Download JEE Main 2025 Question Paper (22 Jan - Shift 1) Mathematics

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23 Jan 2025:Shift-1

23 Jan 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 (1 – 5) 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 number of non-empty equivalence relations on the set {1,2,3} is :
   
      
   1. 6
   2. 7
   3. 5
   4. 4
2. Let $ƒ : R \to R$ be a twice differentiable function such that $ƒ(x + y)$ = $ƒ(x) ƒ(y)$ for all $x, y \in R$. If $ƒ'(0)$ = $4a$ and $ƒ$ staisfies $ƒ''(x)$ – $3a ƒ'(x)$ – $ƒ(x)$ = 0, $a > 0$, then the area of the region $R$ = {$(x,y)$ | $0 \leq y \leq ƒ(ax)$, $0 \leq x \leq 2$} is :
   
      
   1. $e^2-1$
   2. $e^4+1$
   3. $e^4-1$
   4. $e^2+1$
3. Let the triangle $PQR$ be the image of the triangle with vertices (1,3), (3,1) and (2, 4) in the line $x + 2y$ = 2. If the centroid of $\Delta PQR$ is the point $(\alpha, \beta)$, then $15(\alpha – \beta)$ is equal to :
   
      
   1. 24
   2. 19
   3. 21
   4. 22
4. Let $z_1$, $z_2$ and $z_3$ be three complex numbers on the circle $|z|$ = 1 with $arg(z_1)$=$\frac{-\pi}{4}$, $arg(z_2)$=0 and $arg(z_3)$=$\frac{\pi}{4}$. If $|z_1\bar{z_2}+z_2 \bar{z_3}+z_3 \bar{z_1}|^2$=$\alpha+\beta \sqrt{2}$, $\alpha, \beta \in Z$, then the value of $\alpha^2+\beta^2$ is:
   
      
   1. 24
   2. 41
   3. 31
   4. 29
5. Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of $16((sec^{–1}x)^2 + (cosec^{–1}x)^2)$is :
   
      
   1. $24 \pi^2$
   2. $18 \pi^2$
   3. $31 \pi^2$
   4. $22 \pi^2$
6. A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$, then the value of $64(\mu + \sigma^2)$ is :
   
      
   1. 51
   2. 48
   3. 32
   4. 64
7. Let $a_1$, $a_2$, $a_3$..... be a $G.P.$ of increasing positive terms. If $a_1a_5$ = 28 and $a_2 + a_4$ = 29, the $a_6$is equal to
   
      
   1. 628
   2. 526
   3. 784
   4. 812
8. Let $L_1$:$\frac{x-1}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{4}$ and $L_2$:$\frac{x-2}{3}$=$\frac{y-4}{4}$=$\frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$and $L_2$?
   
      
   1. $\left(-\frac{5}{3}, -7, 1\right)$
   2. $\left(2, 3, \frac{1}{3}\right)$
   3. $\left(\frac{8}{3}, -1, \frac{1}{3}\right)$
   4. $\left(\frac{14}{3}, -3, \frac{22}{3}\right)$
9. The product of all solutions of the equation $e^{5\log_ex)^2+3}=x^8$, $x > 0$, is:
   
      
   1. $e^{8/5}$
   2. $e^{6/5}$
   3. $e^2$
   4. $e$
10. If $\sum \limits_{r=1}^{n} T_r$=$\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim \limits_{n \to \infty} \sum \limits_{r=1}^{n}\left(\frac{1}{T_r}\right)$ is equal to:
    
       
    1. 1
    2. 0
    3. $\frac{2}{3}$
    4. $\frac{1}{3}$
11. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is $‘M’$, is :
    
       
    1. 14950
    2. 6084
    3. 4356
    4. 5148
12. Let $x = x(y)$ be the solution of the differential equation $y^2dx$+$\left(x-\frac{1}{x}\right)dy$=0. If $x(1)$=1, then $x\left(\frac{1}{2}\right)$ is to
    
       
    1. $\frac{1}{2}+e$
    2. $\frac{3}{2}+e$
    3. $3-e$
    4. $3+e$
13. Let the parabola $y = x^2 + px – 3$, meet the coordinate axes at the points $P$, $Q$ and $R$. If the circle $C$ with centre at $(–1, –1)$ passes through the points $P$, $Q$ and $R$, then the area of $\Delta PQR$ is :
    
