-an

slug:
consider-the-following-statements-1-the-mean-and-median-are-equal-in-symmetric-distribution-2-the-range-is-the-difference-between-the-maximum-value-an-40145

question:
<div class="question">Consider the following statements:<br/> 1 . The mean and median are equal in symmetric distribution.<br/> 2 . The range is the difference between the maximum value and the minimum value in the data.<br/> 3 . The sum of the areas of the rectangles in the histogra is equal to the total area bounded by the frequency polygon and the horizontal axis. Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 and 3 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 3 only</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1,2 and 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>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">1,2 and 3</span> </div>

answer:
<div class="solution">Mean = Median (in symmetric distribution) Range $=$ (Max. value-Min. value) And sum of areas of rectangles in the histogram is always equal to the total area bounded by frequency polygon and the horizontal axis</div>


tags: ['Mathematics', 'Probability', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-probability-that-there-are-53-sundays-in-a-leap-year-is-twice-the-probability-that-there-are-53-sundays-in-a-n-1

slug:
consider-the-following-statements-1-the-probability-that-there-are-53-sundays-in-a-leap-year-is-twice-the-probability-that-there-are-53-sundays-in-a-n-1-92605

question:
<div class="question">Consider the following statements:<br/>$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.<br/>$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Statement 1:<br/>Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},<br/>\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}<br/>In total, there are 7 possibilities and 1 possibility is Sunday.<br/>Required probability $=\frac{1}{7}$<br/>A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,<br/>\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.<br/>In total, there are 7 possibilities and 2 possibilities have Sundays.<br/>Required probability $=\frac{2}{7}$<br/>So, statement 1 is correct. Statement 2 :<br/>March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$<br/>$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$<br/>In total 7 possibilities, Monday can come in 3 possibilities<br/>$\therefore$ Required probabilities $=\frac{3}{7}$<br/>April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$<br/>$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$<br/>In total 7 possibilities, Monday can come in 2 possibilities.<br/>$\therefore$ Required probability $=\frac{2}{7}$<br/>$\therefore$ Statement 2 is wrong.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-probability-that-there-are-53-sundays-in-a-leap-year-is-twice-the-probability-that-there-are-53-sundays-in-a-n

slug:
consider-the-following-statements-1-the-probability-that-there-are-53-sundays-in-a-leap-year-is-twice-the-probability-that-there-are-53-sundays-in-a-n-65111

question:
<div class="question">Consider the following statements:<br/>$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.<br/>$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Statement 1:<br/>Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},<br/>\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}<br/>In total, there are 7 possibilities and 1 possibility is Sunday.<br/>Required probability $=\frac{1}{7}$<br/>A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,<br/>\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.<br/>In total, there are 7 possibilities and 2 possibilities have Sundays.<br/>Required probability $=\frac{2}{7}$<br/>So, statement 1 is correct. Statement 2 :<br/>March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$<br/>$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$<br/>In total 7 possibilities, Monday can come in 3 possibilities<br/>$\therefore$ Required probabilities $=\frac{3}{7}$<br/>April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$<br/>$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$<br/>In total 7 possibilities, Monday can come in 2 possibilities.<br/>$\therefore$ Required probability $=\frac{2}{7}$<br/>$\therefore$ Statement 2 is wrong.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-product-of-two-nonzero-matrices-can-never-be-identity-matrix-2-the-product-of-two-nonzero-matrices-can-never-b

slug:
consider-the-following-statements-1-the-product-of-two-nonzero-matrices-can-never-be-identity-matrix-2-the-product-of-two-nonzero-matrices-can-never-b-28559

question:
<div class="question">Consider the following statements:<br/>$1.$ The product of two non-zero matrices can never be identity matrix.<br/>$2.$ The product of two non-zero matrices can never be zeromatrix. <br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither lnor 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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Neither lnor 2</span> </div>

answer:
<div class="solution">If B $=\mathrm{A}^{-1}$, then $\mathrm{AB}=\mathrm{I}$ (identity matrix) Therefore, statement 1 is false.<br/>$\mathrm{I}=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ll}0 &amp; 2 \\ 2 &amp; 0\end{array}\right]$, then $\mathrm{IA}=\left[\begin{array}{ll}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$<br/>$\left[\begin{array}{ll}0 &amp; 2 \\ 2 &amp; 0\end{array}\right]=0$<br/>Therefore, statement 2 is not correct.</div>


tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-sum-of-cubes-of-first-20-natural-numbers-is-44400-2-the-sum-of-squares-of-first-20-natural-numbers-is-2870-whi

slug:
consider-the-following-statements-1-the-sum-of-cubes-of-first-20-natural-numbers-is-44400-2-the-sum-of-squares-of-first-20-natural-numbers-is-2870-whi-93117

question:
<div class="question">Consider the following statements:<br/>1) The sum of cubes of first 20 natural numbers is 44400 .<br/>2) The sum of squares of first 20 natural numbers is 2870 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">Consider the Ist statement $1^{3}+2^{3}+\ldots .+20^{3}$. <br/>$=\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]^{2}$ Here $\mathrm{n}=20$<br/>$=\left[20 \times \frac{21}{2}\right]^{2}=(210)^{2}=44100$<br/>So, statement $\mathrm{I}$ is false. Consider the Ind statement. $\begin{aligned} 1^{2}+2^{2}+3^{2}+\ldots \ldots \ldots+20^{2}=\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6} \\ &amp;=\frac{20 \times 21 \times 41}{6}=2870 \end{aligned}$<br/>So, statement II is true.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2018 (Phase 1)']
answer: <div class="solution">Both the given statements are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-sum-of-deviations-from-mean-is-always-zero-2-the-sum-of-absolute-deviations-is-minimum-when-taken-around-media

slug:
consider-the-following-statements-1-the-sum-of-deviations-from-mean-is-always-zero-2-the-sum-of-absolute-deviations-is-minimum-when-taken-around-media-33104

question:
<div class="question">Consider the following statements<br/> 1 . The sum of deviations from mean is always zero<br/> 2 . The sum of absolute deviations is minimum when taken around median Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and</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">Neither $1 \mathrm{n} \alpha$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and</span> </div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-the-value-of-cos-4-6-sin-4-6-is-positive-2-the-value-of-cos-4-4-sin-4-4-is-negative-which-of-the-above-statement-i

slug:
consider-the-following-statements-1-the-value-of-cos-4-6-sin-4-6-is-positive-2-the-value-of-cos-4-4-sin-4-4-is-negative-which-of-the-above-statement-i-68870

question:
<div class="question">Consider the following statements: <br/>1. The value of $\cos 46^{\circ}-\sin 46^{\circ}$ is positive.<br/>2. The value of $\cos 44^{\circ}-\sin 44^{\circ}$ is negative.<br/>Which of the above statement is/are correct?</div>

answer:
<div class="solution">Since $\cos \theta&gt;\sin \theta$, in $\left[0, \frac{\pi}{4}\right]$<br/>and $\cos \theta   &lt; \sin \theta$, in $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$<br/>$\therefore \quad \cos 46^{\circ}-\sin 46^{\circ}=-\mathrm{ve}$<br/>and $\cos 44^{\circ}-\sin 44^{\circ}=+\mathrm{ve}$<br/>So, both the above statements are incorrect.</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-there-exists-2-2-for-which-tan-1-tan-2-sin-1-3-1-sin-1-5-1-sin-1-15-2-2-3-1-which-of-the-above-statements-isare-co

slug:
consider-the-following-statements-1-there-exists-2-2-for-which-tan-1-tan-2-sin-1-3-1-sin-1-5-1-sin-1-15-2-2-3-1-which-of-the-above-statements-isare-co-51377

question:
<div class="question">Consider the following statements: <br/>1) There exists $\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which $\tan ^{-1}(\tan \theta) \neq \theta$.<br/>2) $\quad \sin ^{-1}\left(\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{5}\right)=\sin ^{-1}\left(\frac{2 \sqrt{2}(\sqrt{3}-1)}{15}\right)$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which $\tan ^{-1}(\tan \theta)=\theta$.<br/>Hence, statement (1) is incorrect.<br/>if $x \leq 1 ; y \leq 1 \&amp; x^{2}+y^{2} \leq 1$<br/>$\therefore \sin ^{-1}(x)-\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right]$<br/>$\Rightarrow \sin ^{-1} \frac{1}{3}-\sin ^{-1} \frac{1}{5}$<br/>$\quad=\sin ^{-1}\left[\frac{1}{3} \sqrt{1-\frac{1}{25}}-\frac{1}{5} \sqrt{1-\frac{1}{9}}\right]$<br/>$=\sin ^{-1}\left[\frac{1}{3 \times 5} \sqrt{24}-\frac{1}{5 \times 3} \sqrt{8}\right]$<br/>$=\sin ^{-1}\left[\frac{\sqrt{8 \times 3}-\sqrt{8}}{15}\right]=\sin ^{-1}\left[\frac{2 \sqrt{2}(\sqrt{3}-1)}{15}\right]$<br/>Hence, statement ( 2 ) is correct.</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-there-exists-no-triangle-a-bc-for-which-sin-a-sin-b-sin-c-2-if-the-angle-of-a-triangle-are-in-the-ratio-1-2-3-then

