JEE Main 2025 April 2 Shift 1 Maths Question Paper, Exam Analysis, and Answer Keys (Available)

The largest \( n \in \mathbb{N} \) such that \( 3^n \) divides 50! is:

(1) 21
(2) 22
(3) 23
(4) 25

Let one focus of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be at \( (\sqrt{10}, 0) \), and the corresponding directrix be \( x = \frac{\sqrt{10}}{2} \). If \( e \) and \( l \) are the eccentricity and the latus rectum respectively, then \( 9(e^2 + l) \) is equal to:

(1) 14
(2) 16
(3) 18
(4) 12

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1’s and exactly three 2’s, is equal to:

(1) 360
(2) 45
(3) 2520
(4) 1820

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) \]
for all \( x, y \in \mathbb{R} \). If \( f(0) = 1 \), then the value of \( 24f^{(4)}\left(\frac{5\pi}{3}\right) \) is:
 

(1) 2
(2) –3
(3) 1
(4) 3

Let \( A = \begin{bmatrix} \alpha & -1
6 & \beta \end{bmatrix},\ \alpha > 0 \), such that \( \det(A) = 0 \) and \( \alpha + \beta = 1 \). If \( I \) denotes the \( 2 \times 2 \) identity matrix, then the matrix \( (1 + A)^5 \) is:
 

• (1)  matrix 4   -1
                   6   -1 
• (2) matrix  257  -64
                  514  -127
• (3) matrix 1025   -511
                   2024  -1024 
• (4) matrix} 766   -255
                   1530  -509

The term independent of \( x \) in the expansion of \[ \left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10} \]
for \( x > 1 \) is:

(1) 210
(2) 150
(3) 240
(4) 120

If \( \theta \in [-2\pi,\ 2\pi] \), then the number of solutions of \[ 2\sqrt{2} \cos^2\theta + (2 - \sqrt{6}) \cos\theta - \sqrt{3} = 0 \] is:

(1) 12
(2) 6
(3) 8
(4) 10

Let \( a_1, a_2, a_3, \ldots \) be in an A.P. such that \[ \sum_{k=1}^{12} 2a_{2k - 1} = \frac{72}{5}, \quad and \quad \sum_{k=1}^{n} a_k = 0, \]
then \( n \) is:

(1) 11
(2) 10
(3) 18
(4) 17

If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a > 0 \), attains its local maximum and minimum at \( p \) and \( q \), respectively, such that \( p^2 = q \), then \( f(3) \) is equal to:

(1) 55
(2) 10
(3) 23
(4) 37

Let \( z \) be a complex number such that \( |z| = 1 \). If \[ \frac{2 + kz}{k + z} = kz,\ k \in \mathbb{R}, \]
then the maximum distance of \( k + ik^2 \) from the circle \( |z - (1 + 2i)| = 1 \) is:

(1) \( \sqrt{5} + 1 \)
(2) 2
(3) 3
(4) \( \sqrt{5} + \sqrt{1} \)

If \( \vec{a} \) is a non-zero vector such that its projections on the vectors \( 2\hat{i} - \hat{j} + 2\hat{k},\ \hat{i} + 2\hat{j} - 2\hat{k} \), and \( \hat{k} \) are equal, then a unit vector along \( \vec{a} \) is:

• (1) \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} + 5\hat{k}) \) 
• (2) \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} - 5\hat{k}) \) 
• (3) \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} + 5\hat{k}) \) 
• (4) \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} - 5\hat{k}) \)

Let \( A \) be the set of all functions \( f: \mathbb{Z} \to \mathbb{Z} \) and \( R \) be a relation on \( A \) such that \[ R = \{ (f, g) : f(0) = g(1) and f(1) = g(0) \} \]
Then \( R \) is:

• (1) Symmetric and transitive but not reflective 
• (2) Symmetric but neither reflective nor transitive
• (3) Reflexive but neither symmetric nor transitive
• (4) Transitive but neither reflexive nor symmetric

