Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: 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
Options:
$\frac{80}{3}$
$\frac{84}{3}$
$\frac{32}{3}$
$\frac{17}{3}$
Answer: $\frac{84}{3}$
Question: The sum, of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$, is
Options:
$3(3 - \sqrt{2})$
$6(3 - \sqrt{2})$
$6(2 - \sqrt{2})$
$3(2 - \sqrt{2})$
Answer: $6(2 - \sqrt{2})$
Question: Let $T_r$ be the $r^{\text{th}}$ term of an A.P. If for some $m$, $T_m = \frac{1}{25}$, $T_{25} = \frac{1}{20}$, and $20 \sum_{r=1}^{25} T_r = 13$, then $5m \sum_{r=m}^{2m} T_r$ is equal to
Options:
98
126
142
112
Answer: 126
Question: Three defective oranges are accidentally 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
Options:
$\frac{28}{75}$
$\frac{18}{25}$
$\frac{26}{75}$
$\frac{14}{25}$
Answer: $\frac{28}{75}$
Question: 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
Options:
3
6
2
Answer: 1
Question: If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{25 x^2 \cos^2 x}{2(x + e)^2} \, dx = \pi (\alpha \pi^2 + \beta)$, $\alpha, \beta \in \mathbb{Z}$, then $(\alpha + \beta)^2$ equals
Options:
64
196
144
100
Answer: 100
Question: 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
Options:
$3 a_{99} - 100$
$3 a_{100} - 100$
$3 a_{99} + 100$
$3 a_{100} + 100$
Answer: $3 a_{100} - 100$
Question: 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
Options:
4608
5720
5719
4607
Answer: 4607
Question: The relation $R = \{(x, y): x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:
Options:
reflexive and symmetric but not transitive
an equivalence relation
symmetric and transitive but not reflexive
reflexive and transitive but not symmetric
Answer: an equivalence relation
Question: Let $f(x) = \begin{cases} 3x, & x < 0 \\ \min \{1 + x + [x], x + 2[x]\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{cases}$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals
Options:
Answer: 5
Question: 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
Options:
Answer: 1613
Question: If $\alpha = 1 + \sum_{r=1}^6 (-3)^{r-1} {}^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is
Options:
Answer: 5
Question: Let $E_1: \frac{x^2}{5} + \frac{y^2}{4} = 1$ be an ellipse. Ellipses $E_i$ 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
Options:
Answer: 54
Question: 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
Options:
Answer: 6
Question: 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:
Options:
8750
9100
8925
8575
Answer: 8925
Question: If the system of equations $x + 2y - 3z = 2$, $2x + \lambda y + 5z = 5$, $14x + 3y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
Options:
13
10
12
11
Answer: 12
Question: Let $\mathbf{A} = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2 / \pi)} |\sin x| + \log_{(2 / \pi)} |\cos x| = 2\}$ and $\mathbf{B} = \{x \geqslant 0 : \sqrt{x} (\sqrt{x} - 4) - 3 |\sqrt{x} - 2| + 6 = 0\}$. Then $n(\mathbf{A} \cup \mathbf{B})$ is equal to:
Options:
4
8
6
2
Answer: 8
Question: The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$, and the $y$-axis is:
Options:
$1 - \log_e 2$
$\log_e 2$
$1 + \log_e 2$
$2 \log_e 2 - 1$
Answer: $1 - \log_e 2$
Question: The equation of the chord of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:
Options:
$48x + 25y = 169$
$5x + 16y = 31$
$25x + 101y = 176$
$4x + 122y = 134$
Answer: $48x + 25y = 169$
Question: Let the points $\left(\frac{11}{2}, \alpha\right)$ 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:
Options:
44
55
Answer: 33
Question: Let $f : (0, \infty) \to \mathbf{R}$ 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 $2 f(2)$ is equal to:
Options:
39
19
29
23
Answer: 39
Question: If $7 = 5 + \frac{1}{7} (5 + \alpha) + \frac{1}{77} (5 + 2\alpha) + \frac{1}{77} (5 + 3\alpha) + \infty$, then the value of $\alpha$ is:
Options:
$\frac{6}{7}$
6
$\frac{1}{7}$
1
Answer: 6
Question: 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:
Options:
6
8
9
7
Answer: 8
Question: 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 $\mathbf{P}(\mathbf{E})$ is:
