IIT JEE MAIN 2025 January Shift Session-1

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is:
(1) 6
(2) 7
(3) 5
(4) 4
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $f(x+y) = f(x) f(y)$ for all $x, y \in \mathbb{R}$ . If $f'(0) = 4a$ and $f$ satisfies $f''(x) - 3a f'(x) - f(x) = 0$ , $a > 0$ , then the area of the region $R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}$ is:
(1) $e^2 - 1$
(2) $e^4 + 1$
(3) $e^4 - 1$
(4) $e^2 + 1$
Let the triangle PQR be the image of the triangle with vertices $(1,3)$ , $(3,1)$ , and $(2,4)$ in the line $x + 2y = 2$ . If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$ , then $15(\alpha - \beta)$ is equal to:
(1) 24
(2) 19
(3) 21
(4) 22
Let $z_1, z_2, z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = -\frac{\pi}{4}$ , $\arg(z_2) = 0$ , and $\arg(z_3) = \frac{\pi}{4}$ . If $\left| z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} \right|^2 = \alpha + \beta \sqrt{2}$ , $\alpha, \beta \in \mathbb{Z}$ , then the value of $\alpha^2 + \beta^2$ is:
(1) 24
(2) 41
(3) 31
(4) 29
Using the principal values of the inverse trigonometric functions, the sum of the maximum and minimum values of $16 \left( \left( \sec^{-1} x \right)^2 + \left( \csc^{-1} x \right)^2 \right)$ is:
(1) $24 \pi^2$
(2) $18 \pi^2$
(3) $31 \pi^2$
(4) $22 \pi^2$
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$ , then the value of $64 \left( \mu + \sigma^2 \right)$ is:
(1) 51
(2) 48
(3) 32
(4) 64
Let $a_1, a_2, a_3, \ldots$ be a geometric progression of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$ , then $a_6$ is equal to:
(1) 628
(2) 526
(3) 784
(4) 812
Let $L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$ ?
(1) $\left( -\frac{5}{3}, -7, 1 \right)$
(2) $\left( 2, 3, \frac{1}{3} \right)$
(3) $\left( \frac{8}{3}, -1, \frac{1}{3} \right)$
(4) $\left( \frac{14}{3}, -3, \frac{22}{3} \right)$
The product of all solutions of the equation $e^{5 \left( \log_e x \right)^2 + 3} = x^8$ , $x > 0$ , is:
(1) $e^{8/5}$
(2) $e^{6/5}$
(3) $e^2$
(4) $e$
If $\sum_{r=1}^n T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$ , then $\lim_{n \to \infty} \sum_{r=1}^n \left( \frac{1}{T_r} \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac{2}{3}$
(4) $\frac{1}{3}$
From all the English alphabets, five letters are chosen and arranged in alphabetical order. The total number of ways in which the middle letter is ‘M’ is:
(1) 14950
(2) 6084
(3) 4356
(4) 5148
Let $x = x(y)$ be the solution of the differential equation $y^2 dx + \left( x - \frac{1}{y} \right) dy = 0$ . If $x(1) = 1$ , then $x\left( \frac{1}{2} \right)$ is:
(1) $\frac{1}{2} + e$
(2) $\frac{3}{2} + e$
(3) $3 - e$
(4) $3 + e$
Let the parabola $y = x^2 + p x - 3$ meet the coordinate axes at points $P$ , $Q$ , and $R$ . If the circle $C$ with center at $(-1, -1)$ passes through points $P$ , $Q$ , and $R$ , then the area of $\triangle PQR$ is:
(1) 4
(2) 6
(3) 7
(4) 5
A circle $C$ of radius 2 lies in the second quadrant and touches both coordinate axes. Let $r$ be the radius of a circle with center at $(2, 5)$ that intersects circle $C$ at exactly two points. If the set of all possible values of $r$ is the interval $(\alpha, \beta)$ , then $3 \beta - 2 \alpha$ is equal to:
(1) 15
(2) 14
(3) 12
(4) 10
Let $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$ , $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ , and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$ . Then $7 I_1 + 12 I_2$ is equal to:
(1) $2 \pi$
(2) $\pi$
(3) 1
(4) 2
Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x+y) = f(x) f'(y) + f'(x) f(y)$ for all $x, y \in \mathbb{R}$ . Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to:
(1) 2384
(2) 2525
(3) 5220
(4) 2406
Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}$ . Then $n(B)$ is equal to:
(1) 31
(2) 36
(3) 37
(4) 29
The area of the region inside the circle $(x - 2 \sqrt{3})^2 + y^2 = 12$ and outside the parabola $y^2 = 2 \sqrt{3} x$ is:
(1) $6 \pi - 8$
(2) $3 \pi - 8$
(3) $6 \pi - 16$
(4) $3 \pi + 8$
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$ , where $\gcd(m, n) = 1$ , then $m + n$ is equal to:
(1) 14
(2) 4
(3) 11
(4) 13
Let the foci of a hyperbola be $(1, 14)$ and $(1, -12)$ . If it passes through the point $(1, 6)$ , then the length of its latus rectum is:
(1) $\frac{25}{6}$
(2) $\frac{24}{5}$
(3) $\frac{288}{5}$
(4) $\frac{144}{5}$
Let the function $f(x) = \begin{cases} -3 a x^2 - 2, & x 1$ , $b \in \mathbb{R}$ . If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}$ , $\alpha, \beta \in \mathbb{Z}$ , then the value of $\alpha + \beta$ is.
If $\sum_{r=0}^{5} \frac{{11 \choose 2r+1}}{2r+2} = \frac{m}{n}$ , $\gcd(m, n) = 1$ , then $m - n$ is equal to.
Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \operatorname{adj}(-6 \operatorname{adj}(3 A))) = 2^{m+n} \cdot 3^{mn}$ , $m > n$ . Then $4 m + 2 n$ is equal to.
Let $L_1: \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0}$ and $L_2: \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{\alpha}$ , $\alpha \in \mathbb{R}$ , be two lines that intersect at point $B$ . If $P$ is the foot of the perpendicular from point $A(1, 1, -1)$ on $L_2$ , then the value of $26 \alpha (PB)^2$ is.
Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4 \hat{k}$ , $\lambda > 0$ , on the vector $\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If $|\vec{a} + \vec{c}| = 7$ , then the area of the parallelogram formed by vectors $\vec{b}$ and $\vec{c}$ is.
Let $\alpha, \beta, \gamma, \delta$ be the coefficients of $x^7, x^5, x^3, x$ respectively in the expansion of $\left( x + \sqrt{x^3 - 1} \right)^5 + \left( x - \sqrt{x^3 - 1} \right)^5$ , $x > 1$ . If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ , $\gamma u + \delta v = 20$ , then $u + v$ equals:
(1) 5
(2) 4
(3) 3
(4) 8
In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$ . The number of ways in which these girls and boys can stand in a queue such that all girls stand together, all boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:
(1) 144
(2) 72
(3) 96
(4) 120
Let $P(4, 4 \sqrt{3})$ be a point on the parabola $y^2 = 4 a x$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the feet of perpendiculars from $P$ and $Q$ respectively on the directrix, then the area of quadrilateral $PQMN$ is equal to:
(1) $\frac{263 \sqrt{3}}{8}$
(2) $17 \sqrt{3}$
(3) $\frac{343 \sqrt{3}}{8}$
(4) $\frac{34 \sqrt{3}}{3}$
For a $3 \times 3$ matrix $M$ , let $\operatorname{trace}(M)$ denote the sum of all diagonal elements. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and $\operatorname{trace}(A) = 3$ . If $B = \operatorname{adj}(\operatorname{adj}(2 A))$ , then $|B| + \operatorname{trace}(B)$ equals:
(1) 56
(2) 132
(3) 174
(4) 280
Suppose the number of terms in an arithmetic progression is $2k$ , $k \in \mathbb{N}$ . If the sum of all odd terms is 40, the sum of all even terms is 55, and the last term exceeds the first term by 27, then $k$ is equal to:
(1) 5
(2) 8
(3) 6
(4) 4
Let a line pass through points $P(-2, -1, 3)$ and $Q$ , and be parallel to vector $3 \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If the distance of point $Q$ from point $R(1, 3, 3)$ is 5, then the square of the area of $\triangle PQR$ is equal to:
(1) 136
(2) 140
(3) 144
(4) 148
If $\lim_{x \to \infty} \left( \left( \frac{e}{1-e} \right) \left( \frac{1}{e} - \frac{x}{1+x} \right) \right)^x = \alpha$ , then $\frac{\log_e \alpha}{1 + \log_e \alpha}$ equals:
(1) $e$
(2) $e^{-2}$
(3) $e^2$
(4) $e^{-1}$
Let $f(x) = \int_{0}^{x^2} \frac{t^2 - 8 t + 15}{e^t} \, dt$ , $x \in \mathbb{R}$ . Then the numbers of local maximum and local minimum points of $f$ , respectively, are:
(1) 2 and 3
(2) 3 and 2
(3) 1 and 3
(4) 2 and 2
The perpendicular distance of the line $\frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2}$ from point $P(2, -10, 1)$ is:
(1) 6
(2) $5 \sqrt{2}$
(3) $3 \sqrt{5}$
(4) $4 \sqrt{3}$
If $x = f(y)$ is the solution of the differential equation $\left( 1 + y^2 \right) + \left( x - 2 e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0$ , $y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ , with $f(0) = 1$ , then $f\left( \frac{1}{\sqrt{3}} \right)$ is equal to:
(1) $e^{\pi/4}$
(2) $e^{\pi/12}$
(3) $e^{\pi/3}$
(4) $e^{\pi/6}$
If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$ , where $C$ is the constant of integration, then $g\left( \frac{1}{2} \right)$ equals:
(1) $\frac{\pi}{6} \sqrt{\frac{e}{2}}$
(2) $\frac{\pi}{4} \sqrt{\frac{e}{2}}$
(3) $\frac{\pi}{6} \sqrt{\frac{e}{3}}$
(4) $\frac{\pi}{4} \sqrt{\frac{e}{3}}$
Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2 + (\cos \theta) x - 1 = 0$ , $\theta \in (0, 2 \pi)$ . If $m$ and $M$ are the minimum and maximum values of $\alpha_\theta^4 + \beta_\theta^4$ , then $16 (M + m)$ equals:
(1) 24
(2) 25
(3) 27
(4) 17
Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$ . Then the number of many-one functions $f: A \to B$ such that $1 \in f(A)$ is equal to:
(1) 127
(2) 151
(3) 163
(4) 139
If the system of linear equations $x + y + 2 z = 6$ , $2 x + 3 y + a z = a + 1$ , $-x - 3 y + b z = 2 b$ , where $a, b \in \mathbb{R}$ , has infinitely many solutions, then $7 a + 3 b$ is equal to:
(1) 9
(2) 12
(3) 16
(4) 22
Let $\vec{a}$ and $\vec{b}$ be two unit vectors with an angle of $\frac{\pi}{3}$ between them. If $\lambda \vec{a} + 2 \vec{b}$ and $3 \vec{a} - \lambda \vec{b}$ are perpendicular, then the number of values of $\lambda$ in $[-1, 3]$ is:
(1) 3
(2) 2
(3) 1
(4) 0
Let $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , $a > b$ , and $H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ . Let the distance between the foci of $E$ and $H$ be $2 \sqrt{3}$ . If $a - A = 2$ , and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$ , then the sum of the lengths of their latus rectums is equal to:
(1) 10
(2) 7
(3) 8
(4) 9
If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$ , and $P(A \mid B)$ and $P(B \mid A)$ are the roots of $12 x^2 - 7 x + 1 = 0$ , then $\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is:
(1) $\frac{5}{3}$
(2) $\frac{4}{3}$
(3) $\frac{9}{4}$
(4) $\frac{7}{4}$
The sum of all values of $\theta \in [0, 2 \pi]$ satisfying $2 \sin^2 \theta = \cos 2 \theta$ and $2 \cos^2 \theta = 3 \sin \theta$ is:
(1) $\frac{\pi}{2}$
(2) $4 \pi$
(3) $\frac{5 \pi}{6}$
(4) $\pi$
Let the curve $z (1 + i) + \overline{z} (1 - i) = 4$ , $z \in \mathbb{C}$ , divide the region $|z - 3| \leq 1$ into two parts of areas $\alpha$ and $\beta$ . Then $|\alpha - \beta|$ equals:
(1) $1 + \frac{\pi}{2}$
(2) $1 + \frac{\pi}{3}$
(3) $1 + \frac{\pi}{4}$
(4) $1 + \frac{\pi}{6}$
The area of the region enclosed by the curves $y = x^2 - 4 x + 4$ and $y^2 = 16 - 8 x$ is:
(1) $\frac{8}{3}$
(2) $\frac{4}{3}$
(3) 5
(4) 8
Let $y = f(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{x y}{x^2 - 1} = \frac{x^6 + 4 x}{\sqrt{1 - x^2}}$ , $-1 < x < 1$ , such that $f(0) = 0$ . If $6 \int_{-1/2}^{1/2} f(x) \, dx = 2 \pi - \alpha$ , then $\alpha^2$ is equal to.
Let $A(6, 8)$ , $B(10 \cos \alpha, -10 \sin \alpha)$ , and $C(-10 \sin \alpha, 10 \cos \alpha)$ be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ are its orthocenter and centroid, respectively, then $5 a - 3 h + 6 k + 100 \sin 2 \alpha$ is equal to.
Let the distance between two parallel lines be 5 units, and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to.
If $\sum_{r=1}^{30} \frac{r^2 \left( {30 \choose r} \right)^2}{{30 \choose r-1}} = \alpha \times 2^{29}$ , then $\alpha$ is equal to.
Let $A = \{1, 2, 3\}$ . The number of relations on $A$ , containing $(1, 2)$ and $(2, 3)$ , which are reflexive and transitive but not symmetric, is.
The value of $\int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left( \left( \log_e x \right)^2 + 1 \right)^{-1}}}{e^{\left( \left( \log_e x \right)^2 + 1 \right)^{-1}} + e^{\left( \left( 6 - \log_e x \right)^2 + 1 \right)^{-1}}} \right) dx$ is:
(1) $\log_e 2$
(2) 2
(3) 1
(4) $e^2$
Let $I(x) = \int \frac{dx}{(x - 11)^{\frac{11}{13}} (x + 15)^{\frac{15}{13}}}$ . If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}} \right)$ , $b, c \in \mathbb{N}$ , then $3 (b + c)$ is equal to:
(1) 40
(2) 39
(3) 22
(4) 26
If the function $f(x) = \begin{cases} \frac{2}{x} \left\{ \sin (k_1 + 1) x + \sin (k_2 - 1) x \right\}, & x 0 \end{cases}$ is continuous at $x = 0$ , then $k_1^2 + k_2^2$ is equal to:
(1) 8
(2) 20
(3) 5
(4) 10
If the line $3 x - 2 y + 12 = 0$ intersects the parabola $4 y = 3 x^2$ at points $A$ and $B$ , then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to:
(1) $\tan^{-1} \left( \frac{11}{9} \right)$
(2) $\frac{\pi}{2} - \tan^{-1} \left( \frac{3}{2} \right)$
(3) $\tan^{-1} \left( \frac{4}{5} \right)$
(4) $\tan^{-1} \left( \frac{9}{7} \right)$
Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $\left( \log_e 2, k \right)$ . If the curve satisfies the differential equation $2 (3 + y) e^{2 x} dx - \left( 7 + e^{2 x} \right) dy = 0$ , then $k$ is equal to:
(1) 16
(2) 8
(3) 32
(4) 4
Let $f(x) = \log_e x$ and $g(x) = \frac{x^4 - 2 x^3 + 3 x^2 - 2 x + 2}{2 x^2 - 2 x + 1}$ . Then the domain of $f \circ g$ is:
(1) $\mathbb{R}$
(2) $(0, \infty)$
(3) $[0, \infty)$
(4) $[1, \infty)$
Let the arc $AC$ of a circle subtend a right angle at the center $O$ . If point $B$ on arc $AC$ divides the arc such that $\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}$ , and $\overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB}$ , then $\alpha = \sqrt{2} (\sqrt{3} - 1) \beta$ is equal to:
(1) $2 - \sqrt{3}$
(2) $2 \sqrt{3}$
(3) $5 \sqrt{3}$
(4) $2 + \sqrt{3}$
If the first term of an arithmetic progression is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to:
(1) -1200
(2) -1080
(3) -1020
(4) -120
Let $P$ be the foot of the perpendicular from point $Q(10, -3, -1)$ on the line $\frac{x-3}{7} = \frac{y-2}{-1} = \frac{z+1}{-2}$ . Then the area of the right-angled triangle $PQR$ , where $R$ is the point $(3, -2, 1)$ , is:
(1) $9 \sqrt{15}$
(2) $\sqrt{30}$
(3) $8 \sqrt{15}$
(4) $3 \sqrt{30}$
Let $\left| \frac{\overline{z} - i}{2 \overline{z} + i} \right| = \frac{1}{3}$ , $z \in \mathbb{C}$ , be the equation of a circle with center at $C$ . If the area of the triangle with vertices at $(0, 0)$ , $C$ , and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:
(1) 100
(2) 50
(3) $\frac{121}{25}$
(4) $\frac{81}{25}$
Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$ . Then the minimum number of elements needed to be added to $R$ so that $R$ becomes an equivalence relation is:
(1) 10
(2) 8
(3) 9
(4) 7
The number of words that can be formed using all the letters of the word “DAUGHTER” so that all vowels never come together is:
(1) 34000
(2) 37000
(3) 36000
(4) 35000
Let the area of a $\triangle PQR$ with vertices $P(5, 4)$ , $Q(-2, 4)$ , and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left( 2, \frac{14}{5} \right)$ and $C(c, d)$ , respectively, then $c + 2 d$ is equal to:
(1) $\frac{7}{3}$
(2) 3
(3) 2
(4) $\frac{8}{3}$
If $\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}$ , then $\cos^{-1} \left( \frac{12}{13} \cos x + \frac{5}{13} \sin x \right)$ is equal to:
(1) $x - \tan^{-1} \frac{4}{3}$
(2) $x - \tan^{-1} \frac{5}{12}$
(3) $x + \tan^{-1} \frac{4}{5}$
(4)🇼🇵🇼🇵 $x + \tan^{-1} \frac{5}{12}$
The value of $\left( \sin 70^\circ \right) \left( \cot 10^\circ \cot 70^\circ - 1 \right)$ is:
(1) 1
(2) 0
(3) $\frac{3}{2}$
(4) $\frac{2}{3}$
Marks obtained by all students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval $12-18$ and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is:
(1) 48
(2) 44
(3) 40
(4) 52
Let the position vectors of vertices $A$ , $B$ , and $C$ of a tetrahedron $ABCD$ be $\hat{i} + 2 \hat{j} + \hat{k}$ , $\hat{i} + 3 \hat{j} - 2 \hat{k}$ , and $2 \hat{i} + \hat{j} - \hat{k}$ , respectively. The altitude from vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of triangle $ABC$ at point $E$ . If the length of $AD$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6 \sqrt{2}}$ , then the position vector of $E$ is:
(1) $\frac{1}{2} (\hat{i} + 4 \hat{j} + 7 \hat{k})$
(2) $\frac{1}{12} (7 \hat{i} + 4 \hat{j} + 3 \hat{k})$
(3) $\frac{1}{6} (12 \hat{i} + 12 \hat{j} + \hat{k})$
(4) $\frac{1}{6} (7 \hat{i} + 12 \hat{j} + \hat{k})$
If $A$ , $B$ , and $\operatorname{adj}(A^{-1}) + \operatorname{adj}(B^{-1})$ are non-singular matrices of the same order, then the inverse of $A \left( \operatorname{adj}(A^{-1}) + \operatorname{adj}(B^{-1}) \right)^{-1} B$ is equal to:
(1) $AB^{-1} + A^{-1} B$
(2) $\operatorname{adj}(B^{-1}) + \operatorname{adj}(A^{-1})$
(3) $\frac{1}{|AB|} (\operatorname{adj}(B) + \operatorname{adj}(A))$
(4) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$
If the system of equations $(\lambda - 1) x + (\lambda - 4) y + \lambda z = 5$ , $\lambda x + (\lambda - 1) y + (\lambda - 4) z = 7$ , $(\lambda + 1) x + (\lambda + 2) y - (\lambda + 2) z = 9$ has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to:
(1) 10
(2) 12
(3) 6
(4) 20
One die has two faces marked 1, two faces marked 2, one face marked 3, and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3, and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 when both dice are thrown together is:
(1) $\frac{1}{2}$
(2) $\frac{3}{5}$
(3) $\frac{2}{3}$
(4) $\frac{4}{9}$
If the area of the larger portion bounded between the curves $x^2 + y^2 = 25$ and $y = |x - 1|$ is $\frac{1}{4} (b \pi + c)$ , $b, c \in \mathbb{N}$ , then $b + c$ is equal to.
The sum of all rational terms in the expansion of $\left( 1 + 2^{1/3} + 3^{1/2} \right)^6$ is equal to.
Let the circle $C$ touch the line $x - y + 1 = 0$ , have the center on the positive x-axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x + 2 y = 1$ . Let $H$ be the hyperbola $\frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2} = 1$ , whose one of the foci is the center of $C$ and the length of the transverse axis is the diameter of $C$ . Then $2 \alpha^2 + 3 \beta^2$ is equal to.
If the set of all values of $a$ for which the equation $5 x^3 - 15 x - a = 0$ has three distinct real roots is the interval $(\alpha, \beta)$ , then $\beta - 2 \alpha$ is equal to.
If the equation $a (b - c) x^2 + b (c - a) x + c (a - b) = 0$ has equal roots, where $a + c = 15$ and $b = \frac{36}{5}$ , then $a^2 + c^2$ is equal to.
If in the expansion of $(1 + x)^p (1 - x)^q$ , the coefficients of $x$ and $x^2$ are 1 and -2, respectively, then $p^2 + q^2$ is equal to:
(1) 8
(2) 18
(3) 13
(4) 20
Let $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}$ and $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}$ . If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$ , then $\sum_{(x, y) \in C} |x + y|$ is:
(1) 15
(2) 18
(3) 24
(4) 12
The system of equations $x + y + z = 6$ , $x + 2 y + 5 z = 9$ , $x + 5 y + \lambda z = \mu$ has no solution if:
(1) $\lambda = 17, \mu \neq 18$
(2) $\lambda \neq 17, \mu \neq 18$
(3) $\lambda = 15, \mu \neq 17$
(4) $\lambda = 17, \mu = 18$
Let $\int x^3 \sin x \, dx = g(x) + C$ , where $C$ is the constant of integration. If $8 \left( g\left( \frac{\pi}{2} \right) + g'\left( \frac{\pi}{2} \right) \right) = \alpha \pi^3 + \beta \pi^2 + \gamma$ , $\alpha, \beta, \gamma \in \mathbb{Z}$ , then $\alpha + \beta - \gamma$ equals:
(1) 55
(2) 47
(3) 48
(4) 62
A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ , respectively. If the locus of the point $P$ that divides the rod $AB$ internally in the ratio $2:1$ is $9 \left( x^2 + \alpha y^2 + \beta x y + \gamma x + 28 y \right) - 76 = 0$ , then $\alpha - \beta - \gamma$ is equal to:
(1) 24
(2) 23
(3) 21
(4) 22
The distance of the line $\frac{x-2}{2} = \frac{y-6}{3} = \frac{z-3}{4}$ from point $(1, 4, 0)$ along the line $\frac{x}{1} = \frac{y-2}{2} = \frac{z+3}{3}$ is:
(1) $\sqrt{17}$
(2) $\sqrt{14}$
(3) $\sqrt{15}$
(4) $\sqrt{13}$
Let point $A$ divide the line segment joining points $P(-1, -1, 2)$ and $Q(5, 5, 10)$ internally in the ratio $r:1$ ( $r > 0$ ). If $O$ is the origin and $(\overrightarrow{OQ} \cdot \overrightarrow{OA}) - \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10$ , then the value of $r$ is:
(1) 14
(2) 3
(3) $\sqrt{7}$
(4) 7
If the area of the region $\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|} - e^{-x}, a > 0\}$ is $\frac{e^2 + 8 e + 1}{e}$ , then the value of $a$ is:
(1) 7
(2) 6
(3) 8
(4) 5
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81 \, \text{cm}^3/\text{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \, \text{cm}/\text{min}$ . The surface area (in $\text{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is:
(1) $225 \pi$
(2) $128 \pi$
(3) $196 \pi$
(4) $256 \pi$
A board has 16 squares as shown in the figure. Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:
(1) $\frac{4}{5}$
(2) $\frac{7}{10}$
(3) $\frac{3}{5}$
(4) $\frac{23}{30}$
Let $x = x(y)$ be the solution of the differential equation $y = \left( x - y \frac{dx}{dy} \right) \sin \left( \frac{x}{y} \right)$ , $y > 0$ , and $x(1) = \frac{\pi}{2}$ . Then $\cos (x(2))$ is equal to:
(1) $1 - 2 \left( \log_e 2 \right)^2$
(2) $2 \left( \log_e 2 \right)^2 - 1$
(3) $2 \left( \log_e 2 \right) - 1$
(4) $1 - 2 \left( \log_e 2 \right)$
Let the range of the function $f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{\pi}{3} - x \right) \cdot \cos \left( \frac{\pi}{3} + x \right) \sin 3 x \cdot \cos 6 x$ , $x \in \mathbb{R}$ , be $[\alpha, \beta]$ . Then the distance of the point $(\alpha, \beta)$ from the line $3 x + 4 y + 12 = 0$ is:
(1) 11
(2) 8
(3) 10
(4) 9
Let the shortest distance from $(a, 0)$ , $a > 0$ , to the parabola $y^2 = 4 x$ be 4. Then the equation of the circle passing through point $(a, 0)$ and the focus of the parabola, and having its center on the axis of the parabola, is:
(1) $x^2 + y^2 - 6 x + 5 = 0$
(2) $x^2 + y^2 - 4 x + 3 = 0$
(3) $x^2 + y^2 - 10 x + 9 = 0$
(4) $x^2 + y^2 - 8 x + 7 = 0$
Let $X = \mathbb{R} \times \mathbb{R}$ . Define a relation $R$ on $X$ as $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$ .
Statement-I: $R$ is an equivalence relation.
Statement-II: For some $(a, b) \in X$ , the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$ .
Choose the correct answer:
(1) Both Statement-I and Statement-II are false.
(2) Statement-I is true but Statement-II is false.
(3) Both Statement-I and Statement-II are true.
(4) Statement-I is false but Statement-II is true.