slug:
consider-the-following-statements-1-there-exists-no-triangle-a-bc-for-which-sin-a-sin-b-sin-c-2-if-the-angle-of-a-triangle-are-in-the-ratio-1-2-3-then-99830

question:
<div class="question">Consider the following statements: <br/>1) There exists no triangle $A B C$ for which $\sin A+\sin B=$ $\sin C$<br/>2) If the angle of a triangle are in the ratio $1: 2: 3$, then its sides will be in the ratio $1: \sqrt{3}: 2$. <br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">1) Given, $\sin A+\sin B=\sin C$<br/>$a+b=c \quad\left(\because B y\right.$ sine law,$\left.\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=K\right)$<br/>Here, the sum of two sides of $\Delta \mathrm{ABC}$ is equal to the third side, but it is not possible (Because by triangle inequality, the sum of the length of two sides of a triangle is always greater than the length of the third side) $\mathrm{a}+\mathrm{b}&gt;\mathrm{c}$<br/>2) Ratio of angles of a triangle $A: B: C=1: 2: 3$<br/>$\mathrm{A}+\mathrm{B}+\mathrm{C}=180^{\circ}$<br/>$\begin{aligned} \therefore \mathrm{A} &amp;=30^{\circ} \\ \mathrm{B} &amp;=60^{\circ} \end{aligned}$<br/>$\mathrm{C}=90^{\circ}$<br/>the ratio in sides according to sine rule $a: b: c=\sin A: \sin B: \sin C$<br/>$=\sin 30^{\circ}: \sin 60^{\circ}: \sin 90^{\circ}$<br/>$=\frac{1}{2}, \frac{\sqrt{3}}{2}, 1=\frac{1}{2}: \frac{\sqrt{3}}{2}: 1 \Rightarrow 1: \sqrt{3}: 2$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-two-events-are-mutually-exclusive-if-the-occurrence-of-one-event-prevents-the-occurrence-of-the-other-2-the-probab

slug:
consider-the-following-statements-1-two-events-are-mutually-exclusive-if-the-occurrence-of-one-event-prevents-the-occurrence-of-the-other-2-the-probab-23270

question:
<div class="question">Consider the following statements:<br/>$1$ Two events are mutually exclusive if the occurrence of one event prevents the occurrence of the other.<br/>$2$ The probability of the union of two mutually exclusive events is the sum of their individual probabilities. Which of the above statements is/are correct?</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
answer: <div class="solution">It is a property.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-two-independent-variables-are-always-uncorrelated-2-the-coefficient-of-correlation-between-two-variables-x-and-y-i

slug:
consider-the-following-statements-1-two-independent-variables-are-always-uncorrelated-2-the-coefficient-of-correlation-between-two-variables-x-and-y-i-26896

question:
<div class="question">Consider the following statements<br/> 1 . Two independent variables are always uncorrelated.<br/> 2 . The coefficient of correlation between two variables $\mathrm{x}$ and $\mathrm{Y}$ is positive when $\mathrm{X}$ decreases then $\mathrm{Y}$ decreases Which of the above statements is/are correct?</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-value-of-sin-oscillates-between-1-and-1-2-value-of-cos-oscillates-between-0-and-1-which-of-the-above-statements-is

slug:
consider-the-following-statements-1-value-of-sin-oscillates-between-1-and-1-2-value-of-cos-oscillates-between-0-and-1-which-of-the-above-statements-is-97456

question:
<div class="question">Consider the following statements: <br/>1. Value of $\sin \theta$ oscillates between $-1$ and 1 .<br/>2 Value of $\cos \theta$ oscillates between 0 and 1 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\sin \theta \in[-1,1] ; \mathrm{Q} \in \mathrm{R}$, the value of $\sin \theta$ lies between<br/>$-1$ to 1 . $\cos \theta \in[-1,1] ; \mathrm{Q} \in \mathrm{R}$, the value of $\sin \theta$ lies between $-$<br/>1 to 1 .</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
answer: <div class="solution">Only statement 2 is correct</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-variance-is-unaffected-by-change-of-origin-and-change-of-scale-2-coefficient-of-variance-is-independent-of-the-uni

slug:
consider-the-following-statements-1-variance-is-unaffected-by-change-of-origin-and-change-of-scale-2-coefficient-of-variance-is-independent-of-the-uni-72455

question:
<div class="question">Consider the following statements: $\quad$<br/> 1 . Variance is unaffected by change of origin and change of scale.<br/> 2 . Coefficient of variance is independent of the unit of observations. Which of the statements given above is/are correct?</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1-y-2-e-x-e-x-is-an-increasing-function-on-0-2-y-2-e-x-e-x-is-an-increasing-function-on-which-of-the-above-statement

slug:
consider-the-following-statements-1-y-2-e-x-e-x-is-an-increasing-function-on-0-2-y-2-e-x-e-x-is-an-increasing-function-on-which-of-the-above-statement-11123

question:
<div class="question">Consider the following statements: $\quad$<br/>1.$\quad y=\frac{e^{x}+e^{-x}}{2}$ is an increasing function on $[0, \infty)$.<br/>2.$y=\frac{e^{x}-e^{-x}}{2}$ is an increasing function on $(-\infty, \infty)$. Which of the above statements is/are correct?</div>

answer:
<div class="solution">From statement $-1$<br/>$y=\frac{e^{x}+e^{-x}}{2}$<br/>$\therefore \quad \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}\left[\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}\right]$<br/>Now equating $\frac{\mathrm{dy}}{\mathrm{dx}}$ to zero we get, $\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}=0$<br/>$\therefore \quad \mathrm{e}^{\mathrm{x}}=\mathrm{e}^{-\mathrm{x}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/99L6k7hX-b8Phbe9vYZDK6joOY3D7q-iLrFMNS-mnPo.original.fullsize.png"/><br/><br/>$\mathrm{e}^{x}-\mathrm{e}^{-\mathrm{x}}=0$<br/>$\mathrm{e}^{\mathrm{x}}=\mathrm{e}^{-\mathrm{x}}$<br/>$\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{-\mathrm{x}}}=1$<br/>$\mathrm{e}^{2 \mathrm{x}}=1$<br/>$2 \mathrm{x}=0$<br/>$\mathrm{x}=0$<br/>The given function is increasing on interval $[0, \infty]$ From Statement-2<br/>$\mathrm{y}=\frac{\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}}{2}$<br/>$\frac{d y}{d x}=\frac{1}{2}\left[e^{x}-e^{x}\right]$<br/>Now equating $\frac{\mathrm{dy}}{\mathrm{dx}}$ to zero, we get. $\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}=0$<br/>$\therefore \quad \mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}=0$<br/>$\mathrm{e}^{2 \mathrm{x}}+1=0$<br/>$\mathrm{e}^{2 \mathrm{x}}=-1$<br/>Hence, the given function is increacing from $[-\infty, \infty]$<br/>- Both statement are correct Option (c) is correct.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1parallelism-of-lines-is-an-equivalence-relation-2-xry-ifx-is-a-father-of-y-is-an-equivalence-relation-which-of-the-

slug:
consider-the-following-statements-1parallelism-of-lines-is-an-equivalence-relation-2-xry-ifx-is-a-father-of-y-is-an-equivalence-relation-which-of-the-13256

question:
<div class="question">Consider the following statements:<br/>1.Parallelism of lines is an equivalence relation.<br/>2.$\mathrm{xRy}$,ifx is a father of y, is an equivalence relation. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Statement 1: Let $l, \mathrm{~m}, \mathrm{n}$ areparallel line and $\mathrm{R}$ is a relation. $I \| l$, then $\mathrm{R}$ is reflexive and $I \| \mathrm{m}$ and $\mathrm{m} \| l$, the $\mathrm{R}$ is symmetric. also $l\|\mathrm{~m}, \mathrm{~m}\| \mathrm{n} \Rightarrow I \| \mathrm{n}$, then $\mathrm{R}$ is transitive<br/>Hence, Ris an equivalence relation Statement $2: \mathrm{x}$ is father of $\mathrm{y}$ then $\mathrm{x}$ is not the father of $\mathrm{x}$, so relation is not reflexive. Also, $x$ is father of $y$ but yis not father of $x$, so it is not symmetric. And $x$ is father of $y$ and $y$ is father of $z$ does not imply that $\mathrm{x}$ is father of $\mathrm{z}$ so, it is not transitive too. So, this is not an equivalence relation. so, only statement 1 is<br/>correct</div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1the-function-f-x-sin-x-decreases-on-the-interval-0-2-2the-function-f-x-cos-x-increases-on-the-interval-0-2-which-of