Options:
$\frac{3}{16}$
$\frac{5}{8}$
$\frac{3}{8}$
$\frac{1}{8}$
Answer: $\frac{3}{8}$
Question: Let the position vectors of three vertices of a triangle be $4 \vec{p} + \vec{q} - 3 \vec{r}$, $-5 \vec{p} + \vec{q} + 2 \vec{r}$, and $2 \vec{p} - \vec{q} + 2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p} + \vec{q} + \vec{r}}{2}$ and $\alpha \vec{p} + \beta \vec{q} + \gamma \vec{r}$ respectively, then $\alpha + 2 \beta + 5 \gamma$ is equal to:
Options:
3
4
1
6
Answer: 3
Question: 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:
Options:
$2 \sqrt{14}$
$\sqrt{14}$
$3 \sqrt{7}$
$2 \sqrt{7}$
Answer: $2 \sqrt{14}$
Question: The number of real solution(s) of the equation $x^2 + 3x + 2 = \min \{|x - 3|, |x + 2|\}$ is:
Options:
0
2
3
Answer: 2
Question: The function $f : (-\infty, \infty) \to (-\infty, 1)$, defined by $f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}$, is:
Options:
Neither one-one nor onto
Onto but not one-one
Both one-one and onto
One-one but not onto
Answer: One-one but not onto
Question: In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to:
Options:
525
510
515
505
Answer: 515
Question: Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1 + x)^{2n - 1}$. If $2A = 5B$, then $n$ is equal to:
Options:
22
20
21
19
Answer: 21
Question: 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:
Options:
420
360
160
280
Answer: 360
Question: For some $a$, $b$, let $f(x) = \left| \begin{array}{ccc} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to:
Options:
16
25
9
36
Answer: 16
Question: If the equation of the parabola with vertex $\mathbf{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:
Options:
7
9
8
6
Answer: 9
Question: If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{\beta + \frac{(1 + \beta^5)}{(\alpha - \beta)}\right\} + \cot^{-1} \left\{\gamma + \frac{(1 + \gamma^2)}{(\beta - \gamma)}\right\} + \cot^{-1} \left\{\alpha + \frac{(1 + \alpha^2)}{(\gamma - \alpha)}\right\}$ is equal to:
Options:
$\pi$
0
$\frac{\pi}{2} - (\alpha + \beta + \gamma)$
$3 \pi$
Answer: $\pi$
Question: 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:
Options:
Answer: 957
Question: If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C$, where $C$ is the constant of integration, then $\alpha + 2 \beta$ is equal to:
Options:
Answer: 16
Question: Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right) = 0$, then $y'\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right)$ is equal to:
Options:
Answer: 1
Question: 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:
Options:
Answer: 392
Question: 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 $25 e_2^2$ is equal to:
Options:
Answer: 55
Question: Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to:
Options:
628
812
526
784
Answer: 784
Question: Let $x = x(y)$ be the solution of the differential equation $y^2 \mathrm{d}x + \left(x - \frac{1}{y}\right) \mathrm{d}y = 0$. If $x(1) = 1$, then $x\left(\frac{1}{y}\right)$ is:
Options:
$\frac{1}{2} + e$
$3 + e$
$3 - e$
$\frac{3}{2} + e$
Answer: $3 - e$
Question: 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:
Options:
4
14
13
11
Answer: 14
Question: The product of all solutions of the equation $e^{5(\log_e x)^2 + 3} = x^8$, $x > 0$, is:
Options:
$e^{8/5}$
$e^{5/5}$
$e^2$
$e$
Answer: $e^{8/5}$
Question: 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:
Options:
19
24
21
22
Answer: 22
Question: Let $f(x) = 7 \tan^8 x + 7 \tan^8 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_0^{\pi/4} f(x) \mathrm{d}x$ and $I_2 = \int_0^{\pi/4} x f(x) \mathrm{d}x$. Then $7 I_1 + 12 I_2$ is equal to:
Options:
2
1
$2\pi$
$\pi$
Answer: 1
Question: Let the parabola $y = x^2 + p x - 3$ meet the coordinate axes at the points $P$, $Q$, and $R$. If the circle $C$ with centre at $(-1, -1)$ passes through the points $P$, $Q$, and $R$, then the area of $\triangle PQR$ is:
Options:
7
4
6
5
Answer: 6
Question: Let $L_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{6}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?
