Download JEE Main 2025 Question Paper (02 Apr - Shift 1) Mathematics

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03 Apr 2025:Shift-1

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Important Instructions

• Section-A : Attempt all questions.
• Section-B : Attempt all questions.
• Section-A (01 – 20) contains 20 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and –1 mark for wrong answer.
• Section-B (01 – 05) contains 5 Numerical value based questions. The answer to each question is rounded off to the nearest integer. Each question carries +4 marks for correct answer and –1 mark for wrong answer.

SECTION - A
(One Option Correct Type)

This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct

     
1. The largest $n \in N$ such that $3^n$ divides 50! is:
   
      
   1. 21
   2. 22
   3. 20
   4. 23
2. Let one focus of the hyperbola $H$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 be at $(\sqrt{10}, 0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9 (e^2+ l)$ is equal to:
   
      
   1. 14
   2. 15
   3. 16
   4. 12
3. The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to
   
      
   1. 360
   2. 45
   3. 2520
   4. 1820
4. Let $f : R \to R$ be a twice differentiable function such that $(sinx cosy)$$(f(2x+2y) – f(2x – 2y))$ = $(cosx siny)$$(f(2x+2y) + f(2x – 2y))$, for all $x, y \in R$. If $f^{^{,}}(0)=\frac{1}{2}$, then the value of $24 f''\left(\frac{5\pi}{3}\right)$ is
   
      
   1. 2
   2. -3
   3. 3
   4. -2
5. Let $A$=$\begin{equation*}\begin{bmatrix} \alpha & -1 \\ 6 & \beta \end{bmatrix}\end{equation*}$, $\alpha >0$, such that $det(A)$ = 0 and $\alpha$ + $\beta$ = 1. If I denotes 2 × 2 identity matrix, then the matrix $(1 + A)^8$ is:
   
      
   1. $\begin{equation*}\begin{bmatrix} 4 & -1 \\ 6 & -1 \end{bmatrix}\end{equation*}$
   2. $\begin{equation*}\begin{bmatrix} 257 & -64 \\ 514 & -127 \end{bmatrix}\end{equation*}$
   3. $\begin{equation*}\begin{bmatrix} 1025 & -511 \\ 2024 & -1024 \end{bmatrix}\end{equation*}$
   4. $\begin{equation*}\begin{bmatrix} 766 & -255 \\ 1530 & -509 \end{bmatrix}\end{equation*}$
6. The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{(x^{2/3}+1-x^{1/3})}-\frac{(x+1)}{(x-x^{1/2})}\right)^{10}$, $x > 1$ is:
   
      
   1. 210
   2. 150
   3. 240
   4. 120
7. 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 equal to:
   
      
   1. 12
   2. 6
   3. 8
   4. 10
8. Let $a_1$, $a_2$, $a_3$, ... be in an $A.P.$ such that $\sum \limits_{k=1}^{12}a_{2k-1}$=$-\frac{72}{5}a_1$, $a_1 \neq 0$. If $\sum \limits_{k=1}^{n}a_k$=0, then $n$ is:
   
      
   1. 11
   2. 10
   3. 18
   4. 17
9. If the function $f(x)$ = $2x^3$– $9ax^2$ + $12a^2x$ + 1, where $a$ > 0, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $p^2= q$, then $f(3)$ is equal to:
   
      
   1. 55
   2. 10
   3. 23
   4. 37
10. Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^2z}{k+\bar{z}}$=$kz$, $k \in R$, then the maximum distance of $k + ik^2$from the circle $|z –(1 + 2 i)|$ = 1 is:
    
       
    1. $\sqrt{5}+1$
    2. 2
    3. 3
    4. $\sqrt{3}+1$
11. If $\vec{a}$ is nonzero 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. $\frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}-5\hat{k})$
12. Let $A$ be the set of all functions $f : Z \to Z$ 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
13.  For $\alpha$, $\beta$, $\gamma$ $\in R$, if $\lim \limits_{x \to 0} \frac{x^2\sin \alpha x+(\gamma-1)e^{x^2}}{\sin 2x-\beta x}$=3, then $\beta+\gamma-\alpha$ is equal to :
    
       
    1. 7
    2. 4
    3. 6
    4. -1
14. If the system of linear equations
    $3x$ + $y$ + $\beta z$ = 3
    $2x$ + $\alpha y$ – $z$ = –3
    $x$ + $2y$ + $z$ = 4
    has infinitely many solutions, then the value of $22\beta – 9\alpha$ is :
    
       
    1. 49
    2. 31
    3. 43
    4. 37
15. Let $P_n$=$\alpha^n$+$\beta^n$, $n \in N$. If $P_{10}$=123, $P_9$=76, $P_8$=47 and $P_1$=1, then the quadratic equation having roots $\frac{1}{\alpha}$ 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
16. If $S$ and $S'$ are the foci of the ellipse $\frac{x^2}{18}$+$\frac{y^2}{9}$=1and P be a point on the ellipse, then $min(SP.S'P)$ + $max(SP.S'P)$ is equal to :
    
       
    1. $3(1+\sqrt{2})$
    2. $3(6+\sqrt{2})$
    3. 9
    4. 27
17. 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 $PQR$ 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
18. 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 $\Delta BCD$ is equal to :
    
       
    1. $\sqrt{340}$
    2. 12
    3. $\sqrt{110}$
    4. $7\sqrt{3}$
19.  Let $a \in R$ and $A$ be a matrix of order 3×3 such that $det(A)$ = –4 and $A+I$=$\begin{equation*} \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix} \end{equation*}$, where $I$ is the identity matrix of order 3×3. If $det ((a + 1)adj((a–1)A))$ is $2^m3^n$, $m, n \in {0,1,2,…..20}$, then $m + n$ is equal to : 
    
       
    1. 14
    2. 17
    3. 15
    4. 16
20. Let the focal chord $PQ$ of the parabola $y^2$ = $4x$make an angle of 60° 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

SECTION - B
(Numerical Value Type Questions)

This section contains 05 Numerical based questions. The answer to each question is rounded off to the nearest integer.

1. Let [•] denote the greatest integer function. If $\int \limits_{0}^{e^3}\left[\frac{1}{e^{x-1}}\right]dx$=$\alpha-\log_e2$, then $\alpha^3$ is equal to........
2. Let $ƒ : R \to R$ be a thrice differentiable odd function satisfying $ƒ'(x) \geq 0$, $ƒ'(x)=ƒ(x)$, $ƒ(0)=0$, $ƒ'(0)$=3. Then $9ƒ(log_e3)$ is equal to _____.
3. If the area of the region {$(x, y):|4-x^2|\leq y \leq x^2$, $y \leq 4, x \geq 0$} is $\left(\frac{80\sqrt{2}}{\alpha}-\beta\right)$, $\alpha, \beta \in N$, then $\alpha+\beta$ is equal to.........
4. Three distinct numbers are selected randomly from the set {1,2,3,……,40}. If the probability, that the selected numbers are in an increasing G.P. is $\frac{m}{n}$, $gcd(m,n)$ = 1, then $m + n$ is equal to ____.
5. 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 _____.

02 Apr 2025:Shift-2

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