Jyotiradityaaaa/2025-Jee-Mains-Question huggingface dataset:
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 4
Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to:

(1) 283 
(2) 287 
(3) 295 
(4) 299
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 5
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) \( (-\infty, -1] \cup [1, \infty) \) 
(3) \( (-\infty, \infty) \) 
(4) \( (-\infty, \infty) \) \( \setminus \{0\} \)
Correct option: 3
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 6
Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

(1) \( \frac{1}{2} \) 
(2) \( \frac{1}{4} \) 
(3) \( \frac{3}{5} \) 
(4) \( \frac{1}{5} \)
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 7
If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to:

(1) 10 
(2) 4 
(3) 2 
(4) 8
Correct option: 4
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 8
Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is:

(1) 270 
(2) 340 
(3) 320 
(4) 310
Correct option: 4
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 9
Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:(1) 29  
(2) 30  
(3) 31  
(4) 36  
Correct option: 2
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 10
Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is:

(1) $\frac{4}{15}$
(2) $\frac{1}{3}$
(3) $\frac{2}{5}$
(4) $\frac{4}{5}$
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 11
Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to:

(1) 11
(2) 13
(3) 15
(4) 9
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 12
For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is:

(1) 540
(2) 675
(3) 1350
(4) 135
Correct option: 2
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 13
Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is:

(1) 7
(2) 6
(3) 1
(4) 9
Correct option: 2
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 14
If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line:

(1) $x + 9y = 36$
(2) $4x - 9y = 12$
(3) $6x - 9y = 20$
(4) $9x - 9y = 32$
Correct option: 3
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 15
If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to:

(1) $4 \log_e 2 - 2$
(2) $2 - \log_e x$
(3) $\log_e 2 + 2$
(4) $4 \log_e 2 + 2$
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 16
The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is:

(1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$
(2) $\frac{\pi}{2} - \frac{1}{3}$
(3) $\frac{\pi}{2} - \frac{1}{3}$
(4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$
Correct option: 2
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 17
The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is:

(1) 54
(2) 44
(3) 41
(4) 66
Correct option: 4
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 18
If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is:

(1) 20
(2) 22
(3) 18
(4) 26
Correct option: 2
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 19
If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to:
(1) $-2$  
(2) $6$  
(3) $-6$  
(4) $2$
Correct option: 4
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 20
Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is:
(1) $-2\sqrt{10}$  
(2) $12$  
(3) $6$  
(4) $-6$
Correct option: 3
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 21
Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ 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 $y^2 = 4x$, then the value of $(a + d)$ is
Correct option: 14
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 22
The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is
Correct option: 64
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 23
If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to
Correct option: 4
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 24
The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to
Correct option: 20
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 25
Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 1
Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to:

(1) 100  (2) 120  (3) 110  (4) 90
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 2
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:

(1) 90  (2) 84  (3) 122  (4) 108
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 3
The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is:

(1) 2  (2) 3  (3) 1  (4) 4
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 4
Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is:

(1) both reflexive and transitive but not symmetric  (2) an equivalence relation
(3) reflexive but neither symmetric nor transitive  (4) both reflexive and symmetric but not transitive
Correct option: 2
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 5
Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to:

(1) 392  (2) 384  (3) 192  (4) 96
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 6
Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is:

(1) 173  (2) 164  (3) 158  (4) 161
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 7
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 \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) 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) (5, 17, 4)  (2) (2, 8, 5)
(3) (8, 26, 12)  (4) (-1, -1, 1)
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 8
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) 462  (2) 77  (3) 154  (4) 308
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 9
The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to:

(1) 3 \log_e 4  (2) 4 \log_e 3
(3) 6 \log_e 4  (4) 2 \log_e 3
Correct option: 2
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 10
Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals:

\[
\begin{align*}
(1) & \quad 4\sqrt{6} \\
(2) & \quad 6\sqrt{6} \\
(3) & \quad 18\sqrt{6}/5 \\
(4) & \quad 24\sqrt{6}/5 \\
\end{align*}
\]
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 11
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:

\[
\begin{align*}
(1) & \quad 288 \\
(2) & \quad 222 \\
(3) & \quad 242 \\
(4) & \quad 262 \\
\end{align*}
\]
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 12
Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is:

\[
\begin{align*}
(1) & \quad 13 \\
(2) & \quad 10 \\
(3) & \quad 3 \\
(4) & \quad 7 \\
\end{align*}
\]
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 13
Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the 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:

\[
\begin{align*}
(1) & \quad 20 \\
(2) & \quad 18 \\
(3) & \quad 25 \\
(4) & \quad 16 \\
\end{align*}
\]
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 14
Let \( M \) and \( m \) respectively be the maximum and the minimum values of

\[
f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R}
\]

Then \( M^4 - m^4 \) is equal to:

\[
\begin{align*}
(1) & \quad 1280 \\
(2) & \quad 1295 \\
(3) & \quad 1215 \\
(4) & \quad 1040 \\
\end{align*}
\]
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 15
Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to:

\[
\begin{align*}
(1) & \quad 47 \\
(2) & \quad 36 \\
(3) & \quad 47 \\
(4) & \quad 40 \\
\end{align*}
\]
Correct option: 2
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 16
The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is:

\[
\begin{align*}
(1) & \quad 4/3 \\
(2) & \quad 2 \\
(3) & \quad 7/3 \\
(4) & \quad 5/3 \\
\end{align*}
\]
Correct option: 4
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 17
The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is:

\[
\begin{align*}
(1) & \quad 2184 \\
(2) & \quad 2196 \\
(3) & \quad 2148 \\
(4) & \quad 2172 \\
\end{align*}
\]
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 18
Let \( y = y(x) \) be the solution of the differential equation
\[
\cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\]
If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to:

\[
\begin{align*}
(1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\
(2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\
(3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\
(4) & \frac{1}{\log_2 (4)}
\end{align*}
\]
Correct option: 1
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 19
Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:

\[
\begin{align*}
(1) & \sqrt{14} \\
(2) & 3\sqrt{7} \\
(3) & 2\sqrt{14} \\
(4) & 5\sqrt{7}
\end{align*}
\]
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 20
Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to:

\[
\begin{align*}
(1) & 16 \\
(2) & 12 \\
(3) & 14 \\
(4) & 18
\end{align*}
\]
Correct option: 3
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 21
Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.
Correct option: 5
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 22
Let \( f : (0, \infty) \rightarrow \mathbb{R} \) 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 ______.
Correct option: 112
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 23
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 ______.
Correct option: 1405
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 24
Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.
Correct option: 2
JEE Main 2025 (29 Jan Shift 1) — Mathematics, Question 25
Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which
\[
\lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1
\]
is equal to ______.
Correct option: 24