slug:
consider-the-following-statements-1the-function-f-x-sin-x-decreases-on-the-interval-0-2-2the-function-f-x-cos-x-increases-on-the-interval-0-2-which-of-94170

question:
<div class="question">Consider the following statements: <br/>1.The function $f(x)=\sin x$ decreases on the interval $(0, \pi / 2)$<br/>2.The function $f(x)=\cos x$ increases on the interval $(0, \pi / 2)$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\sin x$ increases on the interval $\left(0, \frac{\pi}{2}\right)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/8CJLIHtfvHpsZ_rXVxLEKwbtj9VYByxDBiqqtM6FJYQ.original.fullsize.png"/><br/><br/>$\cos x$ decreases on the interval $\left(0, \frac{\pi}{2}\right)$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/QsnsTuUSKCEnZJuKM7p47frtHI3UzWFC7fp1-aEG1Xw.original.fullsize.png"/></div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-1the-general-solution-of-d-x-d-y-f-x-x-is-of-the-form-y-g-x-c-where-c-is-an-arbitrary-constant-2the-degree-of-d-x-d-

slug:
consider-the-following-statements-1the-general-solution-of-d-x-d-y-f-x-x-is-of-the-form-y-g-x-c-where-c-is-an-arbitrary-constant-2the-degree-of-d-x-d-52577

question:
<div class="question">Consider the following statements:<br/>1.The general solution of $\frac{d y}{d x}=f(x)+x$ is of the form $y$ $=g(x)+c$, where $c$ is an arbitrary constant.<br/>2.The degree of $\left(\frac{d y}{d x}\right)^{2}=f(x)$ is 2 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement 1:<br/>The general solution of $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{x})+\mathrm{x}$<br/>Now integrating on both side<br/>$\int \frac{\mathrm{dy}}{\mathrm{dx}}=\int[\mathrm{f}(\mathrm{x})+\mathrm{x}] \mathrm{dx}$<br/>$\mathrm{y}=\mathrm{f}(\mathrm{x})+\mathrm{C}$<br/>Statement 1 is correct. Statement 2:<br/>$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{x})+\mathrm{c}$<br/>Squaring both sides,<br/>$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=(\mathrm{f}(\mathrm{x})+\mathrm{c})^{2}$<br/>$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=\left[\mathrm{f}(\mathrm{x})^{2}+2 \mathrm{c} \mathrm{f}(\mathrm{x})+\mathrm{c}^{2}\right]$<br/>Hence, the differential equation is of order $-1$ and degree $-2$.<br/>- both Statements 1 and 2 are correct.<br/>$\therefore$ Option (c) is correct.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2009-i-1-the-equation-to-a-straight-line-parallel-to-the-axis-of-x-is-y-d-where-d-is-a-constant-2-the-equation-to-th

slug:
consider-the-following-statements-2009-i-1-the-equation-to-a-straight-line-parallel-to-the-axis-of-x-is-y-d-where-d-is-a-constant-2-the-equation-to-th-69996

question:
<div class="question">Consider the following statements: $\quad[2009-I]$<br/>1) The equation to a straight line parallel to the axis of $x$ is $y=d$, where $d$ is a constant.<br/>2) The equation to the axis of $x$ is $x=0$. Which of the statement (s) given above is/are correct?</div>

answer:
<div class="solution">We know that, the equation of $\mathrm{x}$ -axis is $y=0$ Statement 1 says that the equation to a straight line parallel to the axis of $x$ is $y=d$. Since, $d$ is constant therefore it can be zero. Thus, only statement 1 is correct.</div>


tags: ['Mathematics', 'Properties of Triangles', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2016-i-1-if-abc-is-an-equilateral-triangle-then-3-tan-a-b-tan-c-1-2-if-abc-is-a-triangle-in-which-a-7-8-b-6-6-then-t

slug:
consider-the-following-statements-2016-i-1-if-abc-is-an-equilateral-triangle-then-3-tan-a-b-tan-c-1-2-if-abc-is-a-triangle-in-which-a-7-8-b-6-6-then-t-70559

question:
<div class="question">Consider the following statements: $\quad[2016-I]$<br/>1) If $\mathrm{ABC}$ is an equilateral triangle, then $3 \tan (A+B) \tan C=1$<br/>2) If $\mathrm{ABC}$ is a triangle in which $\mathrm{A}=78^{\circ}, \mathrm{B}=66^{\circ}$, then $\tan \left(\frac{A}{2}+C\right)   &lt; \tan A$<br/>3) If $\mathrm{ABC}$ is any triangle, then<br/>$$<br/>\tan \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \sin \left(\frac{\mathrm{C}}{2}\right) &lt; \cos \left(\frac{\mathrm{C}}{2}\right)<br/>$$<br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 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">C</span> <span class="option-data">1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2 and 3</span> </li> </ul>

answer:
<div class="solution">$\because A B C$ is an equilateral triangle. $\therefore A=B=C=60^{\circ}$<br/>L.H.S. $=3 \tan (A+B) \tan C$<br/>$=3 \tan 120^{\circ} \tan 60^{\circ}$<br/>$\quad=3(-\sqrt{3})(\sqrt{3})$<br/>$\quad=-9 \neq 1$<br/>Hence statement $(1)$ is incorrect. Statement-2 $A B C$ is a triangle such that $A=78^{\circ}$ and $B=66^{\circ}$ $C=180-(78+66)=180-144=36^{\circ}$<br/>$\frac{A}{2}+C=\frac{78}{2}+36^{\circ}$<br/>$=39+36=75^{\circ}$<br/>$\tan \left(\frac{A}{2}+C\right)   &lt; \tan A$<br/>$\Rightarrow \tan 75^{\circ} &lt; \tan 78^{\circ}$<br/>Hence statement $(2)$ is correct. Statement (3) In a triangle $\mathrm{ABC}$ $A+B+C=180^{\circ}$<br/>$A+B=180^{\circ}-C$<br/>$\frac{A+B}{2}=\frac{180^{\circ}-C}{2}$<br/>$\Rightarrow \frac{A+B}{2}=90-\frac{C}{2}$<br/>$\Rightarrow \tan \left(\frac{A+B}{2}\right)=\tan \left(90-\frac{C}{2}\right)$<br/>$\Rightarrow \tan \left(\frac{A+B}{2}\right)=\cot \frac{C}{2}$<br/>$\tan \left(\frac{A+B}{2}\right) \cdot \sin \frac{C}{2}=\cot \frac{C}{2} \cdot \sin \frac{C}{2}=\cos \frac{C}{2}$<br/>$\Rightarrow \tan \left(\frac{A+B}{2}\right) \cdot \sin \frac{C}{2}=\cos \frac{C}{2}$<br/>We can see that statement $(3)$ is not correct. <br/>Hence only 2 nd statement is correct.</div>


tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2016-i-1-tan-1-x-tan-1-x-1-2-there-exist-x-y-1-1-where-x-y-such-that-sin-1-x-cos-1-y-2-which-of-the-above-statements

slug:
consider-the-following-statements-2016-i-1-tan-1-x-tan-1-x-1-2-there-exist-x-y-1-1-where-x-y-such-that-sin-1-x-cos-1-y-2-which-of-the-above-statements-23778

question:
<div class="question">Consider the following statements: $\quad[2016-I]$<br/>1) $\quad \tan ^{-1} x+\tan ^{-1}\left(\frac{1}{x}\right)=\pi$<br/>2) There exist $\mathrm{x}, \mathrm{y} \in[-1,1]$, where $\mathrm{x} \neq \mathrm{y}$ such that<br/>$$<br/>\sin ^{-1} x+\cos ^{-1} y=\frac{\pi}{2}<br/>$$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement-1<br/>$\because\left(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}\right)$<br/>$\tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{x+\frac{1}{x}}{1-x \cdot \frac{1}{x}}$<br/>$$<br/>\begin{array}{l}<br/>=\tan ^{-1} \frac{x^{2}+1}{0} \\<br/>=\tan ^{-1} \infty=\tan ^{-1} \tan \frac{\pi}{2}<br/>\end{array}<br/>$$<br/>$\tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2}$<br/>Statement (1) is wrong. Statement 2,<br/>$$<br/>\sin ^{-1} x+\cos ^{-1} y=\frac{\pi}{2}<br/>$$<br/>Only when $x=y$ Here $x \neq y$. <br/>Statement $(2)$ is also wrong.</div>


tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2017-i-1-if-lim-x-a-f-x-and-lim-x-a-g-x-both-exist-then-lim-x-a-f-x-g-x-exists-2-if-lim-x-a-f-x-g-x-exists-then-both

slug:
consider-the-following-statements-2017-i-1-if-lim-x-a-f-x-and-lim-x-a-g-x-both-exist-then-lim-x-a-f-x-g-x-exists-2-if-lim-x-a-f-x-g-x-exists-then-both-56013