For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, \]
then \( \beta + \gamma - \alpha \) is equal to:

• (1) 7 
• (2) 4 
• (3) 6 
• (4) –1

If the system of equations: \[ \begin{aligned} 3x + y + \beta z &= 3
2x + \alpha y + z &= 2
x + 2y + z &= 4 \end{aligned} \]
has infinitely many solutions, then the value of \( 22\beta - 9\alpha \) is:

• (1) 49 
• (2) 31 
• (3) 43
• (4) 37

Let \( P_n = \alpha^n + \beta^n \), \( n \in \mathbb{N} \). If \( P_{10} = 123,\ P_9 = 76,\ P_8 = 47 \) and \( P_1 = 1 \), then the quadratic equation having roots \( \alpha \) and \( \frac{1}{\beta} \) is:

• (1) \( x^2 - x + 1 = 0 \)
• (2) \( x^2 + x - 1 = 0 \) 
• (3) \( x^2 - x - 1 = 0 \) 
• (4) \( x^2 + x + 1 = 0 \)

If \( S \) and \( S' \) are the foci of the ellipse \( \frac{x^2}{18} + \frac{y^2}{9} = 1 \), and \( P \) is a point on the ellipse, then \( \min(\vec{SP} \cdot \vec{S'P}) + \max(\vec{SP} \cdot \vec{S'P}) \) is equal to:

• (1) \( 3(1 + \sqrt{2}) \)
• (2) \( 3(6 + \sqrt{2}) \) 
• (3) 9 
• (4) 27

Let the vertices Q and R of the triangle PQR lie on the line \( \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \), \( QR = 5 \), and the coordinates of the point P be \( (0, 2, 3) \). If the area of the triangle PQR is \( \frac{m}{n} \), then:

• (1) \( m - 5\sqrt{21}n = 0 \)
• (2) \( 2m - 5\sqrt{21}n = 0 \)
• (3) \( 5m - 2\sqrt{21}n = 0 \)
• (4) \( 5m - 21\sqrt{2}n = 0 \)

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles \( ABC, ACD, \) and \( ADB \) be 5, 6 and 7 square units respectively. Then the area (in square units) of the tetrahedron ABCD is equal to:

• (1) \( \sqrt{30} \)
• (2) 12
• (3) \( \sqrt{10} \) 
• (4) 7 \( \sqrt{5} \)

Let \( A \in \mathbb{R} \) be a matrix of order 3x3 such that \[ \det(A) = -4 \quad and \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1
2 & 0 & 1
4 & 1 & 2 \end{array} \right] \]
where \( I \) is the identity matrix of order 3. If \( \det( (A + I) \cdot adj(A + I)) \) is \( 2^m \), then \( m \) is equal to:

• (1) 14
• (2) 31 
• (3) 16 
• (4) 13

Let the focal chord PQ of the parabola \( y^2 = 4x \) make an angle of \( 60^\circ \) with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, \( S \) being the focus of the parabola, touches the y-axis at the point \( (0, \alpha) \), then \( 5\alpha^2 \) is equal to:

• (1) 15 
• (2) 25 
• (3) 30 
• (4) 20

Let \( [.] \) denote the greatest integer function. If \[ \int_1^e \frac{1}{x e^x} dx = \alpha - \log 2, \quad then \quad \alpha^2 is equal to: \]

If the area of the region \[ \{(x, y): |4 - x^2| \leq y \leq x^2, y \geq 0\} \]
is \( \frac{80\sqrt{2}}{\alpha - \beta} \), \( \alpha, \beta \in \mathbb{N} \), then \( \alpha + \beta \) is equal to:

Three distinct numbers are selected randomly from the set \( \{1, 2, 3, \dots, 40\} \). If the probability that the selected numbers are in an increasing G.P. is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is equal to:

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9, 4) \) and touching the lines \( x + y = 3 \) and \( x - y = 3 \), is equal to: