JEE Main 2025 April 3 Shift 1 Maths Question Paper, Exam Analysis, and Answer Keys (Available)

JEE Main 2025 Mathematics Questions with Solutions

Question 1:

Let \( A \) be a matrix of order \( 3 \times 3 \) and \( |A| = 5 \). If \[ |2 \, adj(3A \, adj(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} \]
then \( \alpha + \beta + \gamma \) is equal to

• (1) \( 25 \)
• (2) \( 26 \)
• (3) \( 27 \)
• (4) \( 28 \)

Let a line passing through the point \( (4,1,0) \) intersect the line \( L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \) at the point \( A(\alpha, \beta, \gamma) \) and the line \( L_2: x - 6 = y = -z + 4 \) at the point \( B(a, b, c) \). Then \[ \begin{vmatrix} 1 & 0 & 1
\alpha & \beta & \gamma
a & b & c \end{vmatrix} is equal to \]

• (1) \( 8 \)
• (2) \( 16 \)
• (3) \( 12 \)
• (4) \( 6 \)

Let \( \alpha \) and \( \beta \) be the roots of \( x^2 + \sqrt{3}x - 16 = 0 \), and \( \gamma \) and \( \delta \) be the roots of \( x^2 + 3x - 1 = 0 \). If \( P_n = \alpha^n + \beta^n \) and \( Q_n = \gamma^n + \delta^n \), then \[ \frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}} is equal to \]

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

The sum of all rational terms in the expansion of \( \left( 2 + \sqrt{3} \right)^8 \) is

• (1) \(16923\)
• (2) \(3763\)
• (3) \(33845\)
• (4) \(18817\)

Let A = {-3,-2,-1,0,1,2,3}. Let R be a relation on A defined by xRy if and only if \( 0 \le x^2 + 2y \le 4 \). Let \( l \) be the number of elements in R and m be the minimum number of elements required to be added in R to make it a reflexive relation. then \( l + m \) is equal to

• (1) 19
• (2) 20
• (3) 17
• (4) 18

A line passing through the point P\((\sqrt{5}, \sqrt{5})\) intersects the ellipse \( \frac{x^2}{36} + \frac{y^2}{25} = 1 \) at A and B such that (PA).(PB) is maximum. Then 5(PA\(^2\) + PB\(^2\)) is equal to :

• (1) 218
• (2) 377
• (3) 290
• (4) 338

The sum 1 + 3 + 11 + 25 + 45 + 71 + ... upto 20 terms, is equal to

• (1) 7240
• (2) 7130
• (3) 6982
• (4) 8124

If the domain of the function \( f(x) = \log_e \left( \frac{2x-3}{5+4x} \right) + \sin^{-1} \left( \frac{4+3x}{2-x} \right) \) is \( [\alpha, \beta] \), then \( \alpha^2 + 4\beta \) is equal to

• (1) 5
• (2) 4
• (3) 3
• (4) 7

If \( \sum_{r=1}^{9} \left( \frac{r+3}{2^r} \right) \cdot {^9C_r} = \alpha \left( \frac{3}{2} \right)^9 - \beta \), \( \alpha, \beta \in \mathbb{N} \), then \( (\alpha + \beta)^2 \) is equal to

• (1) 27
• (2) 9
• (3) 81
• (4) 18

The number of solutions of the equation \( 2x + 3\tan x = \pi \), \( x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} \) is

• (1) 6
• (2) 5
• (3) 4
• (4) 3

If \( y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1
27 & 28 & 27
1 & 1 & 1 \end{vmatrix} \), \( x \in \mathbb{R} \), then \( \frac{d^2y}{dx^2} + y \) is equal to

• (1) -1
• (2) 28
• (3) 27
• (4) 1

Let g be a differentiable function such that \( \int_0^x g(t) dt = x - \int_0^x tg(t) dt \), \( x \ge 0 \) and let \( y = y(x) \) satisfy the differential equation \( \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) \), \( x \in \left[ 0, \frac{\pi}{2} \right) \). If \( y(0) = 0 \), then \( y\left( \frac{\pi}{3} \right) \) is equal to

• (1) \( \frac{2\pi}{3\sqrt{3}} \)
• (2) \( \frac{4\pi}{3} \)
• (3) \( \frac{2\pi}{3} \)
• (4) \( \frac{4\pi}{3\sqrt{3}} \)