question:
<div class="question">Consider the following statements: $\quad[2017-I]$<br/>1- If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exist, then $\lim _{x \rightarrow a}\{f(x) g$<br/>(x) $\}$ exists.<br/>2- If $\lim _{x \rightarrow a}\{f(x) g(x)\}$ exists, then both $\lim _{x \rightarrow a} f(x)$ and<br/>$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ must exist.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exists, then<br/>$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists. But if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists, then<br/>it is not necesary that both $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ and $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ exists.</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2018-ii-1-the-angle-between-the-planes-2-x-y-z-1-and-x-y-2-z-3-is-3-2-the-distance-between-the-plancs-6-x-3-y-6-z-2-

slug:
consider-the-following-statements-2018-ii-1-the-angle-between-the-planes-2-x-y-z-1-and-x-y-2-z-3-is-3-2-the-distance-between-the-plancs-6-x-3-y-6-z-2-82162

question:
<div class="question">Consider the following statements:<br/>$[2018-I I]$<br/> 1 . The angle between the planes<br/>$2 x-y+z=1$ and $x+y+2 z=3$ is $\frac{\pi}{3}$<br/> 2 . The distance between the plancs<br/>$6 x-3 y+6 z+2=0$ and $2 x-y+2 z+4=0$ is $\frac{10}{9}$<br/>Which of the above statements is/are correct</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both and 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">Neither 1 nor 2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both and 2</span> </div>

answer:
<div class="solution">Angle between planes<br/>$=\cos ^{-1}\left(\frac{2-1+2}{6}\right)$<br/>$=\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$<br/>Distance between planes<br/>$=\frac{\left|\frac{2}{3}-4\right|}{\sqrt{9}}=\frac{10}{3 \times 3}=\frac{10}{9}$</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-2019-i-1-for-an-equation-of-a-line-x-cos-q-y-sin-q-p-in-normal-form-the-length-of-the-perpendicular-from-the-point-a

slug:
consider-the-following-statements-2019-i-1-for-an-equation-of-a-line-x-cos-q-y-sin-q-p-in-normal-form-the-length-of-the-perpendicular-from-the-point-a-41415

question:
<div class="question">Consider the following statements: $\quad[2019-I]$<br/>1) For an equation of a line, $x \cos q+y \sin q=p$, in normal form, the length of the perpendicular from the point $(a, b)$ to the line is $a \cos q+b \sin q+p \mid$<br/>2) The length of the perpendicular from the point $(a, b)$ to<br/>the line $\frac{x}{a}+\frac{y}{b}=1$ is $\left|\frac{a \alpha+b \beta-a b}{\sqrt{a^{2}+b^{2}}}\right|$.<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">The length of perpendicular from $(\alpha, \beta)$ to line $x \cos \theta+y \sin \theta-p=0$<br/>$|\alpha \cos \theta+\beta \sin \theta-p|$<br/>Statement 1 is false. The length of perpendicular from $(\alpha, \beta)$ to line<br/>$\frac{x}{a}+\frac{y}{b}=1$<br/>$\frac{x}{a}+\frac{y}{b}=1 \Rightarrow b x+a y-a b=0$<br/>So, perpendicular is $\left|\frac{b \alpha+a \beta-a b}{\sqrt{a^{2}+b^{2}}}\right|$<br/>Statement 2 is also false.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2012 (Phase 1)']
answer: <div class="solution">Statement 2 and 3</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-a-continuous-random-variable-can-take-all-values-in-an-interval-2-a-random-variable-which-takes-a-finite-number-of-v

slug:
consider-the-following-statements-a-continuous-random-variable-can-take-all-values-in-an-interval-2-a-random-variable-which-takes-a-finite-number-of-v-43168

question:
<div class="question">Consider the following statements:<br/>A continuous random variable can take all values in an<br/>interval.<br/> 2 . A random variable which takes a finite number of values is necessarily discrete.<br/> 3. Construction of a frequency distribution is based on data which are discrete Which of the above statements are correct? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 and 3 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">C</span> <span class="option-data">1 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 3 only</span> </div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-both-the-regression-coefficients-have-same-sign-2-if-one-of-the-regression-coefficients-is-greater-thar-unity-the-ot

slug:
consider-the-following-statements-both-the-regression-coefficients-have-same-sign-2-if-one-of-the-regression-coefficients-is-greater-thar-unity-the-ot-38662

question:
<div class="question">Consider the following statements : Both the regression coefficients have same sign 2 . If one of the regression coefficients is greater thar unity, the other must be less than unity. Which of the above statements is/are correct</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only $y$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and</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">Neither $[\mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and</span> </div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-for-non-empty-sets-a-b-and-c-1-a-b-c-a-b-c-2-a-b-c-a-b-c-which-of-the-statements-given-above-isare-correct

slug:
consider-the-following-statements-for-non-empty-sets-a-b-and-c-1-a-b-c-a-b-c-2-a-b-c-a-b-c-which-of-the-statements-given-above-isare-correct-68379

question:
<div class="question">Consider the following statements:<br/>For non empty sets A, B and C<br/>1.$\quad \mathrm{A}-(\mathrm{B}-\mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup \mathrm{C}$<br/>2.$\quad$ A- $(B \cup C)=(A-B)-C$<br/>Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Let there be three non empty, non overlapping sets; inside a universal set U. This creates 8 regionsmarked as: $a, b, c, d, e, f, g, h$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/F-x7RpeSiR7exoD0Ukq3e0eCYktYdYzOfbYuEQDPmpc.original.fullsize.png"/><br/><br/>Statement 1: $\mathrm{A}-(\mathrm{B}-\mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup \mathrm{C}$<br/>LHS represent region a, RHS represents a, $\mathrm{d}, \mathrm{g}$. Hence, this is not correct. Statement $2: \mathrm{A}-(\mathrm{B} \cup \mathrm{C})=(\mathrm{A}-\mathrm{B})-\mathrm{C}$<br/>LHS represents, region 'a' RHS also represents a. Hence, only statement 2 is correct.</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-for-the-two-nonempty-sets-a-and-b-1-a-b-a-b-a-b-aub-2-a-a-b-aub-which-of-the-above-statements-isare-correct

slug:
consider-the-following-statements-for-the-two-nonempty-sets-a-and-b-1-a-b-a-b-a-b-aub-2-a-a-b-aub-which-of-the-above-statements-isare-correct-39842

question:
<div class="question">Consider the following statements for the two non-empty sets A and B:<br/>(1) $(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A} \cap \overline{\mathrm{B}}) \cup(\overline{\mathrm{A}} \cap \mathrm{B})=\mathrm{AUB}$<br/>(2) $\quad(\mathrm{A} \cup(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}))=\mathrm{AUB}$<br/>Which of the above statements is/are correct?</div>

answer:
<div class="solution">1. $(A \cap B) \cup(A \cap \bar{B}) \cup(\bar{A} \cap B)=A \cup B$<br/>Let us draw Venn diagram<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/U0p7c-alsVWR0B3bmL7OaozxIBe48SeUmPnMMddCsnI.original.fullsize.png"/><br/><br/>$-1$ is correct.<br/>2.$A \cup(\bar{A} \cap \bar{B})=A$<img src="https://cdn-question-pool.getmarks.app/pyq/nda/XSUQFpOBXYaBmix1ERuFyzvW3amVAzsJ9am5W-SSiXM.original.fullsize.png"/><br/><br/>$\therefore 2$ is not correct</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-i-f-x-x-3-is-continuous-at-x-0-ii-f-x-x-3-is-differentiable-at-x-0-which-of-the-statements-given-above-isare-correct

slug:
consider-the-following-statements-i-f-x-x-3-is-continuous-at-x-0-ii-f-x-x-3-is-differentiable-at-x-0-which-of-the-statements-given-above-isare-correct-70578

question:
<div class="question">Consider the following statements: <br/>I. $\quad f(x)=|x-3|$ is continuous at $x=0$.<br/>II. $\quad f(x)=|x-3|$ is differentiable at $x=0$. Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both I and II</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">Neither Inor II</span> </li> </ul>