Options:
$\left(\frac{14}{3}, -3, \frac{22}{3}\right)$
$\left(-\frac{5}{3}, -7, 1\right)$
$\left(2, 3, \frac{1}{3}\right)$
$\left(\frac{5}{3}, -1, \frac{1}{3}\right)$
Answer: $\left(\frac{14}{3}, -3, \frac{22}{3}\right)$
Question: 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:
Options:
2525
5220
2384
2406
Answer: 2525
Question: From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
Options:
5148
6084
4356
14950
Answer: 5148
Question: Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1} x)^2 + (\csc^{-1} x)^2\right)$ is:
Options:
$24\pi^2$
$22\pi^2$
$31\pi^2$
$18\pi^2$
Answer: $22\pi^2$
Question: Let $f: \mathbb{R} \to \mathbb{R}$ 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 $\mathbb{R} = \{(x, y) \mid 0 \leq y \leq f(a x), 0 \leq x \leq 2\}$ is:
Options:
$e^2 - 1$
$e^2 + 1$
$e^4 + 1$
$e^4 - 1$
Answer: $e^2 - 1$
Question: 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:
Options:
$3\pi + 8$
$6\pi - 16$
$3\pi - 8$
$6\pi - 8$
Answer: $6\pi - 16$
Question: 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:
Options:
$\frac{24}{5}$
$\frac{25}{6}$
$\frac{144}{5}$
$\frac{288}{5}$
Answer: $\frac{288}{5}$
Question: 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:
Options:
0
$\frac{2}{3}$
1
$\frac{1}{3}$
Answer: $\frac{2}{3}$
Question: 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 $64(\mu + \sigma^2)$ is:
Options:
51
64
32
48
Answer: 48
Question: The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is:
Options:
6
5
7
4
Answer: 5
Question: A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $(2, 5)$ and intersects the 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:
Options:
10
15
12
14
Answer: 15
Question: 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:
Options:
36
31
37
29
Answer: 31
Question: Let $z_1, z_2$, and $z_3$ 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 $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is:
Options:
24
29
41
31
Answer: 29
Question: 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 $4m + 2n$ is equal to:
Options:
Answer: 34
Question: If $\sum_{x=0}^5 \frac{{ }^{11} C_{2x}}{2x + 2} = \frac{m}{n}$, $\gcd(m, n) = 1$, then $m - n$ is equal to:
Options:
Answer: 2035
Question: 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 the vectors $\vec{b}$ and $\vec{c}$ is:
Options:
Answer: 16
Question: Let the function, $f(x) = \begin{cases} -3 a x^2 - 2, & x < 1 \\ a^2 + b x, & x \geqslant 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. 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:
Options:
Answer: 34
Question: Let $L_1: \frac{x-1}{3} = \frac{y-1}{4} = \frac{z+1}{0}$ and $L_2: \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{0}$, $\alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26 \alpha (PB)^2$ is:
Options:
Answer: 216
Question: The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{4}=\frac{y-2}{2}=\frac{z+3}{3}$ is:
Options:
$\sqrt{17}$
$\sqrt{15}$
$\sqrt{14}$
$\sqrt{13}$
Answer: $\sqrt{14}$
Question: Let $\mathbf{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x+y| \geqslant 3\}$ and $\mathbf{B}=\{(x, y) \in \mathbf{R} \times \mathbf{R} : |x|+|y| \leq 3\}$. If $\mathbf{C}=\{(x, y) \in \mathbf{A} \cap \mathbf{B} : x=0 \text{ or } y=0\}$, then $\sum_{(x, y) \in \mathbf{C}}|x+y|$ is:
Options:
15
24
18
12
Answer: 12
Question: Let $\boldsymbol{X}=\mathbf{R} \times \mathbf{R}$. Define a relation $R$ on $X$ as: $\left(a_1, b_1\right) R \left(a_2, b_2\right) \Leftrightarrow b_1=b_2$. Statement I: $\mathbf{R}$ 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 $y=x$. In the light of the above statements, choose the correct answer from the options given below:
Options:
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Both Statement I and Statement II are true
Statement I is false but Statement II is true
Answer: Statement I is true but Statement II is false
Question: 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:
Options:
48
55
62
47
Answer: 55
Question: 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 $9\left(x^2 + \alpha y^2 + \beta xy + \gamma x + 28 y\right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to:
Options:
22
21
23
24
Answer: 23
Question: 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:
Options:
21
14
6
10
Answer: 14
Question: $\lim_{x \to \infty} \frac{(2x^2 - 3x + 5)(3x - 1)^{\frac{2}{x}}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^2}}$ is equal to:
Options:
$\frac{2}{\sqrt{3x}}$
$\frac{2e}{\sqrt{3}}$
$\frac{2}{3\sqrt{e}}$
$\frac{2e}{3}$
Answer: $\frac{2}{3\sqrt{e}}$
Question: Let the point $A$ divide the line segment joining the 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:
Options:
$\sqrt{7}$
14
3
7
Answer: 7
Question: The length of the chord of the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$, whose mid-point is $\left(1, \frac{1}{2}\right)$, is:
Options:
$\frac{5}{3}\sqrt{15}$
$\frac{1}{3}\sqrt{15}$
$\frac{2}{3}\sqrt{15}$
$\sqrt{15}$
Answer: $\frac{2}{3}\sqrt{15}$
Question: The system of equations $x + y + z = 6$, $x + 2y + 5z = 9$, $x + 5y + \lambda z = \mu$ has no solution if:
Options:
$\lambda = 15, \mu \neq 17$
$\lambda \neq 17, \mu \neq 18$
$\lambda = 17, \mu \neq 18$
$\lambda = 17, \mu = 18$
Answer: $\lambda = 17, \mu \neq 18$
Question: Let the range of the function $f(x) = 6 + 16 \cos x \cdot \cos \left(\frac{\pi}{3} - x\right) \cdot \cos \left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is:
Options:
11
8
10
9
Answer: 11
Question: 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:
Options:
$1 - 2(\log_e 2)^2$
$1 - 2(\log_e 2)$
$2(\log_e 2) - 1$
$2(\log_e 2)^2 - 1$
Answer: $2(\log_e 2)^2 - 1$
Question: A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is $1 \, \text{cm}$, 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:
Options:
$196\pi$
$256\pi$
$225\pi$
$128\pi$
Answer: $256\pi$
Question: The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\frac{z}{2} + \frac{2}{z}\right| = 1$, is:
Options:
4
8
10
6
Answer: 8
Question: 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:
Options:
-1
2
1
0
Answer: -1
Question: If $I = \int_0^{\frac{\pi}{2}} \frac{\sin \frac{1}{2} x}{\sin^2 x + \cos^2 x} \, dx$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin^2 x + \cos^4 x} \, dx$ equals:
Options:
$\frac{\pi^2}{12}$
$\frac{\pi^2}{4}$
$\frac{\pi^2}{16}$
$\frac{\pi^2}{8}$
Answer: $\frac{\pi^2}{16}$
Question: 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:
Options:
$\frac{7}{10}$
$\frac{4}{5}$
$\frac{23}{30}$
$\frac{3}{5}$
Answer: $\frac{4}{5}$
Question: Let the shortest distance from $(a, 0)$, $a > 0$, to the parabola $y^2 = 4x$ be 4. Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is:
Options:
$x^2 + y^2 - 10x + 9 = 0$
$x^2 + y^2 - 6x + 5 = 0$
$x^2 + y^2 - 4x + 3 = 0$
$x^2 + y^2 - 8x + 7 = 0$
Answer: $x^2 + y^2 - 6x + 5 = 0$
Question: 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:
Options:
18
13
8
20
Answer: 13
Question: 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 $\frac{\pi^2 + 8e + 1}{e}$, then the value of $a$ is:
Options:
8
7
5
6
Answer: 5
Question: The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is:
Options:
Answer: 8788
Question: The roots of the quadratic equation $3x^2 - px + q = 0$ are $10^{\text{th}}$ and $11^{\text{th}}$ 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 $q - 2p$ is equal to:
Options:
Answer: 474
Question: 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:
Options:
Answer: 17280