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31fb90ca-bd84-4ca9-9b71-f32add59351d | If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: | 3 | null | 8 | 7 | 5 | 6 | MCQ |
f3c8fe19-bd8c-4f37-9e11-2b5930e3c8c6 | The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is | 3 | null | 900 | 920 | 930 | 950 | MCQ |
c355123b-d38e-40f8-942f-38ad395686a8 | The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is | 17,280 | null | null | null | null | null | Numerical |
9c014fa2-7f21-45b7-9d92-468f1b26fbdc | The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to | 15 | null | null | null | null | null | Numerical |
25c19494-d9a2-408c-ade3-38e0a55aa8f2 | Let \( \alpha, \beta \) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to | 31 | null | null | null | null | null | Numerical |
716dfedd-e03b-472b-b096-06aa266e154c | Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to | 2 | null | $3 + \sqrt{3}$ | $4$ | $4 - \sqrt{3}$ | $3$ | MCQ |
8ea3fe25-f9be-4a1f-bda3-27f56bb14b73 | Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: | 2 | null | $17$ | $21$ | $56$ | $42$ | MCQ |
2a299a1a-0c06-4a60-ac2f-6bdaa40b0fc8 | Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is | 2 | null | $\frac{1 - \sqrt{3}}{\sqrt{2}}$ | $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ | $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ | $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$ | MCQ |
d46a273c-4f2e-406e-8349-7ebd51134569 | For some $n
eq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: | 3 | null | $20$ | $10$ | $35$ | $70$ | MCQ |
1df5bc52-fb7d-4f2b-85ce-28e0f668fd1e | The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to | 1 | null | $14$ | $21$ | $28$ | $7$ | MCQ |
36122fa4-0d5f-4c76-bd83-d0725f7934a9 | Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is | 2 | null | $5$ | $5\sqrt{5}$ | $5\sqrt{6}$ | $10$ | MCQ |
9fb4b7b3-0406-4aa9-b10b-3911e5b9f686 | Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to | 3 | null | $84$ | $113$ | $91$ | $101$ | MCQ |
70403eb0-a5a1-4b96-aa65-89a15344632f | For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is | 3 | null | 9 | 5 | 7 | 4 | MCQ |
c826c0a1-6d4c-48c3-8879-78bd54e24c1d | The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to | 4 | null | 7 | 5 | 24/5 | 20/3 | MCQ |
0b48edf8-5c2a-437c-9ccd-a97230a90e3d | Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is | 4 | null | 20 | 90 | 45 | 25 | MCQ |
0eb94d01-6c3e-47e4-84b4-ea1767364fb9 | Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to | 3 | null | 5 | 3 | 4 | 6 | MCQ |
1eced83e-3ea5-474f-9a8a-033ba116a923 | If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is | 4 | null | \(I(19, 27)\) | \(I(9, 1)\) | \(I(1, 13)\) | \(I(9, 13)\) | MCQ |
4a14b483-58b6-4bca-9bf7-426433b14c59 | \(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is | 2 | null | \(\frac{9}{17}\) | \(\frac{9}{17}\) | \(\frac{9}{17}\) | \(\frac{8}{17}\) | MCQ |
0c6f607d-54d7-466d-be1c-db64fc917a2e | Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4x + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to | 2 | null | 92 | 118 | 102 | 108 | MCQ |
a857fdf6-b070-4f2d-a94c-7f592eeb3779 | Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to | 2 | null | \(\sqrt{15} \div 2\) | \(\frac{1}{2} \sqrt{\frac{3}{2}}\) | \(2\sqrt{2}\) | \(\sqrt{\frac{14}{3}}\) | MCQ |