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

answer:
<div class="solution">Let $f(x)=|x-3|$ $\begin{aligned} \text { L.H.L } &amp;=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0}|0-h-3| \\ &amp;=\lim _{h \rightarrow 0}(h+3)=3 \\ \text { R.H.L } &amp;=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0}|0+h-3| \\ &amp;=\lim _{h \rightarrow 0}|h-3|=3 \end{aligned}$<br/>Since. L.H.L $=$ R.H.L at $x=0$ $\therefore f(x)=|x-3|$ is continuous at $x=0$. Now, LHD $=f^{\prime}\left(0^{-}\right)=\lim _{h \rightarrow 0} \frac{f(0)-f(0-h)}{h}$<br/>$=\lim _{h \rightarrow 0} \frac{3-(3-h)}{h}=1$<br/>and $\mathrm{RHD}=f^{\prime}\left(0^{+}\right)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$<br/>$=\lim _{h \rightarrow 0} \frac{3+h-3}{h}=1$<br/>$\Rightarrow \mathrm{LHD}=\mathrm{RHD}$<br/>$\therefore \quad f(x)$ is differentiable Hence, both statements (I) and (II) are correct.</div>


tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">I only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-i-the-coefficient-of-the-middle-term-in-the-expansion-of-1-x-8-is-equal-to-the-middle-term-of-x-x-1-8-ii-the-coeffic

slug:
consider-the-following-statements-i-the-coefficient-of-the-middle-term-in-the-expansion-of-1-x-8-is-equal-to-the-middle-term-of-x-x-1-8-ii-the-coeffic-98769

question:
<div class="question">Consider the following statements<br/>I. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is equal to the middle term of $\left(x+\frac{1}{x}\right)^{8}$.<br/>II. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is less than the coefficient of the fifth term in the expansion of $(1+x)^{7}$. Which of the above statements is/ are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">I 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">II only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both I and II</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither Inor II</span> </li> </ul>

answer:
<div class="solution">Statement I : Given expansion is $(1+x)^{8}$ Since, $n=8$ is even<br/>$\therefore\left(\frac{n}{2}+1\right)$ th term is the middle term.<br/>ie- $\left(\frac{8}{4}+1\right)^{\text {th }}=5^{\text {th }}$ term $=$ middle term<br/>Now, $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4} 1^{4}={ }^{8} C_{4} x^{4}$<br/>Coeff of $5^{\text {th }}$ term (middle term) $={ }^{8} C_{4}$<br/>Now, consider the expansion $\left(x+\frac{1}{x}\right)^{8}$<br/>It's middle term $=5^{\text {th }}$ term<br/>and $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4}\left(\frac{1}{x}\right)^{4}={ }^{8} C_{4}$<br/>Hence, statement I is correct.<br/>Statement-II : Coeff. of middle term in $(x+1)^{8}$ is<br/>${ }^{8} C_{4}=\frac{8 !}{4 ! 4 !}=70$<br/>Coeff of $5^{\text {th }}$ term in $(1+x)^{7}={ }^{7} C_{4}=\frac{7 !}{4 ! 3 !}=35$<br/>Hence, statement II is incorrect.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2009 (Phase 2)']
answer: <div class="solution">Both the given statements $\mathrm{I}$ and $\mathrm{II}$ are true.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-i-the-data-which-are-collected-from-the-unit-or-individual-respondents-directly-for-the-purpose-of-certain-study-or-

slug:
consider-the-following-statements-i-the-data-which-are-collected-from-the-unit-or-individual-respondents-directly-for-the-purpose-of-certain-study-or-12443

question:
<div class="question">Consider the following statements<br/>I. The data, which are collected from the unit or individual respondents directly for the purpose of certain study or information are known as primary data.<br/>II. The data obtained in a census study are primary data. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both I and II</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">Neither Inor II</span> </li> </ul>

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


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Only I</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-i-the-inverse-of-a-square-matrix-if-it-exists-is-unique-ii-if-a-and-b-are-singular-matrices-of-order-n-then-a-b-is-a

slug:
consider-the-following-statements-i-the-inverse-of-a-square-matrix-if-it-exists-is-unique-ii-if-a-and-b-are-singular-matrices-of-order-n-then-a-b-is-a-13060

question:
<div class="question">Consider the following statements. <br/>I. The inverse of a square matrix, if it exists, is unique.<br/>II. If $A$ and $B$ are singular matrices of order $n$, then $A B$ is also a singular matrix of order $n$. Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Only I</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">Only II</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both I and II</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither I nor II</span> </li> </ul>

answer:
<div class="solution">The inverse of a square matrix if it exists, is unique but if $A$ and $B$ are singular matrices of order $n$, then $A B$ is not a singular matrices of order $n$. Hence, only statement I is correct.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-if-a-and-b-are-independent-events-then-1-a-and-b-are-independent-2-a-and-b-are-independent-3-a-and-b-are-independent

slug:
consider-the-following-statements-if-a-and-b-are-independent-events-then-1-a-and-b-are-independent-2-a-and-b-are-independent-3-a-and-b-are-independent-54018

question:
<div class="question">Consider the following statements:<br/>If $A$ and $B$ are independent events, then<br/>$1.$ $A$ and $\bar{B}$ are independent.<br/>$2.$ $\bar{A}$ and $\mathrm{B}$ are independent.<br/>$3.$ $\bar{A}$ and $\bar{B}$ are independent. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1 and 2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1,2 and 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">None of these</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{A}$ and $\mathrm{B}$ are independent events. $\Rightarrow \quad P(A \cap B)=P(A) \cdot P(B)$<br/>$$<br/>\text { 1. } \begin{aligned}<br/>\text { Consider } \mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}}) &amp;=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\<br/>&amp;=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})<br/>\end{aligned}<br/>$$<br/>(from A)<br/>$$<br/>=\mathrm{P}(\mathrm{A})[1-\mathrm{P}(\mathrm{B})]=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})<br/>$$<br/>Hence, $\mathrm{A}$ and $\overline{\mathrm{B}}$ are independent.<br/>$2.$ Similarly, $\overline{\mathrm{A}}$ and $\mathrm{B}$ are independent.<br/>$3.$<br/>$$<br/>\begin{aligned}<br/>\text { Consider } \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) &amp;=\mathrm{P}(\overline{\mathrm{A} \cup \mathrm{B}}) \\<br/>&amp;=1-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\<br/>=&amp; 1-[\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cup \mathrm{B})] \\<br/>=&amp; 1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \\<br/>=&amp;[1-\mathrm{P}(\mathrm{A})][1-\mathrm{P}(\mathrm{B})] \\<br/>=&amp; \mathrm{P}(\overline{\mathrm{A}}) . \mathrm{P}(\overline{\mathrm{B}})<br/>\end{aligned}<br/>$$<br/>Hence, $\overline{\mathrm{A}}$ and $\overline{\mathrm{B}}$ are independent.</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-a-function-f-x-1-f-x-is-continuous-at-x-a-iff-lim-x-a-f-x-exists-2-if-f-x-is-continuous-at-a-point-the

slug:
consider-the-following-statements-in-respect-of-a-function-f-x-1-f-x-is-continuous-at-x-a-iff-lim-x-a-f-x-exists-2-if-f-x-is-continuous-at-a-point-the-28740

question:
<div class="question">Consider the following statements in respect of a function $\mathrm{f}(\mathrm{x}):$<br/>1- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=$ a iff $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ exists.<br/>2- If $\mathrm{f}(\mathrm{x})$ is continuous at a point, then $\frac{1}{\mathrm{f}(\mathrm{x})}$ is also continuous at that point. <br/>Which of the above, statements is/are correct?</div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})$ is continuous if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ exists and equals $\mathrm{f}(\mathrm{a})$<br/>So, statement $-1$ is not correct. Also, statement $-2$ incorrect.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-a-histogram-1-the-total-area-of-the-rectangles-in-a-histogram-is-equal-to-the-total-area-bounded-by-th

slug:
consider-the-following-statements-in-respect-of-a-histogram-1-the-total-area-of-the-rectangles-in-a-histogram-is-equal-to-the-total-area-bounded-by-th-27890

question:
<div class="question">Consider the following statements in respect of a histogram:<br/> 1 . The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequnecy polygon and the $x$ -axis.<br/> 2 . When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency. Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement (1) is correct because total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and $x$ -axis. Statement $(2)$ is also correct</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-a-square-matrix-a-and-its-transpose-a-t-1-a-a-t-is-always-symmetric-2-a-a-t-is-always-antisymmetric-wh

slug:
consider-the-following-statements-in-respect-of-a-square-matrix-a-and-its-transpose-a-t-1-a-a-t-is-always-symmetric-2-a-a-t-is-always-antisymmetric-wh-40977

question:
<div class="question">Consider the following statements in respect of a square matrix $A$ and its transpose $A^{\mathrm{T}} .$<br/>$1.$ $A+A^{\mathrm{T}}$ is always symmetric.<br/>$2.$ $A-A^{\mathrm{T}}$ is always anti-symmetric Which of the statements given above is/are correct?</div>

answer:
<div class="solution">We know, that A matrix 'A' is said to be symmetric $\mathrm{A}=\mathrm{A}^{\mathrm{T}}$ and anti- symmetric if $\mathrm{A}=-\mathrm{A}^{\mathrm{T}}$<br/>Now, consider $\left(\mathrm{A}+\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{A}$<br/>$\Rightarrow \mathrm{A}+\mathrm{A}^{\mathrm{T}}$ is always symmetric<br/>Now, consider $\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}-\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}-\mathrm{A}$<br/>$=-\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)$<br/>$\Rightarrow \mathrm{A}-\mathrm{A}^{\mathrm{T}}$ is always anti- symmetric.</div>


