CK0607/2025-Jee-Mains-Question huggingface dataset:
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 1

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to:

(1) 628  
(2) 812  
(3) 526  
(4) 784

Correct option: 4
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 2

Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is:

(1) $\frac{1}{3} + e$  
(2) $3 + e$  
(3) $3 - e$  
(4) $\frac{3}{2} + e$

Correct option: 3
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 3

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) 4  
(2) 14  
(3) 13  
(4) 11

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 4

The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is:

(1) $e^{8/5}$  
(2) $e^{6/5}$  
(3) $e^{2}$  
(4) $e$

Correct option: 1
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 5

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) 19  
(2) 24  
(3) 21  
(4) 22

Correct option: 4
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 6

Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to:

(1) 2  
(2) 1  
(3) $2\pi$  
(4) $\pi$

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 7

Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is:

(1) 7  
(2) 4  
(3) 3  
(4) 5

Correct option: 3
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 8

Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

(1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$  
(2) $\left( -\frac{5}{3}, -7, 1 \right)$  
(3) $\left( 2, 3, \frac{1}{2} \right)$  
(4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$

Correct option: 1
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 9

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 $\sum_{n=1}^{100} \log_2 f(n)$ is equal to:

(1) 2525  
(2) 5220  
(3) 2384  
(4) 2406

Correct option: 1
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 10

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:(1) 5148  
(2) 6084  
(3) 4356  
(4) 14950  

Correct option: 1
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 11

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is:

(1) \(24\pi^2\)  
(2) \(22\pi^2\)  
(3) \(31\pi^2\)  
(4) \(18\pi^2\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 12

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 \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), 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) \(e^2 + 1\)  
(3) \(e^4 + 1\)  
(4) \(e^4 - 1\)

Correct option: 1
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 13

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) \(3\pi + 8\)  
(2) \(6\pi - 16\)  
(3) \(3\pi - 8\)  
(4) \(6\pi - 8\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 14

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) \(\frac{24}{5}\)  
(2) \(\frac{25}{9}\)  
(3) \(\frac{144}{5}\)  
(4) \(\frac{288}{5}\)

Correct option: 4
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 15

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) \(0\)  
(2) \(\frac{4}{3}\)  
(3) \(1\)  
(4) \(\frac{1}{2}\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 16

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:

(1) \(51\)  
(2) \(64\)  
(3) \(32\)  
(4) \(48\)

Correct option: 4
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 17

The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is:

(1) \(6\)  
(2) \(5\)  
(3) \(7\)  
(4) \(4\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 18

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:

(1) \(10\)  
(2) \(15\)  
(3) \(12\)  
(4) \(14\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 19

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) \(36\)  
(2) \(31\)  
(3) \(37\)  
(4) \(29\)

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 20

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 $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is:

(1) 24  (2) 29  (3) 41  (4) 31

Correct option: 2
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 21

Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.

Correct option: 34
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 22

If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.

Correct option: 2035
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 23

Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.

Correct option: 16
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 24

Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ 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 ________.

Correct option: 34
JEE Main 2025 (22 Jan Shift 1) — Mathematics, Question 25

Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, 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 $26\alpha(\text{PB})^2$ is ________.

Correct option: 216
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 1

Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals:

(1) 253  (2) 154  (3) 125  (4) 157

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 2

Let \( \vec{a} \) be a unit vector perpendicular to the 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 \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is:

(1) \( \sqrt{6} \)  (2) \( -\sqrt{6} \)  (3) \( -\sqrt{3} \)  (4) \( \sqrt{3} \)

Correct option: 2
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 3

If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to:

(1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \)  (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \)  (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \)  (4) \( \frac{5 - \sqrt{3}}{14} \)

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 4

Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to:

(1) 25  (2) 19  (3) 29  (4) 27

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 5

Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the 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) 224  (3) 210  (4) 168

Correct option: 2
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 6

Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is:

(1) 2  (2) \( \frac{5}{2} \)  (3) \( \frac{1}{2} \)  (4) \( \frac{13}{6} \)

Correct option: 1
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 7

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged in a dictionary, then the word at 440th position in this arrangement, is:

(1) PRNAUK  (2) PRKANU  (3) PRKAUN  (4) PRNAUK

Correct option: 3
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 8

If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, 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) 2119  (4) 2139

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 9

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 the matrix \( A \). Then, the variance of \( X \) is:(1) \( \frac{3}{8} \)  
(2) \( \frac{5}{8} \)  
(3) \( \frac{3}{8} \)  
(4) \( \frac{1}{4} \)  