74a5b313-5de1-4b42-afac-e4b11c1d75e0 | \(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is: | 4 | null | \(\frac{1}{2}\) | \(-\frac{1}{2}\) | \(-1\) | Does not exist | MCQ |
4eabea93-8e5f-433c-914f-3757e0201d82 | Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: | 4 | null | \( \frac{90}{11} \) | \( \frac{85}{11} \) | \( \frac{61}{12} \) | \( \frac{567}{121} \) | MCQ |
a093afec-18f3-4da4-ae18-f21d8f60edb8 | Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to | 1 | null | \( \sqrt{\frac{11}{6}} \) | \( \frac{1}{3\sqrt{2}} \) | \( 16 \) | \( 18 \) | MCQ |
aa607395-f954-4395-99bd-bf683bb6e0f6 | Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______. | 5,120 | null | null | null | null | null | Numerical |
ef0e84d8-2585-4467-b206-30704260a218 | Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: | 3 | null | 8750 | 9100 | 8925 | 8575 | MCQ |
42221b8f-55a0-4bd7-a037-6e5173757fe9 | If the system of equations $2x + \\lambda y + 5z = 5$ has infinitely many solutions, then $\\lambda + \\mu$ is equal to:
$4x + 3y + \\mu z = 3$ | 3 | null | 13 | 10 | 12 | 11 | MCQ |
6d3d3618-c7da-4b3a-8383-66b5b182ab6b | Let $A = \{ x \in (0, \pi) - \{ \frac{\pi}{2} \} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: | 2 | null | 4 | 8 | 6 | 2 | MCQ |
c8897713-8999-444b-84d6-3a54ba0b823d | The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: | 1 | null | $1 - \\\log_e 2$ | $\\\\\log_e 2$ | $1 + \\\\log_e 2$ | $2 \\\\log_e 2 - 1$ | MCQ |
35b0bdd6-4e7e-4ed5-86e3-484752574845 | The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:
$25x + 101y = 176$ | 1 | null | 48x + 25y = 169 | 5x + 16y = 31 | 4x + 122y = 134 | 4x + 122y = 134 | MCQ |
f602ed0f-1b06-458b-8ca3-2bf6c12b4f42 | Let the point $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: | 3 | null | 44 | 22 | 33 | 55 | MCQ |
9e392e8e-9769-4730-8c3c-be055a34abcb | Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: | 1 | null | 39 | 19 | 29 | 23 | MCQ |
9ea00ba8-51c4-4d92-b9fa-4f615e9937b2 | If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: | 2 | null | \frac{6}{7} | 6 | \frac{1}{7} | 1 | MCQ |
9f7ee42f-a422-42c0-a0cd-ff4f14da835a | Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: | 2 | null | 6 | 8 | 9 | 7 | MCQ |
ff241288-2eaf-4e9a-8c75-5e9c5bbf48ec | Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is: | 3 | null | $\frac{3}{16}$ | $\frac{9}{16}$ | $\frac{11}{16}$ | $\frac{5}{16}$ | MCQ |
01a34c0e-d55e-41c9-881a-c5007a131d39 | Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: | 1 | null | 3 | 4 | 1 | 6 | MCQ |
b8677182-5ae1-4aed-ba9a-f5b13c00f01a | Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: | 1 | null | 2\sqrt{14} | \mathbf{v} | \sqrt{7} | 2\sqrt{7} | MCQ |
92212052-fd2a-4fdb-9567-d15d7e04b3e2 | The number of real solution(s) of the equation \(x^2 + 3x + 2 =
\min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: | 3 | null | 1 | 0 | 2 | 3 | MCQ |
66422792-577f-48e3-8577-6ce01f4feeb0 | The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: | 4 | null | Neither one-one nor onto | Onto but not one-one | Both one-one and onto | One-one but not onto | MCQ |
8b3060d6-97c3-427e-802a-831bef7af864 | In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: | 3 | null | 525 | 510 | 515 | 505 | MCQ |
37be8789-ed18-4591-910c-57c0774c29c8 | Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: | 3 | null | 22 | 20 | 21 | 19 | MCQ |