tags: ['Mathematics', 'Circle', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-circles-x-2-y-2-2-x-2-y-0-and-x2y21-1-the-radius-of-the-first-circle-is-twice-that-of-the-second-circl

slug:
consider-the-following-statements-in-respect-of-circles-x-2-y-2-2-x-2-y-0-and-x2y21-1-the-radius-of-the-first-circle-is-twice-that-of-the-second-circl-74088

question:
<div class="question">Consider the following statements in respect of circles $x^{2}+y^{2}-2 x-2 y=0$ and $x^{2}+y^{2}=1 <br/>1. The radius of the first circle is twice that of the second circle.<br/>2. Both the circles pass through the origin. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">The equation of first circle is $x^{2}+y^{2}-2 x-2 y=0$.<br/>Radius $=\sqrt{(-1)^{2}+(-1)^{2}}=\sqrt{2}$<br/>and equation of second circle is $x^{2}+y^{2}=1$.<br/>Radius $=1$<br/>From above it is clear that the radius of first circle is not twice that of second circle.<br/>$\therefore$ Statement 1 is not correct.<br/>Also, first circle passes through the origin while second circle does not pass through the origin. Hence, neither 1 nor 2 statement is correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-class-intervals-of-grouped-frequency-distribution-1-class-intervals-need-not-be-mutually-exclusive-2-c

slug:
consider-the-following-statements-in-respect-of-class-intervals-of-grouped-frequency-distribution-1-class-intervals-need-not-be-mutually-exclusive-2-c-36119

question:
<div class="question">Consider the following statements in respect of class intervals of grouped frequency distribution:<br/> 1 . Class intervals need not be mutually exclusive.<br/> 2 . Class intervals should be exhaustive<br/> 3. Class intervals need not be of equal width Which of the above statements are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only 2 only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2 and 3 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">C</span> <span class="option-data">1 and 3 only</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1,2 and 3</span> </li> </ul>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-histogram-1-the-histogram-is-a-suitable-representation-of-frequency-distribution-of-a-continuous-varia

slug:
consider-the-following-statements-in-respect-of-histogram-1-the-histogram-is-a-suitable-representation-of-frequency-distribution-of-a-continuous-varia-24213

question:
<div class="question">Consider the following statements in respect of histogram<br/> 1 . The histogram is a suitable representation of frequency distribution of a continuous variable<br/> 2 . The area included under the whole histogram is the total frequency. Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 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">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neithe</span> </li> </ul>

answer:
<div class="solution">1 . It is true that, the histogram is a suitable representation of a frequency distribution of continuous variable Hence, Statement 1 is correct 2 . We know that, the area of histogram is proportional to the frequency, so it is not true statement.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2009 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-symmetric-matrices-a-and-b-1-a-b-is-symmetric-2-a-2-b-2-is-symmetric-which-of-the-above-statements-isa

slug:
consider-the-following-statements-in-respect-of-symmetric-matrices-a-and-b-1-a-b-is-symmetric-2-a-2-b-2-is-symmetric-which-of-the-above-statements-isa-18216

question:
<div class="question">Consider the following statements in respect of symmetric matrices $A$ and $B$<br/>$1. A B$ is symmetric.<br/>$2. A^{2}+B^{2}$ is symmetric. <br/>Which of the above statement(s) is/are correct?</div>

answer:
<div class="solution">We know, a matrix $\mathrm{A}$ is said to be symmetric matrix if $A^{\prime}=A$ where ' ' ' represents the transpose.<br/>Consider $(A B)^{\prime}=B^{\prime} A^{\prime}$<br/>Since, $(A B)^{\prime} \neq A B$<br/>$\therefore \mathrm{AB}$ is not symmetric.<br/>and consider $\left(A^{2}+B^{2}\right)^{\prime}=\left(A^{\prime}\right)^{2}+\left(B^{\prime}\right)^{2}=A^{2}+B^{2}$<br/>$\therefore \mathrm{A}^{2}+\mathrm{B}^{2}$ is symmetric.</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-determinant-cos-2-2-sin-2-2-sin-2-2-cos-2-2-where-are-complementary-angles-1-the-value-of-the-dete

slug:
consider-the-following-statements-in-respect-of-the-determinant-cos-2-2-sin-2-2-sin-2-2-cos-2-2-where-are-complementary-angles-1-the-value-of-the-dete-32922

question:
<div class="question">Consider the following statements in respect of the determinant $\left|\begin{array}{ll}\cos ^{2} \frac{\alpha}{2} &amp; \sin ^{2} \frac{\alpha}{2} \\ \sin ^{2} \frac{\beta}{2} &amp; \cos ^{2} \frac{\beta}{2}\end{array}\right|$<br/>where $\alpha, \beta$ are complementary angles<br/>$1.$ The value of the determinant is $\frac{1}{\sqrt{2}} \cos \left(\frac{\alpha-\beta}{2}\right)$.<br/>$2.$ The maximum value of the determinant is $\frac{1}{\sqrt{2}}$.<br/>Which of the above statements is/ are correct?</div>

answer:
<div class="solution">$\alpha+\beta=90^{\circ}$<br/>$\left|\begin{array}{rr}\cos ^{2} \frac{\alpha}{2} &amp; \sin ^{2} \frac{\alpha}{2} \\ \sin ^{2} \frac{\beta}{2} &amp; \cos ^{2} \frac{\beta}{2}\end{array}\right|$<br/>$=\cos ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\beta}{2}-\sin ^{2} \frac{\alpha}{2} \sin ^{2} \frac{\beta}{2}$<br/>$=\left(\cos \frac{\alpha}{2} \cos \frac{\beta}{2}+\sin \frac{\alpha}{2} \sin \frac{\beta}{2}\right)\left(\cos \frac{\alpha}{2} \cos \frac{\beta}{2}-\sin \frac{\alpha}{2} \sin \frac{\beta}{2}\right)$<br/>$=\cos \frac{(\alpha-\beta)}{2} \times \cos \frac{(\alpha+\beta)}{2}$<br/>$=\cos \frac{(\alpha-\beta)}{2} \times \cos \frac{\left(90^{\circ}\right)}{2}$<br/>$=\cos \frac{(\alpha-\beta)}{2} \times \frac{1}{\sqrt{2}}$<br/>Maximum value of $\cos \left(\frac{\alpha-\beta}{2}\right)$ is $1 .$ Somaximum value<br/>of determinent is $\left(\frac{1}{\sqrt{2}}\right)$<br/>So both 1 and 2 are correct.</div>


tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-differential-equation-d-x-2-d-2-y-cos-d-x-d-y-0-1the-degree-of-the-differential-equation-is-not-de

slug:
consider-the-following-statements-in-respect-of-the-differential-equation-d-x-2-d-2-y-cos-d-x-d-y-0-1the-degree-of-the-differential-equation-is-not-de-49575

question:
<div class="question">Consider the following statements in respect of the differential equation $\quad$<br/>$\frac{d^{2} y}{d x^{2}}+\cos \left(\frac{d y}{d x}\right)=0$<br/>1.The degree of the differential equation is not defined.<br/>2.The order of the differential equation is 2 . Which of the above statements is/are correct?</div>

answer:
<div class="solution">Statement 1: Differential equation is not a polynomial equation in its derivatives. So, its degree is not defined. Statement 2 : The highest order derivative in the given polynomial is 2 .</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2011 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">OnlyI</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-function-f-x-x-3-1-x-1-1-i-f-x-is-increasing-in-1-1-ii-f-x-has-noroot-in-1-1-which-of-the-statemen

slug:
consider-the-following-statements-in-respect-of-the-function-f-x-x-3-1-x-1-1-i-f-x-is-increasing-in-1-1-ii-f-x-has-noroot-in-1-1-which-of-the-statemen-99263

question:
<div class="question">Consider the following statements in respect of the function $f(x)=x^{3}-1, x \in[-1,1] \quad$<br/>I $f(x)$ is increasing in $[-1,1]$<br/>II $f(x)$ has noroot in $(-1,1)$. Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">OnlyI</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">Only II</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both I and II</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither I nor II</span> </li> </ul>

answer:
<div class="solution">Since $f(x)$ is an increasing function in $[-1,1]$ and $i t$ has a root in $(-1,1)$. $\therefore \quad$ Only statement I is correct.</div>


tags: ['Mathematics', 'Quadratic Equation', 'NDA', 'NDA 2015 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-given-equation-x-2-2-2-8-x-2-6-x-x-2-2-1all-the-roots-of-the-equation-are-complex-2the-sum-of-all-

slug:
consider-the-following-statements-in-respect-of-the-given-equation-x-2-2-2-8-x-2-6-x-x-2-2-1all-the-roots-of-the-equation-are-complex-2the-sum-of-all-73128