The length of the chord of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ , whose midpoint is $\left( 1, \frac{1}{2} \right)$ , is:
(1) $\frac{2}{3} \sqrt{15}$
(2) $\frac{5}{3} \sqrt{15}$
(3) $\frac{1}{3} \sqrt{15}$
(4) $\sqrt{15}$
Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ , $A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ , and $A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ , then $a_{23}$ equals:
(1) -1
(2) 0
(3) 2
(4) 1
The number of complex numbers $z$ satisfying $|z| = 1$ and $\left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1$ is:
(1) 6
(2) 4
(3) 10
(4) 8
If the square of the shortest distance between the lines $\frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3}$ and $\frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5}$ is $\frac{m}{n}$ , where $m, n$ are coprime numbers, then $m + n$ is equal to:
(1) 6
(2) 9
(3) 21
(4) 14
If $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} \, dx$ , then $\int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx$ equals:
(1) $\frac{\pi^2}{16}$
(2) $\frac{\pi^2}{4}$
(3) $\frac{\pi^2}{8}$
(4) $\frac{\pi^2}{12}$
$\lim_{x \to \infty} \frac{\left( 2 x^2 - 3 x + 5 \right) (3 x - 1)^{\frac{x}{2}}}{\left( 3 x^2 + 5 x + 4 \right) \sqrt{(3 x + 2)^x}}$ is equal to:
(1) $\frac{2}{\sqrt{3 e}}$
(2) $\frac{2 e}{\sqrt{3}}$
(3) $\frac{2 e}{3}$
(4) $\frac{2}{3 \sqrt{e}}$
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.
Let $\alpha, \beta$ be the roots of $x^2 - a x - b = 0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$ . 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.
The focus of the parabola $y^2 = 4 x + 16$ is the center of the circle $C$ of radius 5. If the values of $\lambda$ for which $C$ passes through the point of intersection of the lines $3 x - y = 0$ and $x + \lambda y = 4$ are $\lambda_1$ and $\lambda_2$ , $\lambda_1 < \lambda_2$ , then $12 \lambda_1 + 29 \lambda_2$ is equal to.
The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is.
The roots of the quadratic equation $3 x^2 - p x + q = 0$ are the 10th and 11th terms of an arithmetic progression with common difference $\frac{3}{2}$ . If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2 q$ is equal to.
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 $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$ , then $|\vec{c}|$ is equal to:
(1) $\frac{1}{3 \sqrt{2}}$
(2) 18
(3) 16
(4) $\sqrt{\frac{11}{6}}$
In $I(m, n) = \int_{0}^{1} x^{m-1} (1 - x)^{n-1} \, dx$ , $m, n > 0$ , then $I(9, 14) + I(10, 13)$ is:
(1) $I(9, 1)$
(2) $I(19, 27)$
(3) $I(1, 13)$
(4) $I(9, 13)$
Let $f: \mathbb{R} - \{0\} \to \mathbb{R}$ be a function such that $f(x) - 6 f\left( \frac{1}{x} \right) = \frac{35}{3 x} - \frac{5}{2}$ . If $\lim_{x \to 0} \left( \frac{1}{\alpha x} + f(x) \right) = \beta$ , $\alpha, \beta \in \mathbb{R}$ , then $\alpha + 2 \beta$ is equal to:
(1) 3
(2) 5
(3) 4
(4) 6
Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an arithmetic progression with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$ , then the absolute difference between the 20th and 15th terms of the arithmetic progression is:
(1) 25
(2) 90
(3) 20
(4) 45
Let $f(x) = \frac{2^{x+2} + 16}{2^{2 x + 1} + 2^{x + 4} + 32}$ . 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:
(1) 118
(2) 92
(3) 102
(4) 108
If $\alpha$ and $\beta$ are the roots of $2 z^2 - 3 z - 2 i = 0$ , where $i = \sqrt{-1}$ , then $16 \cdot \operatorname{Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot \operatorname{Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right)$ is equal to:
(1) 398
(2) 312
(3) 409
(4) 441
$\lim_{x \to 0} \operatorname{cosec} x \left( \sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4} \right)$ is:
(1) 0
(2) $\frac{1}{2 \sqrt{5}}$
(3) $\frac{1}{\sqrt{15}}$
(4) $-\frac{1}{2 \sqrt{5}}$
Let in a $\triangle ABC$ , the length of side $AC$ be 6, the vertex $B$ be $(1, 2, 3)$ , and the vertices $A, C$ lie on the line $\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$ . Then the area (in sq. units) of $\triangle ABC$ is:
(1) 42
(2) 21
(3) 56
(4) 17
Let $y = y(x)$ be the solution of the differential equation $\left( x y - 5 x^2 \sqrt{1 + x^2} \right) dx + \left( 1 + x^2 \right) dy = 0$ , $y(0) = 0$ . Then $y(\sqrt{3})$ is equal to:
(1) $\frac{5 \sqrt{3}}{2}$
(2) $\sqrt{\frac{14}{3}}$
(3) $2 \sqrt{2}$
(4) $\sqrt{\frac{15}{2}}$
Let the product of the focal distances of the point $\left( \sqrt{3}, \frac{1}{2} \right)$ 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:
(1) $\frac{3 - 2 \sqrt{2}}{3 \sqrt{2}}$
(2) $\frac{1 - \sqrt{3}}{\sqrt{2}}$
(3) $\frac{3 - 2 \sqrt{2}}{2 \sqrt{3}}$
(4) $\frac{1 - 2 \sqrt{2}}{\sqrt{3}}$
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:
(1) $\frac{9}{17}$
(2) $\frac{9}{19}$
(3) $\frac{8}{17}$
(4) $\frac{8}{19}$
Consider the region $R = \left\{ (x, y) : x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}$ . The area of the largest rectangle with sides parallel to the coordinate axes and inscribed in $R$ is:
(1) $\frac{625}{111}$
(2) $\frac{730}{119}$
(3) $\frac{567}{121}$
(4) $\frac{821}{123}$
The area of the region $\left\{ (x, y) : x^2 + 4 x + 2 \leq y \leq |x + 2| \right\}$ is equal to:
(1) 7
(2) $\frac{24}{5}$
(3) $\frac{20}{3}$
(4) 5
For 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 incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is:
(1) 7
(2) 4
(3) 9
(4) 5
Let circle $C$ be the image of $x^2 + y^2 - 2 x + 4 y - 4 = 0$ in the line $2 x - 3 y + 5 = 0$ , and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the center $O$ of $C$ . If $B(\alpha, \beta)$ , with $\beta < 4$ , lies on $C$ such that the length of arc $AB$ is $\frac{1}{6}^{\text{th}}$ of the perimeter of $C$ , then $\beta - \sqrt{3} \alpha$ is equal to:
(1) 3
(2) $3 + \sqrt{3}$
(3) $4 - \sqrt{3}$
(4) 4
For some $n \neq 10$ , let the coefficients of the 5th, 6th, and 7th terms in the binomial expansion of $(1 + x)^{n+4}$ be in arithmetic progression. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
(1) 70
(2) 35
(3) 20
(4) 10
The product of all rational roots of the equation $\left( x^2 - 9 x + 11 \right)^2 - (x - 4)(x - 5) = 3$ is equal to:
(1) 14
(2) 7
(3) 28
(4) 21
Let the line passing through the point $(-1, 2, 1)$ and parallel to the line $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{1}$ at point $P$ . Then the distance of $P$ from point $Q(4, -5, 1)$ is:
(1) 5
(2) 10
(3) $5 \sqrt{6}$
(4) $5 \sqrt{5}$
Let the lines $3 x - 4 y - \alpha = 0$ , $8 x - 11 y - 33 = 0$ , and $2 x - 3 y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2 x - 3 y + \lambda = 0$ is $\left( \frac{57}{13}, \frac{-40}{13} \right)$ , then $|\alpha \lambda|$ is equal to:
(1) 84
(2) 91
(3) 113
(4) 101
If the system of equations $2 x - y + z = 4$ , $5 x + \lambda y + 3 z = 12$ , $100 x - 47 y + \mu z = 212$ has infinitely many solutions, then $\mu - 2 \lambda$ is equal to:
(1) 56
(2) 59
(3) 55
(4) 57
Let $f$ be a differentiable function such that $2 (x + 2)^2 f(x) - 3 (x + 2)^2 = 10 \int_{0}^{x} (t + 2) f(t) \, dt$ , $x \geq 0$ . Then $f(2)$ is equal to.
If for some $\alpha, \beta$ ; $\alpha \leq \beta$ , $\alpha + \beta = 8$ , and $\sec^2 \left( \tan^{-1} \alpha \right) + \csc^2 \left( \cot^{-1} \beta \right) = 36$ , then $\alpha^2 + \beta$ is.
The number of 3-digit numbers that are divisible by 2 and 3, but not divisible by 4 and 9, is.
Let $A$ be a $3 \times 3$ matrix such that $X^T A X = 0$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ . If $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}$ , $A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}$ , and $\det(\operatorname{adj}(2 (A + I))) = 2^\alpha 3^\beta 5^\gamma$ , $\alpha, \beta, \gamma \in \mathbb{N}$ , then $\alpha^2 + \beta^2 + \gamma^2$ is.
Let $S = \{ p_1, p_2, \ldots, p_{10} \}$ be the set of the 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.