8999ad58-5192-4896-bbd5-131e2f36c4a7 | Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_e(x - 2) - x^2 + ax + 1$ is strictly increasing and $(b, c)$ be the largest open interval, in which the function $g(x) = (x - 1)^3(x + 2 - a)^2$ is strictly decreasing. Then $100(a + b - c)$ is equal to: | 2 | null | 420 | 360 | 160 | 280 | MCQ |
5dc239df-f2e9-4f64-956d-7cf33138be50 | For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: | 1 | null | 16 | 25 | 9 | 36 | MCQ |
fb786c03-50bb-45a1-8a07-5ba97cb76d37 | If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: | 1 | null | $\pi$ | $0$ | $\pi - (\alpha + \beta + \gamma)$ | $3\pi$ | MCQ |
984ecf93-7530-44c2-b9fc-40badc1010a5 | Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________. | 55 | null | null | null | null | null | Numerical |
f0c34e35-de11-40b6-b844-4aceda9c9c64 | Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then | 2 | null | area of triangle ABO is \( \frac{11}{3} \) | ABO is an obtuse angled isosceles triangle | area of triangle ABO is \( \frac{11}{4} \) | ABO is a scalene triangle | MCQ |
386008a7-9f2d-417a-bc5e-9da6c7c9c48a | Let $ f : \mathbb{R} \to \mathbb{R} $ be a function defined by $ f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. $ If $ f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy $, then the value of $ 28 \sum_{i=1}^{5} |f(i)| $ is | 4 | null | 545 | 715 | 735 | 675 | MCQ |
3abb0ea3-69b6-4fe5-bc92-c24c1341f8de | Let $ ABCD $ be a trapezium whose vertices lie on the parabola $ y^2 = 4x $. Let the sides $ AD $ and $ BC $ of the trapezium be parallel to $ y $-axis. If the diagonal $ AC $ is of length $ \frac{25}{4} $ and it passes through the point $ (1, 0) $, then the area of $ ABCD $ is | 1 | null | $ \frac{73}{8} $ | $ \frac{25}{9} $ | $ \frac{16}{8} $ | $ \frac{75}{8} $ | MCQ |
ce69f97f-67ad-4564-96b5-d19b006f6e1b | The sum of all local minimum values of the function
\[
f(x) = \begin{cases}
1 - 2x, & x < -1 \\
\frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\
\frac{1}{12}(x - 4)(x - 5), & x > 2
\end{cases}
\]
is | 1 | null | \( \frac{137}{72} \) | \( \frac{131}{72} \) | \( \frac{137}{72} \) | \( \frac{167}{72} \) | MCQ |
7a54308e-bc39-4255-b8dc-4b50afb12022 | Let $^nC_{r-1} = 28, ^nC_r = 56$ and $^nC_{r+1} = 70$. Let $A(4
obreak{\cos t}, 4
obreak{\sin t}), B(2
obreak{\sin t}, -2
obreak{\cos t})$ and $C(3r - n, r^2 - n - 1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$, is the locus of the centroid of triangle $ABC$, then $\alpha$ equals | 4 | null | 6 | 18 | 8 | 20 | MCQ |
fcb2d563-8ac4-452f-be89-ce259c8146c1 | If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to | 4 | null | 1.81\sqrt{2} | 41 | 82 | \frac{81}{2} | MCQ |
d323f007-a281-4f15-8c52-e3b78f1b9fb8 | Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals | 2 | null | $\frac{1}{16}$ | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{15}{16}$ | MCQ |
2400477c-a743-4733-95d2-a4f76884f8f5 | If the image of the point \( (4, 4, 3) \) in the line \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3} \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to | 1 | null | 9 | 12 | 7 | 8 | MCQ |
13d54bd0-0372-4112-a983-0f35933d161b | \[\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\] is equal to: | 2 | null | \(1\) | \(0\) | \(\frac{32}{65}\) | \(\frac{33}{65}\) | MCQ |
