Download JEE Main 2025 Question Paper (03 Apr - Shift 2) Mathematics

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04 Apr 2025:Shift-1

04 Apr 2025:Shift-2

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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. Let $f : R \to R$ be a function defined by $f(x)$=$||x+2|-2|x||$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m + n$ is
   
      
   1. 25
   2. 26
   3. 27
   4. 28
2. Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y–$ and $z–$axes, respectively, is half of the angle that this line makes with the positive $x-$axes. Then the sum of all possible values of the angle $\beta$ is
   
      
   1. $\frac{3\pi}{4}$
   2. $\pi$
   3. $\frac{\pi}{2}$
   4. $\frac{3\pi}{2}$
3. If the four distinct points (4, 6), (–1,5), (0,0) and $(k, 3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to
   
      
   1. 32
   2. 33
   3. 34
   4. 35
4. Let the Mean and Variance of five observations $x_1$= 1, $x_2$ = 3, $x_3$ = $a$, $x_4$ = 7 and $x_5$ = $b$, $a > b$, be 5 and 10 respectively. Then the Variance of the observations $n + x_n$, $n$ = 1, 2, …….. 5 is
   
      
   1. 16923
   2. 3763
   3. 33845
   4. 18817
5. Consider the lines $x(3\lambda + 1)$ + $y (7\lambda+2)$ = $17\lambda$ + 5, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point (3, 6) is $d$, then the value of $d^2$is
   
      
   1. 20
   2. 30
   3. 10
   4. 15
6. Let $A$ = {–2,–1,0, 1, 2, 3}. let $R$ be a relation on $A$ defined by $x R y$ if and only if $y$ = max{$x, 1$}. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to
   
      
   1. 12
   2. 11
   3. 13
   4. 14
7. let the equation $x(x + 2)(12–k)$ = 2 have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $3x + 4y$ + 5 = 0 is
   
      
   1. 15
   2. $5\sqrt{3}$
   3. $15\sqrt{5}$
   4. 12
8. Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant, $P_1$, $P_2$,….$P_{12}$ are 12 points on line $L_1$ and $Q_1$, $Q_2$, …..$Q_9$ are 9 points on line $L_2$. Then the total number of triangles, that can be formed having vertices at three of the 22 points $O$, $P_1$, $P_2$,… $P_{12}$, $Q_1$, $Q_2$,….$Q_9$, is:
   
      
   1. 5
   2. 4
   3. 3
   4. 7
9. The integral $\int \limits_{0}^{\pi}\frac{8x dx}{4\cos^2x+\sin^2x}$ is equal to
   
      
   1. $2\pi^2$
   2. $4\pi^2$
   3. $\pi^2$
   4. $\frac{3\pi^2}{2}$
10. Let $f$ be a function such that $f(x)+3f\left(\frac{24}{x}\right)$=$4x$, $x \neq 0$. Then $f(3)$+$f(8)$ is equal to
    
       
    1. 11
    2. 10
    3. 12
    4. 13
11. The area of the region $\left\{(x, y):|x-y| \leq y \leq 4\sqrt{x} \right\}$ is
    
       
    1. 512
    2. $\frac{1024}{3}$
    3. $\frac{512}{3}$7
    4. $\frac{2048}{3}$
12. If the domain of the function $f(x)$ = $log_7(1 – log_4(x^2– 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$, then $\alpha$ + $\beta$ + $\gamma$ + $\delta$ is equal to
    
       
    1. 18
    2. 16
    3. 15
    4. 17
13.  If the probability that the random variable $X$ takes the value $x$ is given by $P(X = x)$ = $k(x + 1)3^{–x}$, $x$ = 0,1,2,3......, where $k$ is a constant, then $P(X \geq 3)$ is equal to
    
       
    1. $\frac{7}{27}$
    2. $\frac{4}{9}$
    3. $\frac{8}{27}$
    4. $\frac{1}{9}$
14. Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx}$+$3(\tan^2x)y$+$3y$=$\sec^2x$, $y(0)$=$\frac{1}{3}+e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to
    
       
    1. $\frac{2}{3}$
    2. $\frac{4}{3}$
    3. $\frac{4}{3}+e^3$
    4. $\frac{2}{3}+e^3$
15. If $z_1$, $z_2$, $z_3 \in C$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\sum \limits_{k=1}^{3}(z_k-z_0)^2$ is equal to
    
       
    1. 0
    3. $i$
    4. $-i$
16. The number of solutions of equation $(4 – \sqrt{3})\sin x$$-2\sqrt{3} \cos^2 x$=$-\frac{4}{1+\sqrt{3}}$, $x \in \left[-2\pi, \frac{5\pi}{2}\right]$ is
    
       
    1. 4
    2. 3
    3. 6
    4. 5
17. Let $C$ be the circle of minimum area enclosing the ellipse $E$ : $\frac{x^2}{a^2}+\frac{y^2}{b^2}$= 1 with eccentricity $\frac{1}{2}$ and foci(±2, 0). Let $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length 29 is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y-$axis. Then the maximum area of the triangle $PQR$ is :
    
       
    1. $6(3+\sqrt{2})$
    2. $8(3+\sqrt{2})$
    3. $6(2+\sqrt{3})$
    4. $8(2+\sqrt{3})$
18. The shortest distance between the curves $y^2$ = $8x$ and $x^2$ + $y^2$+ $12y$ + 35 = 0 is :
    
       
    1. $\frac{7\sqrt{2}}{2}$
    2. $\frac{7\sqrt{2}}{16}$
    3. $\frac{7\sqrt{2}}{4}$
    4. $\frac{7\sqrt{2}}{8}$
19.  Let $f(x)$=$\left\{ \begin{array}{ccc} (1+ax)^{1/x} &, x<0 \\ 1+b &, x=0 \\ \frac{(x+4)^{1/2}-2}{(x+c)^{1/3}-2} &, x>0 \end{array} \right.$ be continuous at $x = 0$. Then $e^abc$ is equal to
    
       
    1. 64
    2. 72
    3. 48
    4. 36
20. 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

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. 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 $P(CDBEA)$=$\frac{2^{\alpha}}{2^{\beta}-1}$, $\alpha, \beta \in N$, then $\alpha +\beta$ is equal to :
2. 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 ……
3. 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 $\vec{a}×\hat{d}$=$\vec{b}×\hat{d}$ and $\vec{c}•\hat{d}$=1. If $\vec{c}$ is perpendicular to $\vec{a}$, then $|3\lambda \hat{d}+\mu \vec{c}|^2$ is equal to........
4. If the number of seven-digit numbers, such that the sum of their digits is even, is $m·n·10^n$; $m, n \in$ {$1, 2, 3, …., 9$}, then $m + n$ is equal to
5. The area of the region bounded by the curve $y$ = max{$|x|, x|x-2|$}, then $x-$axis and the lines $x = –2$ and $x = 4$ is equal to ______.

03 Apr 2025:Shift-1

03 Apr 2025:Ans-Keys

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