The equation of the chord of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ , whose midpoint is $(3, 1)$ , is:
(1) $48 x + 25 y = 169$
(2) $4 x + 122 y = 134$
(3) $25 x + 101 y = 176$
(4) $5 x + 16 y = 31$
The function $f: (-\infty, \infty) \to (-\infty, 1)$ , defined by $f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}$ , is:
(1) One-one but not onto
(2) Onto but not one-one
(3) Both one-one and onto
(4) Neither one-one nor onto
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) $\frac{\pi}{2} - (\alpha + \beta + \gamma)$
(2) $3 \pi$
(3) 0
(4) $\pi$
Let $f: (0, \infty) \to \mathbb{R}$ be a function that is differentiable at all points of its domain and satisfies $x^2 f'(x) = 2 x f(x) + 3$ , with $f(1) = 4$ . Then $2 f(2)$ is equal to:
(1) 29
(2) 19
(3) 39
(4) 23
Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{\frac{2}{\pi}} |\sin x| + \log_{\frac{2}{\pi}} |\cos x| = 2 \right\}$ and $B = \left\{ x \geq 0 : \sqrt{x} (\sqrt{x} - 4) - 3 |\sqrt{x} - 2| + 6 = 0 \right\}$ . Then $n(A \cup B)$ is equal to:
(1) 4
(2) 2
(3) 8
(4) 6
Let the position vectors of three vertices of a triangle be $4 \overrightarrow{p} + \overrightarrow{q} - 3 \overrightarrow{r}$ , $-5 \overrightarrow{p} + \overrightarrow{q} + 2 \overrightarrow{r}$ , and $2 \overrightarrow{p} - \overrightarrow{q} + 2 \overrightarrow{r}$ . If the position vectors of the orthocenter and circumcenter of the triangle are $\frac{\overrightarrow{p} + \overrightarrow{q} + \overrightarrow{r}}{4}$ and $\alpha \overrightarrow{p} + \beta \overrightarrow{q} + \gamma \overrightarrow{r}$ , respectively, then $\alpha + 2 \beta + 5 \gamma$ is equal to:
(1) 3
(2) 1
(3) 6
(4) 4
Let $[x]$ denote the greatest integer function, and let $m$ and $n$ be the numbers of points where the function $f(x) = [x] + |x - 2|$ , $-2 < x < 3$ , is not continuous and not differentiable, respectively. Then $m + n$ is equal to:
(1) 6
(2) 9
(3) 8
(4) 7
Let the point $\left( \frac{11}{2}, \alpha \right)$ lie on or inside the triangle with sides $x + y = 11$ , $x + 2 y = 16$ , and $2 x + 3 y = 29$ . Then the product of the smallest and largest values of $\alpha$ is equal to:
(1) 22
(2) 44
(3) 33
(4) 55
In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$ , then $S_{30} - S_{10}$ is equal to:
(1) 510
(2) 515
(3) 525
(4) 505
If $7 = 5 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2 \alpha) + \frac{1}{7^3} (5 + 3 \alpha) + \ldots \infty$ , then the value of $\alpha$ is:
(1) 1
(2) $\frac{6}{7}$
(3) 6
(4) $\frac{1}{7}$
If the system of equations $x + 2 y - 3 z = 2$ , $2 x + \lambda y + 5 z = 5$ , $14 x + 3 y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
(1) 13
(2) 10
(3) 11
(4) 12
Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2 \log_e (x - 2) - x^2 + a x + 1$ 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:
(1) 280
(2) 360
(3) 420
(4) 160
Suppose $A$ and $B$ are the coefficients of the 30th and 12th terms, respectively, in the binomial expansion of $(1 + x)^{2n-1}$ . If $2 A = 5 B$ , then $n$ is equal to:
(1) 22
(2) 21
(3) 20
(4) 19
Let $\vec{a} = 3 \hat{i} - \hat{j} + 2 \hat{k}$ , $\vec{b} = \vec{a} \times (\hat{i} - 2 \hat{k})$ , and $\vec{c} = \vec{b} \times \hat{k}$ . Then the projection of $\vec{c} - 2 \hat{j}$ on $\vec{a}$ is:
(1) $3 \sqrt{7}$
(2) $\sqrt{14}$
(3) $2 \sqrt{14}$
(4) $2 \sqrt{7}$
For some $a, b$ , let $f(x) = \left| \begin{array}{ccc} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{array} \right|$ , $x \neq 0$ . If $\lim_{x \to 0} f(x) = \lambda + \mu a + \nu b$ , then $(\lambda + \mu + \nu)^2$ is equal to:
(1) 25
(2) 9
(3) 36
(4) 16
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:
(1) 8575
(2) 9100
(3) 8925
(4) 8750
The area of the region enclosed by the curves $y = e^x$ , $y = |e^x - 1|$ , and the y-axis is:
(1) $1 + \log_e 2$
(2) $\log_e 2$
(3) $2 \log_e 2 - 1$
(4) $1 - \log_e 2$
The number of real solutions of the equation $x^2 + 3 x + 2 = \min \{ |x - 3|, |x + 2| \}$ is:
(1) 2
(2) 0
(3) 3
(4) 1
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:
(1) $\frac{5}{8}$
(2) $\frac{3}{16}$
(3) $\frac{1}{8}$
(4) $\frac{3}{8}$
If the equation of the parabola with vertex $V \left( \frac{3}{2}, 3 \right)$ and directrix $x + 2 y = 0$ is $\alpha x^2 + \beta y^2 - \gamma x y - 30 x - 60 y + 225 = 0$ , then $\alpha + \beta + \gamma$ is equal to:
(1) 6
(2) 8
(3) 7
(4) 9
Number of functions $f: \{1, 2, \ldots, 100\} \to \{0, 1\}$ that assign 1 to exactly one of the positive integers less than or equal to 98 is equal to.
Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L: \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ , and $R(5, p, q)$ be a point on $L$ . Then the square of the area of $\triangle PQR$ is.
Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2 x - 4 y \sin x$ , $x \in \left( 0, \frac{\pi}{2} \right)$ . If $y \left( \frac{\pi}{3} \right) = 0$ , then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to.
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 with lengths 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 $25 e_2^2$ is equal to.
If $\int \frac{2 x^2 + 5 x + 9}{\sqrt{x^2 + x + 1}} \, dx = x \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C$ , where $C$ is the constant of integration, then $\alpha + 2 \beta$ is equal to.
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 is not more than 8, is:
(1) 4608
(2) 5720
(3) 5719
(4) 4607
Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^2 = 4 x$ . Let sides $AD$ and $BC$ be parallel to the y-axis. If the diagonal $AC$ has length $\frac{25}{4}$ and passes through point $(1, 0)$ , then the area of $ABCD$ is:
(1) $\frac{75}{4}$
(2) $\frac{25}{2}$
(3) $\frac{125}{8}$
(4) $\frac{75}{8}$
Two numbers $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1} + i^{k_2}$ , $(i = \sqrt{-1})$ , is non-zero, equals:
(1) $\frac{1}{2}$
(2) $\frac{1}{4}$
(3) $\frac{3}{4}$
(4) $\frac{2}{3}$
If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$ , $x \in \mathbb{R}$ , then $\sum_{k=1}^{81} f \left( \frac{k}{82} \right)$ is equal to:
(1) 41
(2) $\frac{81}{2}$
(3) 82
(4) $81 \sqrt{2}$
Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = (2 + 3 a) x^2 + \left( \frac{a + 2}{a - 1} \right) x + b$ , $a \neq 1$ . If $f(x + y) = f(x) + f(y) + 1 - \frac{2}{7} x y$ , then the value of $28 \sum_{i=1}^5 |f(i)|$ is:
(1) 715
(2) 735
(3) 545
(4) 675
Let $A(x, y, z)$ be a point in the xy-plane, equidistant from 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
(S2) The area of $\triangle ABC$ is $\frac{9 \sqrt{2}}{2}$
(1) Both are true
(2) Only (S1) is true
(3) Only (S2) is true
(4) Both are false
The relation $R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:
(1) Reflexive and transitive but not symmetric
(2) Reflexive and symmetric but not transitive
(3) An equivalence relation
(4) Symmetric and transitive but not reflexive
Let the equation of the circle that touches the x-axis at point $(a, 0)$ , $a > 0$ , and cuts off an intercept of length $b$ on the y-axis be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$ . If the circle lies below the x-axis, then the ordered pair $(2 a, b^2)$ is equal to:
(1) $\left( \alpha, \beta^2 + 4 \gamma \right)$
(2) $\left( \gamma, \beta^2 - 4 \alpha \right)$
(3) $\left( \gamma, \beta^2 + 4 \alpha \right)$
(4) $\left( \alpha, \beta^2 - 4 \gamma \right)$
Let $\langle a_n \rangle$ be a sequence such that $a_0 = 0$ , $a_1 = \frac{1}{2}$ , and $2 a_{n+2} = 5 a_{n+1} - 3 a_n$ , $n = 0, 1, 2, 3, \ldots$ . Then $\sum_{k=1}^{100} a_k$ is equal to:
(1) $3 a_{99} - 100$
(2) $3 a_{100} - 100$
(3) $3 a_{100} + 100$
(4) $3 a_{99} + 100$
$\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac{33}{65}$
(4) $\frac{32}{65}$
Let $T_r$ be the $r$ th term of an arithmetic progression. If for some $m$ , $T_m = \frac{1}{25}$ , $T_{25} = \frac{1}{20}$ , and $20 \sum_{r=1}^{25} T_r = 13$ , then $5 m \sum_{r=m}^{2m} T_r$ is equal to:
(1) 112
(2) 126
(3) 98
(4) 142
If the image of the point $(4, 4, 3)$ in the line $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3}$ is $(\alpha, \beta, \gamma)$ , then $\alpha + \beta + \gamma$ is equal to:
(1) 9
(2) 12
(3) 8
(4) 7
If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos^2 x}{1 + e^x} \, dx = \pi \left( \alpha \pi^2 + \beta \right)$ , $\alpha, \beta \in \mathbb{Z}$ , then $(\alpha + \beta)^2$ equals:
(1) 144
(2) 196
(3) 100
(4) 64
The sum of all local minimum values of the function $f(x) = \begin{cases} 1 - 2 x, & x 2 \end{cases}$ is:
(1) $\frac{171}{72}$
(2) $\frac{131}{72}$
(3) $\frac{157}{72}$
(4) $\frac{167}{72}$
The sum of the squares of all roots of the equation $x^2 + |2 x - 3| - 4 = 0$ is:
(1) $3 (3 - \sqrt{2})$
(2) $6 (3 - \sqrt{2})$
(3) $6 (2 - \sqrt{2})$
(4) $3 (2 - \sqrt{2})$
Let for some function $y = f(x)$ , $\int_{0}^{x} t f(t) \, dt = x^2 f(x)$ , $x > 0$ , and $f(2) = 3$ . Then $f(6)$ is equal to:
(1) 1
(2) 2
(3) 6
(4) 3
Let ${n \choose r-1} = 28$ , ${n \choose r} = 56$ , and ${n \choose r+1} = 70$ . Let $A(4 \cos t, 4 \sin t)$ , $B(2 \sin t, -2 \cos t)$ , and $C(3 r - n, r^2 - n - 1)$ be the vertices of triangle $ABC$ , where $t$ is a parameter. If $(3 x - 1)^2 + (3 y)^2 = \alpha$ is the locus of the centroid of triangle $ABC$ , then $\alpha$ equals:
(1) 20
(2) 8
(3) 6
(4) 18
Let $O$ be the origin, point $A$ be $z_1 = \sqrt{3} + 2 \sqrt{2} i$ , point $B(z_2)$ be such that $\sqrt{3} |z_2| = |z_1|$ and $\arg(z_2) = \arg(z_1) + \frac{\pi}{6}$ . Then:
(1) Area of triangle $ABO$ is $\frac{11}{\sqrt{3}}$
(2) $ABO$ is a scalene triangle
(3) Area of triangle $ABO$ is $\frac{11}{4}$
(4) $ABO$ is an obtuse-angled isosceles triangle
Three defective oranges are accidentally mixed with seven good ones, and it is not possible to differentiate between them. Two oranges are drawn at random. If $x$ denotes the number of defective oranges, then the variance of $x$ is:
(1) $\frac{28}{75}$
(2) $\frac{14}{25}$
(3) $\frac{26}{75}$
(4) $\frac{18}{25}$
The area (in sq. units) of the region $\left\{ (x, y) : 0 \leq y \leq 2 |x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \right\}$ is:
(1) $\frac{80}{3}$
(2) $\frac{64}{3}$
(3) $\frac{17}{3}$
(4) $\frac{32}{3}$
Let $M$ denote the set of all real matrices of order $3 \times 3$ , and let $S = \{-3, -2, -1, 1, 2\}$ . Let $S_1 = \left\{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \right\}$ , $S_2 = \left\{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \right\}$ , $S_3 = \left\{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \right\}$ . If $n(S_1 \cup S_2 \cup S_3) = 125 \alpha$ , then $\alpha$ equals.
If $\alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} {12 \choose 2r-1}$ , then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is.
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.
Let $f(x) = \begin{cases} 3 x, & x 2 \end{cases}$ , where $[.]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points where $f$ is not continuous and not differentiable, respectively, then $\alpha + \beta$ equals.
Let $E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ ‘s are constructed such that their centers and eccentricities are the same as those of $E_1$ , and the length of the minor axis of $E_i$ is the length of the major axis of $E_{i+1}$ ( $i \geq 1$ ). If $A_i$ is the area of ellipse $E_i$ , then $\frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right)$ is equal to.
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 bag is selected at random, and a ball is drawn from it. If the ball is white, then the probability that it is drawn from Bag $B_2$ is:
(1) $\frac{1}{3}$
(2) $\frac{4}{15}$
(3) $\frac{2}{3}$
(4) $\frac{2}{5}$
Let $A$ , $B$ , and $C$ be three points in the xy-plane with position vectors $\sqrt{3} \hat{i} + \hat{j}$ , $\hat{i} + \sqrt{3} \hat{j}$ , and $a \hat{i} + (1 - a) \hat{j}$ , respectively, with respect to origin $O$ . If the distance of point $C$ from the line bisecting the angle between vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\frac{9}{\sqrt{2}}$ , then the sum of all possible values of $a$ is:
(1) 1
(2) $\frac{9}{2}$
(3) 0
(4) 2
If the components of $\vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$ along and perpendicular to $\vec{b} = 3 \hat{i} + \hat{j} - \hat{k}$ are $\frac{16}{11} (3 \hat{i} + \hat{j} - \hat{k})$ and $\frac{1}{11} (-4 \hat{i} - 5 \hat{j} - 17 \hat{k})$ , respectively, then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:
(1) 23
(2) 18
(3) 16
(4) 26
If $\alpha + i \beta$ and $\gamma + i \delta$ are the roots of $x^2 - (3 - 2 i) x - (2 i - 2) = 0$ , $i = \sqrt{-1}$ , then $\alpha \gamma + \beta \delta$ is equal to:
(1) 6
(2) 2
(3) -2
(4) -6
If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $\left( \sqrt{2}, \frac{4}{3} \right)$ , and the length of the chord is $\frac{2 \sqrt{\alpha}}{3}$ , then $\alpha$ is:
(1) 18
(2) 22
(3) 26
(4) 20
Let $S$ be the set of all words formed by arranging all letters of the word GARDEN. From set $S$ , one word is selected at random. The probability that the selected word does not have vowels in alphabetical order is:
(1) $\frac{1}{4}$
(2) $\frac{2}{3}$
(3) $\frac{1}{3}$
(4) $\frac{1}{2}$
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 $g(x^3) = x^6 + x^7$ , then the value of $\sum_{r=1}^{15} f(r^3)$ is:
(1) 320
(2) 340
(3) 270
(4) 310
The square of the distance of the point $\left( \frac{15}{7}, \frac{32}{7}, 7 \right)$ 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:
(1) 54
(2) 41
(3) 66
(4) 44
The area of the region bounded by the curves $x (1 + y^2) = 1$ and $y^2 = 2 x$ is:
(1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$
(2) $\frac{\pi}{4} - \frac{1}{3}$
(3) $\frac{\pi}{2} - \frac{1}{3}$
(4) $\frac{1}{2} \left( \frac{\pi}{2} - \frac{1}{3} \right)$
Let $A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix}$ and $P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ , $\theta > 0$ . If $B = P A P^T$ , $C = P^T B^{10} P$ , and the sum of the diagonal elements of $C$ is $\frac{m}{n}$ , where $\gcd(m, n) = 1$ , then $m + n$ is:
(1) 65
(2) 127
(3) 258
(4) 2049
If $f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx$ , $f(0) = -6$ , then $f(1)$ is equal to:
(1) $\log_e 2 + 2$
(2) $4 (\log_e 2 - 2)$
(3) $2 - \log_e 2$
(4) $4 (\log_e 2 + 2)$
Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$ . If $F(x) = x f(x)$ for all $x \in \mathbb{R}$ , $\int_{0}^{2} x F'(x) \, dx = 6$ , and $\int_{0}^{2} x^2 F''(x) \, dx = 40$ , then $F'(2) + \int_{0}^{2} F(x) \, dx$ is equal to:
(1) 11
(2) 15
(3) 9
(4) 13
For positive integers $n$ , if $4 a_n = n^2 + 5 n + 6$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right)$ , then the value of $507 S_{2025}$ is:
(1) 540
(2) 1350
(3) 675
(4) 135
Let $f: [0, 3] \to A$ be defined by $f(x) = 2 x^3 - 15 x^2 + 36 x + 7$ and $g: [0, \infty) \to B$ be defined by $g(x) = \frac{x^{2025}}{x^{2025} + 1}$ . If both functions are onto and $S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$ , then $n(S)$ is equal to:
(1) 30
(2) 36
(3) 29
(4) 31
Let $[x]$ denote the greatest integer less than or equal to $x$ . Then the domain of $f(x) = \sec^{-1} (2 [x] + 1)$ is:
(1) $(-\infty, -1] \cup [0, \infty)$
(2) $(-\infty, -\infty)$
(3) $(-\infty, -1] \cup [1, \infty)$
(4) $(-\infty, \infty) - \{0\}$
If $\sum_{r=1}^{13} \left\{ \frac{1}{\sin \left( \frac{\pi}{4} + (r-1) \frac{\pi}{6} \right) \sin \left( \frac{\pi}{4} + \frac{r \pi}{6} \right)} \right\} = a \sqrt{3} + b$ , $a, b \in \mathbb{Z}$ , then $a^2 + b^2$ is equal to:
(1) 10
(2) 2
(3) 8
(4) 4
Two equal sides of an isosceles triangle are along $-x + 2 y = 4$ and $x + y = 4$ . If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is:
(1) -6
(2) 12
(3) 6
(4) $-2 \sqrt{10}$
Let the coefficients of three consecutive terms $T_r$ , $T_{r+1}$ , and $T_{r+2}$ in the binomial expansion of $(a + b)^{12}$ be in a geometric progression, 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 $\left( \sqrt[4]{3} + \sqrt[3]{4} \right)^{12}$ . Then $p + q$ is equal to:
(1) 283
(2) 295
(3) 287
(4) 299
If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8 x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ , and a point $P$ moves on the line $2 x - 3 y + 4 = 0$ , then the centroid of $\triangle PAB$ lies on the line:
(1) $4 x - 9 y = 12$
(2) $x + 9 y = 36$
(3) $9 x - 9 y = 32$
(4) $6 x - 9 y = 20$
Let $f: \mathbb{R} - \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$ . If $f(K) = -2 K$ , then the sum of squares of all possible values of $K$ is:
(1) 1
(2) 6
(3) 7
(4) 9
The number of natural numbers between 212 and 999 such that the sum of their digits is 15 is.