669b1282-46df-4444-b097-abc0e94d1c6f | Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), | 3 | null | \(\text{both are true}\) | \(\text{only (S2) is true}\) | \(\text{only (S1) is true}\) | \(\text{both are false}\) | MCQ |
e8e525f0-719e-4737-b196-ec4c9442a18e | The area (in sq. units) of the region \( \{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \} \) is | 2 | null | \( \frac{80}{3} \) | \( \frac{44}{3} \) | \( \frac{32}{3} \) | \( \frac{17}{3} \) | MCQ |
3c8d4be8-c367-4856-bdc1-bfe37e60d677 | The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is | 3 | null | 3(3 - \sqrt{2}) | 6(3 - \sqrt{2}) | 6(2 - \sqrt{2}) | 3(2 - \sqrt{2}) | MCQ |
577660f5-d79b-4d00-9af8-d50b7849743f | Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{m}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then
\(5m \sum_{r=m}^{m+2} T_r\) is equal to | 2 | null | 98 | 126 | 142 | 112 | MCQ |
d688540d-d072-44a1-b481-a459af38593f | Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is | 1 | null | \(\frac{28}{75}\) | \(\frac{18}{25}\) | \(\frac{26}{75}\) | \(\frac{14}{25}\) | MCQ |
403a27b0-6d1e-49c4-924d-8722f6a2915f | Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to | 1 | null | 1 | 3 | 6 | 2 | MCQ |
f15c71bb-d3e0-42f2-b7da-a93f795013ef | If $\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}$, then $(\alpha + \beta)^2$ equals | 4 | null | 64 | 196 | 144 | 100 | MCQ |
bec8e309-4e7f-4767-9f29-a2c36dac2786 | Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to | 2 | null | $$\frac{300}{-2}$$ | $$\frac{300}{2}$$ | $$\frac{300}{4}$$ | $$\frac{300}{-4}$$ | MCQ |
4dfa6e24-a0d7-4203-b672-58d09c63870b | The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is | 4 | null | 4608 | 5720 | 5719 | 4607 | MCQ |
01291000-d3a5-41f5-aac9-1a90237dadf5 | The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: | 2 | null | reflexive and symmetric but not transitive | an equivalence relation | symmetric and transitive but not reflexive | reflexive and transitive but not symmetric | MCQ |
5df8a224-1e66-4ab3-a733-446cb3d5df54 | Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals | 5 | null | null | null | null | null | Numerical |
70401e79-9f72-4a97-9c2d-6ff0463d1a59 | Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let
\[
S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\
S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\
S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}.
\]
If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals | 1,613 | null | null | null | null | null | Numerical |
70575064-0b51-4839-8155-1945b70779e0 | Let $E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$'s are constructed such that their centres and eccentricities are same as that of $E_1$, and the length of minor axis of $E_i$ is the length of major axis of $E_{i+1}(i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right)$ is equal to | 54 | null | null | null | null | null | Numerical |
25f7b0a3-85b7-4842-bc13-f36f8a07d9a2 | Let $ \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{d} = \vec{a} \times \vec{b} $. If $ \vec{c} $ is a vector such that $ \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 $ and the angle between $ \vec{d} $ and $ \vec{c} $ is $ \frac{\pi}{4} $, then $ |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 $ is equal to | 6 | null | null | null | null | null | Numerical |
