JEE Main 2025 April 2 Shift 2 Maths Question Paper, Exam Analysis, and Answer Keys (Available)

If the image of the point \( P(1, 0, 3) \) in the line joining the points \( A(4, 7, 1) \) and \( B(3, 5, 3) \) is \( Q(\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

(1) 47/3
(2) 46/3
(3) 18
(4) 13

Let \( f : [1, \infty) \to [2, \infty) \) be a differentiable function,
If \( \int_{1}^{x} f(t) \, dt = 5x f(x) - x^5 - 9 \) for all \( x \geq 1 \), then the value of \( f(3) \) is :

• (1) 18
  (2) 32
  (3) 22
  (4) 26

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by \( \frac{21}{2} \). Then the number of terms which are integers in the A.P. is:

• (1) 4
  (2) 10
  (3) 6
  (4) 8

Let \( A = \{1, 2, 3, \dots, 10\} \) and \( R \) be a relation on \( A \) such that \( R = \{(a, b) : a = 2b + 1\} \). Let \( (a_1, a_2), (a_3, a_4), (a_5, a_6), \dots, (a_k, a_{k+1}) \) be a sequence of \( k \) elements of \( R \) such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer \( k \), for which such a sequence exists, is equal to:

• (1) 6
  (2) 7
  (3) 5
  (4) 8

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

• (1) \( \frac{4}{\sqrt{17}} \)
  (2) \( \frac{\sqrt{5}}{16} \)
  (3) \( \frac{3}{\sqrt{19}} \)
  (4) \( \frac{\sqrt{5}}{7} \)

The line \( L_1 \) is parallel to the vector \( \mathbf{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} \) and passes through the point \( (7, 6, 2) \), and the line \( L_2 \) is parallel to the vector \( \mathbf{b} = 2\hat{i} + \hat{j} + 3\hat{k} \) and passes through the point \( (5, 3, 4) \). The shortest distance between the lines \( L_1 \) and \( L_2 \) is:

• (1) \( \frac{23}{\sqrt{38}} \)
  (2) \( \frac{21}{\sqrt{57}} \)
  (3) \( \frac{23}{\sqrt{57}} \)
  (4) \( \frac{21}{\sqrt{38}} \)

Let \( (a, b) \) be the point of intersection of the curve \( x^2 = 2y \) and the straight line \( y = 2x - 6 \) in the second quadrant. Then the integral \[ I = \int_a^b \frac{9x^2}{1 + 5x^3} \, dx \]
is equal to:

• (1) 24
  (2) 27
  (3) 18
  (4) 21

If the system of equation \[ 2x + \lambda y + 3z = 5
3x + 2y - z = 7
4x + 5y + \mu z = 9 \]
has infinitely many solutions, then \( \lambda^2 + \mu^2 \) is equal to:

• (1) 22
  (2) 18
  (3) 26
  (4) 30

If \( \theta \in \left[ \frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of \[ \sqrt{3} \csc^2\theta - 2(\sqrt{3} - 1) \csc\theta - 4 = 0, \]
is equal to:



(1) 6
(2) 8
(3) 10
(4) 7

Given three identical bags each containing 10 balls, whose colours are as follows:


\begin{tabbing
Bag I \hspace{0.7cm \= 3 Red \hspace{0.5cm 2 Blue \hspace{0.5cm 5 Green

Bag II \hspace{0.7cm \= 4 Red \hspace{0.5cm 3 Blue \hspace{0.5cm 3 Green

Bag III \hspace{0.5cm \= 5 Red \hspace{0.5cm 1 Blue \hspace{0.5cm 4 Green

\end{tabbing

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is \( p \) and if the ball is Green, the probability that it is from Bag III is \( q \), then the value of \( \frac{1}{p} + \frac{1}{q} \) is:



(1) 6
(2) 9
(3) 7
(4) 8

If the mean and the variance of 6, 4, 8, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \( a + b + ab \) is equal to:



(1) 105
(2) 103
(3) 100
(4) 106

If the domain of the function \( f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} \) is \( (a, b) \), then \( (1 + a)^2 + b \) is equal to:



(1) 26
(2) 29
(3) 25
(4) 30

\[ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) \]
is equal to:



(1) \( 2 + \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)
(2) \( 2 - \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)
(3) \( 2 + \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)
(4) \( 2 - \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)

If \( \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} \) is finite, then \( (a + b) \) is equal to:

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

If \( \sum_{r=0}^{10} \left( 10^{r+1} - 1 \right) \, \binom{10}{r} = \alpha^{11} - 1 \right) \), then \( \alpha \) is equal to :

• (1) 15
  (2) 11
  (3) 24
  (4) 20

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

• (1) 5880
  (2) 960
  (3) 840
  (4) 5760

Let the point \( P \) of the focal chord \( PQ \) of the parabola \( y^2 = 16x \) be \( (1, -4) \). If the focus of the parabola divides the chord \( PQ \) in the ratio \( m : n \), gcd(\(m, n\)) = 1, then \( m^2 + n^2 \) is equal to:

• (1) 17
  (2) 10
  (3) 37
  (4) 26

Let \( \mathbf{a} = 2\hat{i} - 3\hat{j} + \hat{k}, \, \mathbf{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \) and a vector \( \mathbf{c} \) be such that \( (\mathbf{a} - \mathbf{c}) \times \mathbf{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \) and \( \mathbf{a} \cdot \mathbf{c} = 3 \). If \( \mathbf{b} \times \mathbf{c} = \mathbf{a} \), then \( |\mathbf{a} \cdot \mathbf{c}| \) is equal to:

• (1) 18
  (2) 12
  (3) 9
  (4) 15

Let the area of the triangle formed by a straight line \( L: x + by + c = 0 \) with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line \( L \) makes an angle of \( 45^\circ \) with the positive x-axis, then the value of \( b^2 + c^2 \) is:

• (1) 90
  (2) 93
  (3) 97
  (4) 83

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( A^2(A - 2I) - 4(A - I) = O \), where \( I \) and \( O \) are the identity and null matrices, respectively. If \( A^3 = \alpha A^2 + \beta A + \gamma I \), where \( \alpha \), \( \beta \), and \( \gamma \) are real constants, then \( \alpha + \beta + \gamma \) is equal to:

• (1) 12
  (2) 20
  (3) 76
  (4) 4

Let \( y = y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \sec^2 x \]
such that \( y(0) = \frac{5}{4} \). Then \( 12 \left( y \left( \frac{\pi}{4} \right) - e^2 \right) \) is equal to:

If the sum of the first 10 terms of the series \[ \frac{4.1}{1 + 4.1^4} + \frac{4.2}{1 + 4.2^4} + \frac{4.3}{1 + 4.3^4} + \cdots \]
is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is equal to ........

If \( y = \cos \left( \frac{\pi}{3} + \cos^{-1} \frac{x}{2} \right) \), then \( (x - y)^2 + 3y^2 \) is equal to ______.

Let \( A(4, -2), B(1, 1) \) and \( C(9, -3) \) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is ______.

If the set of all \( a \in \mathbb{R} \setminus \{1\} \), for which the roots of the equation \( (1 - a)x^2 + 2(a - 3)x + 9 = 0 \) are positive is \( (-\infty, -\alpha] \cup [\beta, \gamma] \), then \( 2\alpha + \beta + \gamma \) is equal to ...........