Let $f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{\tan \left( \frac{x}{2^{r+1}} \right) + \tan^3 \left( \frac{x}{2^{r+1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{r+1}} \right)} \right)$ . Then $\lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)}$ is equal to.
The interior angles of a polygon with $n$ sides are in an arithmetic progression with common difference $6^\circ$ . If the largest interior angle is $219^\circ$ , then $n$ is equal to.
Let $A$ and $B$ be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y^2 = 4 x$ with respect to the line $x + y + 4 = 0$ . If $d$ denotes the distance between $A$ and $B$ , and $a$ denotes the area of $\triangle SAB$ , where $S$ is the focus of the parabola $y^2 = 4 x$ , then the value of $a + d$ is.
If $y = y(x)$ is the solution of the differential equation $\sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right)$ , $-2 \leq x \leq 2$ , $y(2) = \frac{\pi^2 - 8}{4}$ , then $y^2(0)$ is equal to.
Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at points $A$ and $B$ . If the line perpendicular to $AB$ and passing through the midpoint of chord $AB$ intersects the circle at $C$ and $D$ , then the area of quadrilateral $ADBC$ is equal to:
(1) $3 \sqrt{7}$
(2) $2 \sqrt{14}$
(3) $5 \sqrt{7}$
(4) $\sqrt{14}$
Let $M$ and $m$ be the maximum and minimum values of $f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 4 x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 4 x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 4 x \end{array} \right|$ , $x \in \mathbb{R}$ . Then $M^4 - m^4$ is equal to:
(1) 1280
(2) 1295
(3) 1040
(4) 1215
Two parabolas have the same focus $(4, 3)$ , and their directrices are the x-axis and y-axis, respectively. If these parabolas intersect at points $A$ and $B$ , then $(AB)^2$ is equal to:
(1) 192
(2) 384
(3) 96
(4) 392
Let $ABC$ be a triangle formed by the lines $7 x - 6 y + 3 = 0$ , $x + 2 y - 31 = 0$ , and $9 x - 2 y - 19 = 0$ . Let point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3 x + 6 y - 53 = 0$ . Then $h^2 + k^2 + h k$ is equal to:
(1) 37
(2) 47
(3) 40
(4) 36
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:
(1) 77
(2) 462
(3) 308
(4) 154
Let $P$ be the set of seven-digit numbers with the sum of their digits equal to 11. If the numbers in $P$ are formed using digits 1, 2, and 3 only, then the number of elements in set $P$ is:
(1) 158
(2) 173
(3) 164
(4) 161
Let the area of the region $\left\{ (x, y) : 2 y \leq x^2 + 3, y + |x| \leq 3, y \geq |x - 1| \right\}$ be $A$ . Then $6 A$ is equal to:
(1) 16
(2) 12
(3) 18
(4) 14
The least value of $n$ for which the number of integral terms in the binomial expansion of $\left( \sqrt[3]{7} + \sqrt[12]{11} \right)^n$ is 183 is:
(1) 2184
(2) 2148
(3) 2172
(4) 2196
The number of solutions of the equation $\left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0$ is:
(1) 2
(2) 4
(3) 1
(4) 3
Let $y = y(x)$ be the solution of the differential equation $\cos x \left( \log_e (\cos x) \right)^2 dy + \left( \sin x - 3 y \sin x \log_e (\cos x) \right) dx = 0$ , $x \in \left( 0, \frac{\pi}{2} \right)$ . If $y \left( \frac{\pi}{4} \right) = \frac{-1}{\log_e 2}$ , then $y \left( \frac{\pi}{6} \right)$ is:
(1) $\frac{2}{\log_e 3 - \log_e 4}$
(2) $\frac{1}{\log_e 4 - \log_e 3}$
(3) $-\frac{1}{\log_e 4}$
(4) $\frac{1}{\log_e 3 - \log_e 4}$
Define a relation $R$ on the interval $\left[ 0, \frac{\pi}{2} \right)$ by $x R y$ if and only if $\sec^2 x - \tan^2 y = 1$ . Then $R$ is:
(1) An equivalence relation
(2) Both reflexive and transitive but not symmetric
(3) Both reflexive and symmetric but not transitive
(4) Reflexive but neither symmetric nor transitive
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 the same eccentricity $\frac{1}{\sqrt{3}}$ . Let the product of their latus rectum lengths be $\frac{32}{\sqrt{3}}$ , and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at points $A, B, C, D$ , then the area of quadrilateral $ABCD$ equals:
(1) $6 \sqrt{6}$
(2) $\frac{18 \sqrt{6}}{5}$
(3) $\frac{12 \sqrt{6}}{5}$
(4) $\frac{24 \sqrt{6}}{5}$
Consider an arithmetic progression 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) 84
(2) 122
(3) 90
(4) 108
Let $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{b} = 2 \hat{i} + 7 \hat{j} + 3 \hat{k}$ . Let $L_1: \vec{r} = (-\hat{i} + 2 \hat{j} + \hat{k}) + \lambda \vec{a}$ , $\lambda \in \mathbb{R}$ , and $L_2: \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}$ , $\mu \in \mathbb{R}$ , be two lines. If the line $L_3$ passes through the point of intersection of $L_1$ and $L_2$ , and is parallel to $\vec{a} + \vec{b}$ , then $L_3$ passes through the point:
(1) $(8, 26, 12)$
(2) $(2, 8, 5)$
(3) $(-1, -1, 1)$
(4) $(5, 17, 4)$
The value of $\lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^3 + 6 k^2 + 11 k + 5}{(k+3)!} \right)$ is:
(1) $\frac{4}{3}$
(2) 2
(3) $\frac{7}{3}$
(4) $\frac{5}{3}$
The integral $80 \int_{0}^{\frac{\pi}{4}} \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2 \theta} \right) d \theta$ is equal to:
(1) $3 \log_e 4$
(2) $6 \log_e 4$
(3) $4 \log_e 3$
(4) $2 \log_e 3$
Let $L_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2}$ and $L_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1}$ be two lines. Let $L_3$ be a line passing through point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$ . If $L_3$ intersects $L_1$ , then $|5 \alpha - 11 \beta - 8 \gamma|$ equals:
(1) 18
(2) 16
(3) 25
(4) 20
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 variance of $2 (x_1 - 1) + 4 \beta, 2 (x_2 - 1) + 4 \beta, \ldots, 2 (x_{10} - 1) + 4 \beta$ , then $\frac{\beta \mu}{\sigma^2}$ is equal to:
(1) 100
(2) 110
(3) 120
(4) 90
Let $|z_1 - 8 - 2 i| \leq 1$ and $|z_2 - 2 + 6 i| \leq 2$ , $z_1, z_2 \in \mathbb{C}$ . Then the minimum value of $|z_1 - z_2|$ is:
(1) 3
(2) 7
(3) 13
(4) 10
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:
(1) 262
(2) 288
(3) 242
(4) 222
Let $f: (0, \infty) \to \mathbb{R}$ be a twice differentiable function. If for some $a \neq 0$ , $\int_{0}^{1} f(\lambda x) \, d\lambda = a f(x)$ , $f(1) = 1$ , and $f(16) = \frac{1}{8}$ , then $16 - f'\left( \frac{1}{16} \right)$ is equal to.
Let $S = \left\{ m \in \mathbb{Z} : A^{m^2} + A^m = 3 I - A^{-6} \right\}$ , where $A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$ . Then $n(S)$ is equal to.