6a784d07-5204-4c1c-878d-1b5058eb9c9c | Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: | 3 | null | 127 | 258 | 65 | 2049 | MCQ |
f165565c-7d21-4031-8cec-9f41449f2a39 | If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: | 1 | null | 26 | 18 | 23 | 16 | MCQ |
843fe6c2-b6ac-477f-b561-a71dd5df3469 | Let $A, B, C$ be three points in $xy$-plane, whose position vectors are given by $\sqrt{3}\hat{i} + \hat{j}, \hat{i} + \sqrt{3}\hat{j}$ and $\hat{i} + (1 - a)\hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\frac{a}{\sqrt{2}}$, then the sum of all the possible values of $a$ is: | 3 | null | 2 | 9/2 | 1 | 0 | MCQ |
3f46fd12-9c19-49e9-aebd-3fc5ecc67139 | Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: | 1 | null | 283 | 287 | 295 | 299 | MCQ |
56cce03c-cae5-4480-ae4e-2ca17508e9aa | Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: | 3 | null | \( (-\infty, -1] \cup [0, \infty) \) | \( (-\infty, -1] \cup [1, \infty) \) | \( (-\infty, \infty) \) | \( (-\infty, \infty) \setminus \{0\} \) | MCQ |
2470b736-8780-4e71-8c6a-ca9e0fcd4a1f | Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: | 1 | null | \( \frac{1}{2} \) | \( \frac{1}{4} \) | \( \frac{3}{5} \) | \( \frac{1}{5} \) | MCQ |
2459077d-12c1-45e7-83ae-c1d6ea3ab06e | If $ \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} $ = $ a\sqrt{3} + b $, $ a, b \in \mathbb{Z} $, then $ a^2 + b^2 $ is equal to: | 4 | null | 10 | 4 | 2 | 8 | MCQ |
948355f4-6765-49f3-86c3-51028dee123e | Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x) =
\int_0^x t f(t) \, dt$. If $g(x^2) = x^6 + x^7$, then value of $\sum_{r=1}^{15} f(r^3)$ is: | 4 | null | 270 | 340 | 320 | 310 | MCQ |
98f67cfb-1e05-4e92-bb4c-49fdebf46689 | Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to: | 2 | null | 30 | 31 | 32 | 29 | MCQ |
7af307a7-d2ac-4ef8-b48f-a2e6059c95d4 | Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: | 1 | null | \frac{4}{15} | \frac{1}{3} | \frac{2}{5} | \frac{4}{5} | MCQ |
8b5b6bb0-7a08-49a4-af8b-3fea283b05bd | Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \\in \\mathbb{R}$, $\\int_{x}^{2} x F'(x)\,dx = 6$ and $\\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \\int_{x}^{2} F(x)\,dx$ is equal to: | 1 | null | 11 | 13 | 15 | 9 | MCQ |
33f5a9c1-f95c-4b38-b86d-9636f14834e2 | For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right)$, then the value of $507S_{2025}$ is: | 2 | null | 540 | 675 | 1350 | 135 | MCQ |
6a8fac3c-20ca-4da7-be4b-00fdf3d812a1 | Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: | 2 | null | 7 | 6 | 1 | 9 | MCQ |
a1f3abd3-97f9-4a96-8965-6bddf9ac8b33 | If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: | 3 | null | $x + 9y = 36$ | $4x - 9y = 12$ | $6x - 9y = 20$ | $9x - 9y = 32$ | MCQ |
b9a37e33-f873-4d50-b919-9864d6fcfec7 | If $f(x) = \\int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \\frac{1}{1+t^4} dt$, $f(0) = -6$, then $f(1)$ is equal to: | 1 | null | $4 \\log_e 2 - 2$ | $2 - \\log_e 2$ | $\\log_e 2 + 2$ | $4 \\log_e 2 + 2$ | MCQ |
3b235e61-18ba-47aa-a627-bf5a37bdb464 | The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: | 2 | null | 2 \left( \frac{\pi}{2} - \frac{1}{3} \right) | \frac{\pi}{2} - \frac{1}{3} | \frac{\pi}{2} - \frac{1}{3} | \frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right) | MCQ |
a94bc062-64c8-4846-a7f8-6ceb620ae4ee | The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: | 4 | null | 54 | 44 | 41 | 66 | MCQ |