Correct option: 3
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 10

Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to:

(1) 5  
(2) 4  
(3) 3  
(4) 2

Correct option: 3
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 11

Let the area enclosed between the curves \( |y| = 1 - x^2 \) 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) 33  
(3) 15  
(4) 18

Correct option: 2
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 12

The remainder, when \( 7^{10^3} \) is divided by 23, is equal to:

(1) 6  
(2) 17  
(3) 9  
(4) 14

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 13

If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to:

(1) 58  
(2) 46  
(3) 37  
(4) 72

Correct option: 1
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 14

If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

(1) 195  
(2) 179  
(3) 186  
(4) 174

Correct option: 3
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 15

Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is:

(1) \( \sqrt{3} \)  
(2) \( 2\sqrt{2} \)  
(3) \( 2\sqrt{3} \)  
(4) \( 4\sqrt{2} \)

Correct option: 3
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 16

Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is:

(1) \( \sqrt{10} \)  
(2) \( 2\sqrt{3} \)  
(3) 2  
(4) 3

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 17

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) 6  
(2) 3  
(3) 5  
(4) 4

Correct option: 1
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 18

Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to:

(1) 3080
(2) 560
(3) 3410
(4) 440

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 19

Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$

Then, the sum of all the elements in the range of $R$ is equal to:

(1) $\frac{10}{9}$
(2) $\frac{5}{2}$
(3) $\frac{\sqrt{3}}{2}$
(4) $\frac{1}{3}$

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 20

If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then

$(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to:

(1) 4
(2) $\frac{4}{3}$
(3) 3
(4) 2

Correct option: 4
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 21

If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.

Correct option: 12
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 22

Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.

Correct option: 11132
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 23

If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.

Correct option: 64
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 24

Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.

Correct option: 1328
JEE Main 2025 (29 Jan Shift 2) — Mathematics, Question 25

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 $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.

Correct option: 10
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 1

For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals:

1. 56
2. 132
3. 174
4. 280

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 2

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 the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:

1. 96
2. 144
3. 120
4. 72

Correct option: 2
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 3

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals:

1. 5
2. 4
3. 3
4. 8

Correct option: 1
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 4

Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to:

1. 148
2. 136
3. 144
4. 140

Correct option: 2
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 5

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 the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is:

1. $\frac{4}{3}
2. \frac{7}{4}
3. \frac{5}{3}
4. \frac{3}{4}$

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 6

If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals:

1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$
2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$
3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$
4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$

Correct option: 2
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 7

The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is:

1. $\frac{8}{3}$
2. $\frac{4}{3}$
3. 8
4. $\frac{3}{2}$

Correct option: 1
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 8

Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are:

1. 2 and 3
2. 2 and 1
3. 3 and 2
4. 1 and 3

Correct option: 1
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 9

Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to: (1) \( 17\sqrt{3} \)  
(2) \( \frac{263\sqrt{3}}{8} \)  
(3) \( \frac{34\sqrt{3}}{3} \)  
(4) \( \frac{34\sqrt{3}}{8} \) 

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 10

Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is:

1. 2
2. 1
3. 0
4. 3

Correct option: 3
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 11

If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals:

1. \( e^{-1} \)
2. \( e^2 \)
3. \( e^4 \)
4. \( e^6 \)

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 12

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. 151
2. 139
3. 163
4. 127

Correct option: 1
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 13

Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:

1. 6
2. 5
3. 8
4. 4

Correct option: 2
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 14

The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is:

1. \( 4\sqrt{3} \)
2. \( 5\sqrt{2} \)
3. \( 4\sqrt{3} \)
4. \( 3\sqrt{5} \)

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 15

The system of linear equations:

\[
\begin{align*}
x + y + 2z &= 6 \\
-2x + 3y + az &= a + 1 \\
7a + 3b &= 0
\end{align*}
\]

If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:

1. 16
2. 12
3. 22
4. 9

Correct option: 1
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 16

If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to:

1. \( e^{\pi/12} \)
2. \( e^{\pi/4} \)
3. \( e^{\pi/3} \)
4. \( e^{\pi/6} \)

Correct option: 4
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 17

Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals:

1. 24
2. 25
3. 17
4. 27

Correct option: 2
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 18

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) \( 4\pi \)  
(2) \( \frac{5\pi}{6} \)  
(3) \( \pi \)  
(4) \( \frac{\pi}{2} \)  

Correct option: 3
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 19

Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), 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) \( 1 + \frac{\pi}{6} \)
(4) \( 1 + \frac{\pi}{4} \)

Correct option: 1