Jyotiradityaaaa/2025-Jee-Mains-Question huggingface dataset:
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 12
Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is

(1) 20
(2) 90
(3) 45
(4) 25
Correct option: 4
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 13
Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to

(1) 5
(2) 3
(3) 4
(4) 6
Correct option: 3
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 14
If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is

(1) \(I(19, 27)\)
(2) \(I(9, 1)\)
(3) \(I(1, 13)\)
(4) \(I(9, 13)\)
Correct option: 4
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 15
\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is

(1) \(\frac{9}{17}\)
(2) \(\frac{9}{17}\)
(3) \(\frac{9}{17}\)
(4) \(\frac{8}{17}\)
Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 16
Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to

(1) 92
(2) 118
(3) 102
(4) 108
Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 17
Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to

(1) \(\sqrt{15} \div 2\)
(2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\)
(3) \(2\sqrt{2}\)
(4) \(\sqrt{\frac{14}{3}}\)
Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 18
\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is: (1) 0  
(2) \( \frac{1}{\sqrt{15}} \)  
(3) \( \frac{1}{2\sqrt{5}} \)  
(4) \( -\frac{1}{2\sqrt{5}} \) 
Correct option: 4
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 19
Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is:

(1) \( \frac{90}{11} \)  
(2) \( \frac{85}{11} \)  
(3) \( \frac{61}{12} \)  
(4) \( \frac{567}{121} \)
Correct option: 4
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 20
Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to

(1) \( \sqrt{\frac{11}{6}} \)  
(2) \( \frac{1}{3\sqrt{2}} \)  
(3) \( 16 \)  
(4) \( 18 \)
Correct option: 1
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 21
Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.
Correct option: 5120
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 22
If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.
Correct option: 14
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 23
Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If

\[
A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}
\]

and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.
Correct option: 44
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 24
Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.
Correct option: 19
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 25
The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.
Correct option: 125
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 1
Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:

(1) 8750
(2) 9100
(3) 8925
(4) 8575
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 2
If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to:

$14x + 3y + \mu z = 33$

(1) 13
(2) 10
(3) 12
(4) 11
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 3
Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to:

(1) 4
(2) 8
(3) 6
(4) 2
Correct option: 2
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 4
The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is:

(1) $1 - \log_e 2$
(2) $\log_e 2$
(3) $1 + \log_e 2$
(4) $2 \log_e 2 - 1$
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 5
The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:

$25x + 101y = 176$

(1) $48x + 25y = 169$
(2) $5x + 16y = 31$
(3) $4x + 122y = 134$
(4) $4x + 122y = 134$
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 6
Let the points $(\frac{11}{2}, \alpha)$ 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:

(1) 44
(2) 22
(3) 33
(4) 55
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 7
Let $f : (0, \infty) \rightarrow \mathbb{R}$ 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 $2f(2)$ is equal to:

(1) 39
(2) 19
(3) 29
(4) 23
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 8
If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is:

(1) $\frac{6}{7}$
(2) 6
(3) $\frac{1}{7}$
(4) 1
Correct option: 2
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 9
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:

(1) 6
(2) 8
(3) 9
(4) 7
Correct option: 2
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 10
Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is: (1) \( \frac{3}{16} \)  
(2) \( \frac{5}{8} \)  
(3) \( \frac{3}{8} \)  
(4) \( \frac{1}{8} \)  
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 11
Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to:

(1) 3
(2) 4
(3) 1
(4) 6
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 12
Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is:

(1) \(2\sqrt{14}\)
(2) \(\mathbf{v}\)
(3) \(\sqrt{7}\)
(4) \(2\sqrt{7}\)
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 13
The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is:

(1) 1
(2) 0
(3) 2
(4) 3
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 14
The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is:

(1) Neither one-one nor onto
(2) Onto but not one-one
(3) Both one-one and onto
(4) One-one but not onto
Correct option: 4
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 15
In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to:

(1) 525
(2) 510
(3) 515
(4) 505
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 16
Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to:

(1) 22
(2) 20
(3) 21
(4) 19
Correct option: 3
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 17
Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to:

(1) 420
(2) 360
(3) 160
(4) 280
Correct option: 2
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 18
For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to:

(1) 16
(2) 25
(3) 9
(4) 36
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 19
If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) 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: (1) 7  
(2) 9  
(3) 8  
(4) 6  