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines \( L_1 : 2x + y + 6 = 0 \) and \( L_2 : 4x + 2y - p = 0 \), \( p > 0 \), at the points A and B, respectively. If \( AB = \frac{9}{\sqrt{2}} \) and the foot of the perpendicular from the point A on the line \( L_2 \) is M, then \( \frac{AM}{BM} \) is equal to

• (1) 5
• (2) 4
• (3) 2
• (4) 3

Let \( z \in \mathbb{C} \) be such that \( \frac{z+3i}{z-2+i} = 2+3i \). Then the sum of all possible values of \( z \) is

• (1) \( 19 - 2i \)
• (2) \( -19 - 2i \)
• (3) \( 19 + 2i \)
• (4) \( -19 + 2i \)

Let \( f(x) = \int x^3 \sqrt{3-x^2} dx \). If \( 5f(\sqrt{2}) = -4 \), then \( f(1) \) is equal to

• (1) \( -\frac{2\sqrt{2}}{5} \)
• (2) \( -\frac{8\sqrt{2}}{5} \)
• (3) \( -\frac{4\sqrt{2}}{5} \)
• (4) \( -\frac{6\sqrt{2}}{5} \)

Let \( a_1, a_2, a_3, ... \) be a G.P. of increasing positive numbers. If \( a_3 a_5 = 729 \) and \( a_2 + a_4 = \frac{111}{4} \), then \( 24(a_1 + a_2 + a_3) \) is equal to

• (1) 131
• (2) 130
• (3) 129
• (4) 128

Let the domain of the function \( f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2) \) be (a, b). If \( \int_0^{a+b} [x^2] dx = p - q\sqrt{r} \), \( p, q, r \in \mathbb{N} \), gcd(p, q, r) = 1, where [.] is the greatest integer function, then p + q + r is equal to

• (1) 10
• (2) 8
• (3) 11
• (4) 9

The radius of the smallest circle which touches the parabolas \( y = x^2 + 2 \) and \( x = y^2 + 2 \) is

• (1) \( \frac{7\sqrt{2}}{2} \)
• (2) \( \frac{7\sqrt{2}}{16} \)
• (3) \( \frac{7\sqrt{2}}{4} \)
• (4) \( \frac{7\sqrt{2}}{8} \)

Let \( f(x) = \begin{cases} (1+ax)^{1/x} & , x < 0
1+b & , x = 0
\frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , x > 0 \end{cases} \) be continuous at x = 0. Then \( e^a bc \) is equal to

• (1) 64
• (2) 72
• (3) 48
• (4) 36

Line \( L_1 \) passes through the point (1, 2, 3) and is parallel to z-axis. Line \( L_2 \) passes through the point \( (\lambda, 5, 6) \) and is parallel to y-axis. Let for \( \lambda = \lambda_1, \lambda_2, \lambda_2 < \lambda_1 \), the shortest distance between the two lines be 3. Then the square of the distance of the point \( (\lambda_1, \lambda_2, 7) \) from the line \( L_1 \) is

• (1) 40
• (2) 32
• (3) 25
• (4) 37

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number \( n \) be denoted by \( W_n \). Let the probability \( P(W_n) \) of choosing the word \( W_n \) satisfy \( P(W_n) = 2P(W_{n-1}) \), \( n > 1 \). If \( P(CDBEA) = \frac{2^\alpha}{2^\beta - 1} \), \( \alpha, \beta \in \mathbb{N} \), then \( \alpha + \beta \) is equal to :

Let the product of the focal distances of the point P(4, \( 2\sqrt{3} \)) on the hyperbola H: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then \( p^2 + q^2 \) is equal to .....

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k} \), \( \vec{c} = \lambda \hat{j} + \mu \hat{k} \) and \( \hat{d} \) be a unit vector such that \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \) and \( \vec{c} \cdot \hat{d} = 1 \). If \( \vec{c} \) is perpendicular to \( \vec{a} \), then \( |3\lambda \hat{d} + \mu \vec{c}|^2 \) is equal to ____.

If the number of seven-digit numbers, such that the sum of their digits is even, is \( m \cdot n \cdot 10^a \); \( m, n \in \{1, 2, 3, ..., 9\} \), then \( m + n \) is equal to

The area of the region bounded by the curve \( y = \max\{|x|, |x-2|\} \), then x-axis and the lines x = -2 and x = 4 is equal to ____.