47c8b9a2-16cc-4302-be4c-ae7c30ea8d6c | If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: | 2 | null | 20 | 22 | 18 | 26 | MCQ |
911e6993-1f84-423b-9702-25e971cbe392 | If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: | 4 | null | $-2$ | $6$ | $-6$ | $2$ | MCQ |
1f39939e-71c6-4a74-8bea-f6c2d235d37d | Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: | 3 | null | $-2\sqrt{10}$ | $12$ | $6$ | $-6$ | MCQ |
7e9ca87b-0353-425d-b767-83cf4cff50f1 | If $y = y(x)$ is the solution of the differential equation, $\\sqrt{4 - x^2} \\frac{dy}{dx} = \\left(\\sin^{-1}\\left(\\frac{x}{2}\\right)\\right)^2 - y \\sin^{-1}\\left(\\frac{x}{2}\\right)$, $-2 \\leq x \\leq 2$, $y(2) = \\frac{x^2 - 8}{4}$, then $y(0)$ is equal to | 4 | null | 4 | $\\frac{\\pi^2}{6}$ | $\\frac{\\pi^2}{16}$ | $\\frac{\\pi^2}{4}$ | MCQ |
45a8920f-53e0-4308-ba55-52e6797819cf | Let $f(x) = \\lim_{x \\to \\infty} \\sum_{r=0}^{n} \\left(\\frac{\\tan(x/2^{r+1}) + \\tan^2(x/2^{r+1})}{1 - \\tan^2(x/2^{r+1})}\\right)$. Then $\\lim_{x \\to 0} \\frac{x - e^{-f(x)}}{x - f(x)}$ is equal to | 1 | null | $1$ | $0$ | $-1$ | does not exist | MCQ |
87886fd7-10e5-4ed6-acb0-e2f288266f84 | Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: | 1 | null | 100 | 120 | 110 | 90 | MCQ |
4a3af956-2936-4814-a7c5-ad78d38791b4 | Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: | 1 | null | 90 | 84 | 122 | 108 | MCQ |
842577b9-21fa-44e9-8071-82c1a6bb6254 | The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: | 4 | null | 2 | 3 | 1 | 4 | MCQ |
22a47813-1150-4bfb-8c1e-7779eb2311f2 | Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: | 2 | null | both reflexive and transitive but not symmetric | an equivalence relation | reflexive but neither symmetric nor transitive | both reflexive and symmetric but not transitive | MCQ |
9abb4939-e62b-4c95-b6ae-baa8b0f237b8 | Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: | 3 | null | 392 | 384 | 192 | 96 | MCQ |
d70df5b1-fbd0-4589-b004-9693965e1ff7 | Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: | 4 | null | 173 | 164 | 158 | 161 | MCQ |
b858d9a3-549d-424b-a0f5-3f33c1789778 | Let $ \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} $ and $ \vec{c} $ be a vector such that $ \vec{a} \times \vec{c} = \vec{c} \times \vec{b} $ and $ (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. $ Then the maximum value of $ |\vec{c}|^2 $ is: | 4 | null | 462 | 77 | 154 | 308 | MCQ |
a7ad8cd4-2aac-4155-9246-182dfb07b87d | The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: | 2 | null | 3 \log_e 4 | 4 \log_e 3 | 6 \log_e 4 | 2 \log_e 3 | MCQ |
9f3f478f-3e56-42e5-9b3e-6ad35534d3e1 | Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: | 4 | null | 4\sqrt{6} | 6\sqrt{6} | 18\sqrt{6}/5 | 24\sqrt{6}/5 | MCQ |
b1f11888-ceb0-442f-810a-0a51a834e287 | Let $ A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} $. If $ A_{ij} $ is the cofactor of $ a_{ij} $, $ C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 $, and $ C = [C_{ij}] $, then $ 8|C| $ is equal to: | 3 | null | 288 | 222 | 242 | 262 | MCQ |
3dd9de4c-0aa8-46af-be7b-f79cd0ee13e2 | Let $ |z_1 - 8 - 2i| \\\leq 1 $ and $ |z_2 - 2 + 6i| \\leq 2, z_1, z_2 \\in \\mathbb{C} $. Then the minimum value of $ |z_1 - z_2| $ is: | 4 | null | 13 | 10 | 3 | 7 | MCQ |
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