question:
<div class="question">Consider the following statements in respect of the given<br/>equation <br/>$\left(x^{2}+2\right)^{2}+8 x^{2}=6 x\left(x^{2}+2\right)$<br/>1.All the roots of the equation are complex<br/>2.The sum of all the roots of the equation is 6 Which of the above statements is/are correct?</div>

answer:
<div class="solution">$\quad\left(x^{2}+2\right)^{2}+8 x^{2}=$2<br/>$+8 x^{2}$<br/>$y^{2}-6 x y+8 x$<br/>$y=\frac{6 x \pm \sqrt{36 x^{2}-32 x^{2}}}{2}$<br/>$y=4 x$<br/>At<br/>$x^{2}-4 x+2=0$<br/>Discriminant, $\mathrm{D}=$<br/>Roots are real Sum of roots $=-(-4)$<br/>At<br/>$\mathrm{D}=4-8=-$<br/>Roots are complex Sum of roots $=$ Sum of all roots $=4+2=6$ only statement 2 is (b)</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-matrix-a-0-1-2-1-0-3-2-3-0-2014-i-1-the-matrix-a-is-skewsymmetric-2-the-matrix-a-is-symmetric-3-th

slug:
consider-the-following-statements-in-respect-of-the-matrix-a-0-1-2-1-0-3-2-3-0-2014-i-1-the-matrix-a-is-skewsymmetric-2-the-matrix-a-is-symmetric-3-th-35663

question:
<div class="question">Consider the following statements in respect of the matrix<br/>$A=\left[\begin{array}{ccc}0 &amp; 1 &amp; 2 \\ -1 &amp; 0 &amp; -3 \\ -2 &amp; 3 &amp; 0\end{array}\right] \quad[2014-I]$<br/>$1.$ The matrix $A$ is skew-symmetric.<br/>$2.$ The matrix $A$ is symmetric.<br/>$3.$ The matrix $A$ is invertible. <br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 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">3 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 and 3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2 and 3</span> </li> </ul>

answer:
<div class="solution">$A=\left[\begin{array}{ccc}0 &amp; 1 &amp; 2 \\ -1 &amp; 0 &amp; -3 \\ -2 &amp; 3 &amp; 0\end{array}\right]$<br/>$\mathrm{A}^{\mathrm{T}}=\left[\begin{array}{ccc}0 &amp; -1 &amp; -2 \\ 1 &amp; 0 &amp; 3 \\ 2 &amp; -3 &amp; 0\end{array}\right]=-\left[\begin{array}{ccc}0 &amp; 1 &amp; 2 \\ -1 &amp; 0 &amp; -3 \\ -2 &amp; 3 &amp; 0\end{array}\right]=-\mathrm{A}$<br/>Hence, $\mathrm{A}$ is skew symmetric matrix<br/>$|A|=\left|\begin{array}{ccc}0 &amp; 1 &amp; 2 \\ -1 &amp; 0 &amp; -3 \\ -2 &amp; 3 &amp; 0\end{array}\right|=1(-6)-2(-3)=-6+6=0$<br/>Therefore A is non-invertible.</div>


tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2011 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-square-matrices-a-and-b-of-same-order-1-a-and-b-are-nonzero-and-a-b-0-either-a-0-or-b-0-2-a-b-0-a-

slug:
consider-the-following-statements-in-respect-of-the-square-matrices-a-and-b-of-same-order-1-a-and-b-are-nonzero-and-a-b-0-either-a-0-or-b-0-2-a-b-0-a-17999

question:
<div class="question">Consider the following statements in respect of the square matrices $A$ and $B$ of same order: <br/>$1.$ $A$ and $B$ are non-zero and $A B=0 \Rightarrow$<br/>either $|A|=0$ or $|B|=0$<br/>$2.$ $A B=0 \Rightarrow A=0$ or $B=0$<br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 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">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2 .</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">If the matrix $A B$ is zero then it is not necessary that either $A=0$ or $B=0$ therefore statement 2 is incorrect. Let $A B=0$<br/>$\Rightarrow|A B|=0$<br/>$\Rightarrow|A||B|=0 \Rightarrow$ either $|A|=0$ or $|B|=0$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-in-respect-of-the-vectors-u-1-1-2-3-u-2-2-3-1-u-3-1-3-2-and-u-4-4-6-2-i-u-1-is-parallel-to-u-4-ll-ii-u-2-is-parallel

slug:
consider-the-following-statements-in-respect-of-the-vectors-u-1-1-2-3-u-2-2-3-1-u-3-1-3-2-and-u-4-4-6-2-i-u-1-is-parallel-to-u-4-ll-ii-u-2-is-parallel-41684

question:
<div class="question">Consider the following statements in respect of the vectors<br/>$\overrightarrow{\mathbf{u}_{1}}=(1,2,3), \overrightarrow{\mathbf{u}_{2}}=(2,3,1), \overrightarrow{\mathbf{u}_{3}}=(1,3,2)$ and $\overrightarrow{\mathbf{u}_{4}}=(4,6,2)$<br/>I. $\overrightarrow{\mathbf{u}}_{1}$ is parallel to $\overrightarrow{\mathbf{u}_{4}}$.<br/>$\overrightarrow{\mathrm{ll}}$<br/>$\rightarrow$<br/>II. $\mathbf{u}_{2}$ is parallel to $\mathbf{u}_{4}$.<br/>\Pi. $\overrightarrow{\mathbf{u}_{2}}$ is parallel to $\overrightarrow{\mathbf{u}_{3}}$.<br/>Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Only I</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Only II</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">OnlyIII</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both I and III</span> </li> </ul>

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

answer:
<div class="solution">Since, $\vec{u}_{4}=2 \vec{u}_{2}$<br/>$\therefore \vec{u}_{2}$ is parallel to $\vec{u}_{4}$<br/>Hence, only statement II is correct.</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-l-if-120-0-then-sec-tan-1-is-positive-ii-if-120-0-then-cosec-cot-is-negative-which-of-the-statements-given-above-isa

slug:
consider-the-following-statements-l-if-120-0-then-sec-tan-1-is-positive-ii-if-120-0-then-cosec-cot-is-negative-which-of-the-statements-given-above-isa-39789

question:
<div class="question">Consider the following statements <br/>L. If $\theta=1200^{\circ}$, then $(\sec \theta+\tan \theta)^{-1}$ is positive.<br/>II. If $\theta=1200^{\circ}$, then $(\operatorname{cosec} \theta-\cot \theta)$ is negative.<br/>Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">II only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both I and II</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Neither Inor II</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">Neither Inor II</span> </div>

answer:
<div class="solution">Statement-I: Let $\theta=1200^{\circ}$<br/>Consider $(\sec \theta+\tan \theta)^{-1}=\frac{1}{\sec \theta+\tan \theta}$<br/>$=\frac{\sec \theta-\tan \theta}{(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)}$<br/>$=\frac{\sec \theta-\tan \theta}{\left(\sec ^{2} \theta-\tan ^{2} \theta\right)}=\sec \theta-\tan \theta$<br/>When $1200 \div 360$<br/>we get remainder as $120^{\circ}$ Now, put $\theta=120^{\circ}$<br/>$\Rightarrow(\sec \theta+\tan \theta)^{-1}=\sec 120^{\circ}-\tan 120^{\circ}$<br/>$=\frac{1}{\cos 120^{\circ}}-\tan \left(90^{\circ}+30^{\circ}\right)$<br/>$=\frac{1}{\cos (90+30)}-\cot 30^{\circ}$<br/>$=\frac{1}{-\sin 30^{\circ}}+\sqrt{3}$<br/>$=-2+\sqrt{3} \quad$ which is negative<br/>Now, Statement-II :<br/>Consider $\operatorname{cosec} \theta-\cot \theta$<br/>$$<br/>\begin{array}{l}<br/>=\operatorname{cosec} 120^{\circ}-\cot 120^{\circ} \\<br/>=\frac{1}{\sin (90+30)}-\cot (90+30) \\<br/>=\frac{1}{\cos 30^{\circ}}+\tan 30^{\circ} \\<br/>=\frac{2}{\sqrt{3}}+\frac{1}{\sqrt{3}}=\frac{3}{\sqrt{3}}<br/>\end{array}<br/>$$<br/>which is positive. Hence, both statements are incorrect.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2010 (Phase 1)']
answer: <div class="solution">Both (1) and (2) statements are correct.</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-related-to-a-variable-x-having-a-binomial-distribution-b-x-n-p-1-if-p-2-1-then-the-distribution-is-symmetrical-2-p-r

slug:
consider-the-following-statements-related-to-a-variable-x-having-a-binomial-distribution-b-x-n-p-1-if-p-2-1-then-the-distribution-is-symmetrical-2-p-r-91942