Let $[t]$ be the greatest integer less than or equal to $t$ . Then the least value of $p \in \mathbb{N}$ for which $\lim_{x \to 0^+} \left( x \left( \left[ \frac{1}{x} \right] + \left[ \frac{2}{x} \right] + \ldots + \left[ \frac{p}{x} \right] \right) - x^2 \left( \left[ \frac{1}{x^2} \right] + \left[ \frac{2^2}{x^2} \right] + \ldots + \left[ \frac{9^2}{x^2} \right] \right) \right) \geq 1$ is equal to.
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is 4.
Let $S = \left\{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2 x + 1) \right\}$ . Then $\sum_{x \in S} (2 x - 1)^2$ is equal to.
If the set of all $a \in \mathbb{R}$ for which the equation $2 x^2 + (a - 5) x + 15 = 3 a$ has no real roots is the interval $(\alpha, \beta)$ , and $X = \{ x \in \mathbb{Z} : \alpha < x < \beta \}$ , then $\sum_{x \in X} x^2$ is equal to:
(1) 2109
(2) 2129
(3) 2139
(4) 2119
If $\sin x + \sin^2 x = 1$ , $x \in \left( 0, \frac{\pi}{2} \right)$ , then $\left( \cos^{12} x + \tan^{12} x \right) + 3 \left( \cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x \right) + \left( \cos^6 x + \tan^6 x \right)$ is equal to:
(1) 4
(2) 3
(3) 2
(4) 1
Let the area enclosed between the curves $|y| = x^2$ and $x^2 + y^2 = 1$ be $\alpha$ . If $9 \alpha = \beta \pi + \gamma$ , $\beta, \gamma$ are integers, then the value of $|\beta - \gamma|$ equals:
(1) 27
(2) 18
(3) 15
(4) 33
If the domain of the function $\log_5 (18 x - x^2 - 77)$ is $(\alpha, \beta)$ and the domain of the function $\log_{x-1} \left( \frac{2 x^2 + 3 x - 2}{x^2 - 3 x - 4} \right)$ is $(\gamma, \delta)$ , then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:
(1) 195
(2) 174
(3) 186
(4) 179
Let the function $f(x) = \left( x^2 - 1 \right) \left| x^2 - a x + 2 \right| + \cos |x|$ be not differentiable at the two points $x = \alpha = 2$ and $x = \beta$ . Then the distance of the point $(\alpha, \beta)$ from the line $12 x + 5 y + 10 = 0$ is equal to:
(1) 3
(2) 4
(3) 2
(4) 5
Let a straight line $L$ pass through point $P(2, -1, 3)$ and be perpendicular to the lines $\frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2}$ and $\frac{x-3}{1} = \frac{y-2}{3} = \frac{z+2}{4}$ . If the line $L$ intersects the yz-plane at point $Q$ , then the distance between points $P$ and $Q$ is:
(1) 2
(2) $\sqrt{10}$
(3) 3
(4) $2 \sqrt{3}$
Let $S = \mathbb{N} \cup \{0\}$ . Define a relation $R$ from $S$ to $\mathbb{R}$ by $R = \left\{ (x, y) : \log_e y = x \log_e \left( \frac{2}{5} \right), x \in S, y \in \mathbb{R} \right\}$ . Then, the sum of all elements in the range of $R$ is equal to:
(1) $\frac{3}{2}$
(2) $\frac{5}{3}$
(3) $\frac{10}{9}$
(4) $\frac{5}{2}$
Let the line $x + y = 1$ meet the x and y axes at $A$ and $B$ , respectively. A right-angled triangle $AMN$ is inscribed in triangle $OAB$ , where $O$ is the origin and points $M$ and $N$ lie on lines $OB$ and $AB$ , respectively. If the area of triangle $AMN$ is $\frac{4}{9}$ of the area of triangle $OAB$ and $AN : NB = \lambda : 1$ , then the sum of all possible values of $\lambda$ is:
(1) $\frac{1}{2}$
(2) $\frac{13}{6}$
(3) $\frac{5}{2}$
(4) 2
If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ , whose midpoint is $\left( \frac{5}{2}, \frac{1}{2} \right)$ , then $\alpha + \beta$ is equal to:
(1) 37
(2) 46
(3) 58
(4) 72
If all words with or without meaning made using all letters of the word “KANPUR” are arranged as in a dictionary, then the word at the 440th position in this arrangement is:
(1) PRNAKU
(2) PRKANU
(3) PRKAUN
(4) PRNAUK
Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$ for which the equations $x + y + z = 1$ , $x + 2 y + 4 z = m$ , and $x + 4 y + 10 z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} \left( n^\alpha + n^\beta \right)$ is equal to:
(1) 440
(2) 3080
(3) 3410
(4) 560
Let $A = [a_{ij}]$ be a matrix of order $3 \times 3$ , with $a_{ij} = (\sqrt{2})^{i+j}$ . If the sum of all elements in the third row of $A^2$ is $\alpha + \beta \sqrt{2}$ , $\alpha, \beta \in \mathbb{Z}$ , then $\alpha + \beta$ is equal to:
(1) 280
(2) 168
(3) 210
(4) 224
Let $P$ be the foot of the perpendicular from point $(1, 2, 2)$ on the line $L: \frac{x-1}{1} = \frac{y+1}{-1} = \frac{z-2}{2}$ . Let the line $\vec{r} = (-\hat{i} + \hat{j} - 2 \hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k})$ , $\lambda \in \mathbb{R}$ , intersect line $L$ at $Q$ . Then $2 (PQ)^2$ is equal to:
(1) 27
(2) 25
(3) 29
(4) 19
Let a circle $C$ pass through points $(4, 2)$ and $(0, 2)$ , and its center lie on $3 x + 2 y + 2 = 0$ . Then the length of the chord of circle $C$ , whose midpoint is $(1, 2)$ , is:
(1) $\sqrt{3}$
(2) $2 \sqrt{3}$
(3) $4 \sqrt{2}$
(4) $2 \sqrt{2}$
Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$ . Let the random variable $X$ denote the possible values of the determinant of matrix $A$ . Then, the variance of $X$ is:
(1) $\frac{1}{4}$
(2) $\frac{3}{8}$
(3) $\frac{5}{8}$
(4) $\frac{3}{4}$
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is $\frac{29}{45}$ , then $n$ is equal to:
(1) 3
(2) 4
(3) 5
(4) 6
The remainder when $7^{103}$ is divided by 23 is equal to:
(1) 14
(2) 9
(3) 17
(4) 6
Let $f(x) = \int_{0}^{x} t (t^2 - 9 t + 20) \, dt$ , $1 \leq x \leq 5$ . If the range of $f$ is $[\alpha, \beta]$ , then $4 (\alpha + \beta)$ equals:
(1) 157
(2) 253
(3) 125
(4) 154
Let $\vec{a}$ be a unit vector perpendicular to vectors $\vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ and $\vec{c} = 2 \hat{i} + 3 \hat{j} - \hat{k}$ , and makes an angle of $\cos^{-1} \left( -\frac{1}{3} \right)$ with vector $\hat{i} + \hat{j} + \hat{k}$ . If $\vec{a}$ makes an angle of $\frac{\pi}{3}$ with vector $\hat{i} + \alpha \hat{j} + \hat{k}$ , then the value of $\alpha$ is:
(1) $-\sqrt{3}$
(2) $\sqrt{6}$
(3) $-\sqrt{6}$
(4) $\sqrt{3}$
If for the solution curve $y = f(x)$ of the differential equation $\frac{dy}{dx} + (\tan x) y = \frac{2 + \sec x}{(1 + 2 \sec x)^2}$ , $x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$ , $f \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{10}$ , then $f \left( \frac{\pi}{4} \right)$ is equal to:
(1) $\frac{9 \sqrt{3} + 3}{10 (4 + \sqrt{3})}$
(2) $\frac{\sqrt{3} + 1}{10 (4 + \sqrt{3})}$
(3) $\frac{5 - \sqrt{3}}{2 \sqrt{2}}$
(4) $\frac{4 - \sqrt{2}}{14}$
If $24 \int_{0}^{\frac{\pi}{4}} \left( \sin \left| 4 x - \frac{\pi}{12} \right| + [2 \sin x] \right) dx = 2 \pi + \alpha$ , where $[.]$ denotes the greatest integer function, then $\alpha$ is equal to.
If $\lim_{t \to 0} \left( \int_{0}^{1} (3 x + 5)^t \, dx \right)^{\frac{1}{t}} = \frac{\alpha}{5 e} \left( \frac{8}{5} \right)^{\frac{2}{3}}$ , then $\alpha$ is equal to.
Let $a_1, a_2, \ldots, a_{2024}$ be an arithmetic progression such that $a_1 + \left( a_5 + a_{10} + a_{15} + \ldots + a_{2020} \right) + a_{2024} = 2233$ . Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to.
Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$ . Then the number of all possible ordered pairs $(a, b)$ for which $\left| \frac{z - a}{z + b} \right| = 1$ and $\left| \begin{array}{ccc} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{array} \right| = 1$ , $z \in \mathbb{C}$ , where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$ , is equal to.
Let $y^2 = 12 x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ) = \frac{147}{4}$ . Let $C$ be the circle described taking $PQ[/sqrt{3} y = \beta$ , then $\beta - \alpha$ is equal to.