\(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)
Correct option: 2
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 20
If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to:

(1) $\pi$

(2) 0

(3) $\pi - (\alpha + \beta + \gamma)$

(4) $3\pi$
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 21
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 ________.
Correct option: 957
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 22
If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.
Correct option: 16
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 23
Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.
Correct option: 1
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 24
Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.
Correct option: 392
JEE Main 2025 (24 Jan Shift 2) — Mathematics, Question 25
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 $25e_2^2$ is equal to ________.
Correct option: 55
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 1
Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then

(1) area of triangle ABO is \( \frac{11}{3} \) 
(2) ABO is an obtuse angled isosceles triangle 
(3) area of triangle ABO is \( \frac{11}{4} \) 
(4) ABO is a scalene triangle
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 2
Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is

(1) 545 
(2) 715 
(3) 735 
(4) 675
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 3
Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is

(1) \( \frac{73}{8} \) 
(2) \( \frac{25}{9} \) 
(3) \( \frac{16}{8} \) 
(4) \( \frac{75}{8} \)
Correct option: 1
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 4
The sum of all local minimum values of the function

\[
f(x) = \begin{cases} 
1 - 2x, & x < -1 \\
\frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\
\frac{1}{12}(x - 4)(x - 5), & x > 2 
\end{cases}
\]

is

(1) \( \frac{137}{72} \) 
(2) \( \frac{131}{72} \) 
(3) \( \frac{137}{72} \) 
(4) \( \frac{167}{72} \)
Correct option: 1
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 5
Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals

(1) 6 
(2) 18 
(3) 8 
(4) 20
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 6
Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to

(1) \( (\gamma, \beta^2 - 4\alpha) \) 
(2) \( (\alpha, \beta^2 + 4\gamma) \) 
(3) \( (\gamma, \beta^2 + 4\alpha) \) 
(4) \( (\alpha, \beta^2 - 4\gamma) \)
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 7
If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to

(1) \( 1.81\sqrt{2} \) 
(2) \( 41 \) 
(3) \( 82 \) 
(4) \( \frac{81}{2} \)
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 8
Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals:(1) \( \frac{1}{2} \)  
(2) \( \frac{3}{4} \)  
(3) \( \frac{1}{7} \)  
(4) \( \frac{2}{3} \) 
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 9
If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to
\[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8.
\]
Correct option: 1
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 10
\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to:
\[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}.
\]
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 11
Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\),
\[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}.
\]
Correct option: 3
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 12
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
\[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}.
\]
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 13
The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is
\[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2})
\]
Correct option: 3
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 14
Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then
\(5m \sum_{r=m}^{m+2} T_r\) is equal to
\[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112.
\]
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 15
Three defective oranges are accidently 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
\[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}.
\]
Correct option: 1
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 16
Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to
\[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2.
\]
Correct option: 1
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 17
If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals
\[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100.
\]
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 18
Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to : (1) \( 3a_{99} - 100 \)  
(2) \( 3a_{100} + 100 \)  
(3) \( 3a_{99} + 100 \)  
(4) \( 3a_{100} - 100 \)  

Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 19
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

(1) 4608
(2) 5720
(3) 5719
(4) 4607
Correct option: 4
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 20
The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is:

(1) reflexive and symmetric but not transitive
(2) an equivalence relation
(3) symmetric and transitive but not reflexive
(4) reflexive and transitive but not symmetric
Correct option: 2
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 21
Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals

\( \frac{5}{2} \)
Correct option: 5
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 22
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

\( \frac{5}{2} \)
Correct option: 1613
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 23
If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is

\( \frac{5}{2} \)
Correct option: 5
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 24
Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s 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

\( \frac{5}{\pi} \)
Correct option: 54
JEE Main 2025 (28 Jan Shift 1) — Mathematics, Question 25
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

\( \frac{5}{2} \)
Correct option: 6
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 1
Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is:

(1) 127 
(2) 258 
(3) 65 
(4) 2049
Correct option: 3
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 2
If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

(1) 26 
(2) 18 
(3) 23 
(4) 16
Correct option: 1
JEE Main 2025 (28 Jan Shift 2) — Mathematics, Question 3
Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is:

(1) 2 
(2) 9/2 
(3) 1 
(4) 0
Correct option: 3