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JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 4 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) $\frac{3}{2} + e$ | 3 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) $e$ | 1 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 4 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) $\pi$ | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 3 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$ | 1 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 1 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(18\pi^2\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(e^4 - 1\) | 1 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(6\pi - 8\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(\frac{288}{5}\) | 4 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(\frac{1}{2}\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(48\) | 4 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 17 | The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is:
(1) \(6\)
(2) \(5\)
(3) \(7\)
(4) \(4\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(14\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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) \(29\) | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 ________. | 34 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 22 | If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________. | 2,035 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 ________. | 16 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 ________. | 34 |
JEE Main 2025 (22 Jan Shift 1) | Mathematics | 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 ________. | 216 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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) \( \sqrt{3} \) | 2 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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) \( \frac{5 - \sqrt{3}}{14} \) | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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) \( \frac{13}{6} \) | 1 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 3 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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: | 3 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 3 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 12 | The remainder, when \( 7^{10^3} \) is divided by 23, is equal to:
(1) 6
(2) 17
(3) 9
(4) 14 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 3 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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) \( 4\sqrt{2} \) | 3 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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) $\frac{1}{3}$ | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 ________. | 12 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 ________. | 11,132 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 ________. | 64 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 ________. | 1,328 |
JEE Main 2025 (29 Jan Shift 2) | Mathematics | 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 ________. | 10 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. $\frac{4}{3}
2. \frac{7}{4}
3. \frac{5}{3}
4. \frac{3}{4}$ | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ | 2 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. $\frac{3}{2}$ | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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: | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 3 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. \( e^6 \) | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. \( 3\sqrt{5} \) | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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. \( e^{\pi/6} \) | 4 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 2 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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 | 3 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 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) \( 1 + \frac{\pi}{4} \) | 1 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 20 | 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 the foci of \( H \) 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) 9
(3) 8
(4) 7 | 3 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 21 | If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______. | 465 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 22 | 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 _______. | 3 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 23 | Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______. | 145 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 24 | Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______. | 27 |
JEE Main 2025 (22 Jan Shift 2) | Mathematics | 25 | 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 _______. | 28 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 1 | If the first term of an A.P. 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) $-1080$
- (2) $-1020$
- (3) $-1200$
- (4) $-120$ | 1 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 2 | 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 the dice are thrown together, is
- (1) $\frac{3}{16}$
- (2) $\frac{1}{4}$
- (3) $\frac{3}{8}$
- (4) $\frac{5}{8}$ | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 3 | Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is
- (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$
- (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$
- (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$
- (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 4 | If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to
- (1) $AB^{-1} + A^{-1}B$
- (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$
- (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$
- (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$ | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 5 | Marks obtained by all the 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) 52
- (2) 48
- (3) 44
- (4) 40 | 3 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 6 | Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to
- (1) 4
- (2) 32
- (3) 8
- (4) 16 | 3 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 7 | If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to
- (1) 20
- (2) 5
- (3) 8
- (4) 10 | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 8 | If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 9 | Let \( P \) be the foot of the perpendicular from the 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
\begin{align*}
(1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\
(3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30}
\end{align*} | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 10 | Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) 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
\begin{align*}
(1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\
(3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3}
\end{align*} | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 11 | Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is
\begin{align*}
(1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\
(3) \ (0, \infty) & & \quad (4) \ \mathbb{R}
\end{align*} | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 12 | \((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \)
If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to
\begin{align*}
(1) \ 6 & & \quad (2) \ 10 \\
(3) \ 20 & & \quad (4) \ 12
\end{align*} | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 13 | The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
\begin{align*}
(1) \ 36000 & & \quad (2) \ 37000 \\
(3) \ 34000 & & \quad (4) \ 35000
\end{align*} | 1 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 14 | 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 in \( R \) so that \( R \) becomes an equivalence relation, is
\begin{align*}
(1) \ 10 & & \quad (2) \ 7 \\
(3) \ 8 & & \quad (4) \ 9
\end{align*} | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 15 | 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 \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to
\begin{align*}
(1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\
(3) \ 2 & & \quad (4) \ 3
\end{align*} | 4 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 16 | The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is
\begin{align*}
(1) \ 2 & & \quad (2) \ \log_2 2 \\
(3) \ 1 & & \quad (4) \ e^2
\end{align*} | 3 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 17 | Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals:
- (1) 50
- (2) 100
- (3) \( \frac{81}{25} \)
- (4) \( \frac{121}{25} \) | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 18 | The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is
- (1) \( 2/3 \)
- (2) 0
- (3) \( 3/2 \)
- (4) 1 | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 19 | Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to
- (1) 22
- (2) 39
- (3) 40
- (4) 26 | 2 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 20 | If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), 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) \( x + \tan^{-1}\frac{4}{5} \)
- (3) \( x - \tan^{-1}\frac{5}{12} \)
- (4) \( x + \tan^{-1}\frac{5}{12} \) | 3 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 21 | Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to | 19 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 22 | 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 | 117 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 23 | If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to | 30 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 24 | The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to | 612 |
JEE Main 2025 (23 Jan Shift 1) | Mathematics | 25 | If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to | 77 |
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