question:
<div class="question">Consider the following statements related to a variable $X$ having a binomial distribution $b_{x}(n, p)$<br/>$1.$ If $p=\frac{1}{2}$, then the distribution is symmetrical.<br/>$2.$ $\quad p$ remaining constant, $P(X=r)$ increases as $n$ increases. Which of the statements given above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither 1 nor 2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and 2 .</span> </div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-related-to-the-nature-of-bayes-theorem-1-bayes-theorem-is-a-formula-for-computation-of-a-conditional-probability-2-b

slug:
consider-the-following-statements-related-to-the-nature-of-bayes-theorem-1-bayes-theorem-is-a-formula-for-computation-of-a-conditional-probability-2-b-62504

question:
<div class="question">Consider the following statements related to the nature of Bayes' theorem<br/>$1.$ Bayes' theorem is a formula for computation of a conditional probability.<br/>$2.$ Bayes' theorem modifies an assumed probability of an event in the light of a related event which is observed. Which of the statements given above is/are correct?</div>

answer:
<div class="solution">Baye's theorem says If $A_{1}, A_{2}, \ldots, A_{n}$ are $(n)$ mutuallyexclusive and exhaustive events in sample space $S$ and $E$ is any event is $S$ intersecting events $A_{i}($ iz $1,2, \ldots, n)$ such that $\mathrm{P}(\mathrm{E}) \neq 0$ then $P\left(A_{i} / E\right)=\frac{P\left(E / A_{i}\right)}{P(E)}$<br/>$=\frac{P\left(A_{i}\right) P\left(E / A_{i}\right)}{\sum_{|2|}^{n} P\left(A_{i}\right) P\left(E / A_{i}\right)}$<br/>Thus, both statement 1 and 2 are correct.</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2017 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-statement-1-range-is-not-a-good-measure-of-dispersion-statement-2-range-is-highly-affected-by-the-existence-of-extre

slug:
consider-the-following-statements-statement-1-range-is-not-a-good-measure-of-dispersion-statement-2-range-is-highly-affected-by-the-existence-of-extre-60352

question:
<div class="question">Consider the following statements:<br/>Statement $1:$ Range is not a good measure of dispersion. Statement 2 : Range is highly affected by the existence of extreme values Which one of the following is correct in respect of the above statements?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement</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">Both Statement 1 and Statement 2 are correct bui Statement 2 is not the correct explanation of Statement</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Statement 1 is correct but Statement 2 is not correct</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Statement 2 is correct but Statement 1 is not correct</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement</span> </div>


tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-statement-1-the-function-f-ir-ir-such-that-f-x-x-3-for-all-x-ir-is-oneone-statement-2-f-a-f-b-for-all-a-b-ir-if-the-

slug:
consider-the-following-statements-statement-1-the-function-f-ir-ir-such-that-f-x-x-3-for-all-x-ir-is-oneone-statement-2-f-a-f-b-for-all-a-b-ir-if-the-27230

question:
<div class="question">Consider the following statements :<br/>Statement $1:$ The function $\mathrm{f}: \mathrm{IR} \rightarrow \mathrm{IR}$ such that $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}$ for all $\mathrm{x} \in \mathrm{IR}$ is one-one.<br/>Statement $2: \mathrm{f}(\mathrm{a}) \Rightarrow \mathrm{f}(\mathrm{b})$ for all $\mathrm{a}, \mathrm{b} \in \mathrm{IR}$ if the function $\mathrm{f}$ is<br/>one-one. Which one of the following is correct in respect of the above statements?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Both the statements are true and Statement 2 is the correct explanation of Statement 1 .</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">Both the statements are true and Statement 2 is not the correct explanation of Statement 1 .</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Statement 1 is true but Statement 2 is false.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Statement 1 is false but Statement 2 is true.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both the statements are true and Statement 2 is the correct explanation of Statement 1 .</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}$<br/>$\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{x}_{1}^{3}$<br/>$\mathrm{f}\left(\mathrm{x}_{2}\right)=\mathrm{x}_{2}^{3}$<br/>if $\mathrm{x}_{1}=\mathrm{x}_{2}$<br/>then $\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)$<br/>So it is one-one function<br/>Hence option (a) is correct.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-statement-i-if-the-line-segment-joining-the-points-p-m-n-and-q-r-s-subtends-an-angle-at-the-origin-then-cos-m-2-n-2-

slug:
consider-the-following-statements-statement-i-if-the-line-segment-joining-the-points-p-m-n-and-q-r-s-subtends-an-angle-at-the-origin-then-cos-m-2-n-2-79885

question:
<div class="question">Consider the following statements:<br/>Statement $\mathbf{I}:$ If the line segment joining the points $\mathrm{P}(\mathrm{m}, \mathrm{n})$ and $\mathrm{Q}(\mathrm{r}, \mathrm{s})$ subtends an angle $\alpha$ at the<br/>$$<br/>\text { origin, then } \cos \alpha=\frac{\mathrm{ms}-\mathrm{nr}}{\sqrt{\left(\mathrm{m}^{2}+\mathrm{n}^{2}\right)\left(\mathrm{r}^{2}+\mathrm{s}^{2}\right)}} \text { . }<br/>$$<br/>Statements II : In any triangle $\mathrm{ABC}$, it is true that<br/>$$<br/>a^{2}=b^{2}+c^{2}-2 b c \cos A<br/>$$<br/>What of the following is correct in respect of the above two statements?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Both Statement I and Statement II are true, but Statement II is not the correct explanation of Statement I</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Statement I is true, but Statement II is false</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Statement I is false, but Statement II is true</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">Statement I is false, but Statement II is true</span> </div>

answer:
<div class="solution">$\cos \alpha=\frac{m r+n s}{\sqrt{m^{2}+n^{2}} \sqrt{r^{2}+s^{2}}}$<br/>Statement 1 is false, statement 2 is true.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-statement-i-x-sin-x-for-all-x-0-statement-ii-f-x-x-sin-x-is-an-increasing-function-for-all-x-0-which-one-of-the-foll

slug:
consider-the-following-statements-statement-i-x-sin-x-for-all-x-0-statement-ii-f-x-x-sin-x-is-an-increasing-function-for-all-x-0-which-one-of-the-foll-87834

question:
<div class="question">Consider the following statements:<br/>Statement I: $x&gt;\sin x$ for all $x&gt;0$<br/>Statement II: $f(\mathrm{x})=\mathrm{x}-\sin \mathrm{x}$ is an increasing function for all $\mathrm{x}&gt;0$<br/>Which one of the following is correct in respect of the above statements?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Both Statement I and Statement II are true and Statement II is the correct explanation of Statement II.</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">Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement $\mathrm{I}$.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Statement I is true but Statement II is false</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Statement I is false but Statement II is true</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both Statement I and Statement II are true and Statement II is the correct explanation of Statement II.</span> </div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2013 (Phase 2)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-the-algebraic-sum-of-the-deviations-of-a-set-of-n-value-from-its-arithmetic-mean-is-zere-2-in-the-case-of-frequency-

slug:
consider-the-following-statements-the-algebraic-sum-of-the-deviations-of-a-set-of-n-value-from-its-arithmetic-mean-is-zere-2-in-the-case-of-frequency-54642

question:
<div class="question">Consider the following statements<br/>The algebraic sum of the deviations of a set of $n$ value:<br/>from its arithmetic mean is zere 2 . In the case of frequency distribution, mode is the value of variable which corresponds to maximum frequency. Which of the statements above given is/are correct</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and</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">Neither $1 \mathrm{nol}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and</span> </div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-the-appropriate-number-of-classes-while-constructing-a-frequency-distribution-should-be-chosen-such-that-1-the-class

slug:
consider-the-following-statements-the-appropriate-number-of-classes-while-constructing-a-frequency-distribution-should-be-chosen-such-that-1-the-class-70968

question:
<div class="question">Consider the following statements The appropriate number of classes while constructing a frequency distribution should be chosen such that<br/> 1 . the class-frequency first increases to a peak and then declines.<br/> 2 . the class-frequency should cluster around the class mid point Which of the statements given is/are correct?</div>

answer:
<div class="solution">The appropriate number of classes while constructing a frequency distribution should be chosen such that the class frequency should cluster around the class mid point</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch1_P2.db
title:
consider-the-following-statements-there-are-infinitely-many-rational-numbers-between-two-distinct-1integers-2rational-numbers-3real-numbers-which-of-t

slug:
consider-the-following-statements-there-are-infinitely-many-rational-numbers-between-two-distinct-1integers-2rational-numbers-3real-numbers-which-of-t-51317

question:
<div class="question">Consider the following statements There are infinitely many rational numbers between two<br/>distinct<br/>1.integers<br/>2.rational numbers<br/>3.real numbers Which of the statements above are correct</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Only 1 and 2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Only 2 and 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Only 1 and 3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1,2 and 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">There are infinitely many rational numbers between two distinct integers, so, statement 1 is correct. Same is true in case of two distinct rational numbers and real numbers. So, statement $(2)$ and $(3)$ are also correct.</div>