       
    1. 4
    2. 6
    3. 7
    4. 5
14. A circle $C$ of radius $2$ lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point (2, 5) and intersects the circle $C$ at exactly two points. If the set of all possible values of $r$ is the interval $(\alpha, \beta)$, then $3\beta – 2\alpha$ is equal to :
    
       
    1. 15
    2. 14
    3. 12
    4. 10
15. Let for $ƒ(x)$ = $7tan^8 x$ + $7tan^6x$ – $3tan^4x$ – $3tan^2 x$, $I_1$=$\int \limits_{0}^{\pi/4} f(x)dx$ and $I_2$=$\int \limits_{0}^{\pi/4} xf(x)dx$. Then $7I_1$+$12I_2$ is equal to:
    
       
    1. $2 \pi$
    2. $\pi$
    3. 1
    4. 2
16. Let $ƒ(x)$ be a real differentiable function such that $ƒ(0)$ = 1 and $ƒ(x + y)$ = $ƒ(x)ƒ'(y)$ + $ƒ'(x) ƒ(y)$ for all $x, y \in R$. Then $\sum \limits_{n=1}^{100}\log_ef(n)$ is equal to :
    
       
    1. 2384
    2. 2525
    3. 5220
    4. 2406
17. Let $A$ = {1,2,3,.......,10} and $B$=$\left\{\frac{m}{n}: m, n \in A, m < n\text{ and } gcd(m, n)=1 \right\}$. Then $n(B)$ is equal to :
    
       
    1. 31
    2. 36
    3. 37
    4. 29
18. The area of the region, inside the circle $(x-2\sqrt{3})^2$+$y^2$=12 and outside the parabola $y^2$=$2\sqrt{3} x$ is
    
       
    1. $6\pi - 8$
    2. $3\pi - 8$
    3. $6\pi - 16$
    4. $3\pi + 8$
19.  Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $gcd(m, n)$ = 1, then $m + n$ is equal to :
    
       
    1. 14
    2. 4
    3. 11
    4. 13
20. Let the foci of a hyperbola be (1, 14) and (1, –12). If it passes through the point (1, 6), then the length of its latus-rectum is :
    
    1. $\frac{25}{6}$
    2. $\frac{24}{5}$
    3. $\frac{288}{5}$
       
    4. $\frac{144}{5}$

SECTION - B
(Numerical Answer Type)

This section contains 5 Numerical based questions. The answer to each question is rounded off to the nearest integer.

1. Let the function,
   $f(x)$=$\left\{ \begin{array}{cc} -3ax^2-2 &, x < 1 \\ a^2+bx &, x \geq 1 \end{array} \right.$
   be differentiable for all $x \in R$, where $a > 1, b \in R$. If the area of the region enclosed by $y = f(x)$ and the line $y = – 20$ is $\alpha + \beta \sqrt{3}$, $\alpha$, $\beta \in Z$, then the value of $\alpha + \beta$ is ____.
2. If $\sum \limits_{r=0}^{5}\frac{{}^{11}C_{2r+1}}{2r+2}$=$\frac{m}{n}$, $gcd(m, n)$=1, then $m – n$ is equal to ______.
3. Let $A$ be a square matrix of order 3 such that $det(A)$ = –2 and $det(3adj(–6adj(3A)))$=$2^{m+n}•3^{mn}$, $m>n$. Then $4m + 2n$ is equal to _____.
4. Let $L_1$:$\frac{x-1}{3}$=$\frac{y-1}{-1}$=$\frac{z+1}{0}$ and $L_2$:$\frac{x-2}{2}$=$\frac{y}{0}$=$\frac{z+4}{\alpha}$, $\alpha \in R$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, –1)$ on $L_2$, then the value of $26\alpha (PB)^2$ is ____.
5. Let $\vec{c}$be the projection vector of $\vec{b}$=$\lambda \hat{i}$+$4\hat{k}$, $\lambda$ > 0, on the vector $\vec{a}$=$\hat{i}$+$2\hat{j}$+$2\hat{k}$. If $|\vec{a}+\vec{c}|$=7, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$is ______.

22 Jan 2025:Shift-2

22 Jan 2025:Ans-Keys

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