Download JEE Main 2025 Question Paper (04 Apr - Shift 1) Mathematics

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07 Apr 2025:Shift-1

07 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 $f, g: (1, \infty) \to R$ be defined as $f(x)$=$\frac{2x+3}{5x+2}$ and $g(x)$=$\frac{2-3x}{1-x}$. If the range of the function $fog:[2, 4] \to R$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to
   
      
   1. 68
   2. 29
   3. 2
   4. 56
2. Consider the sets $A$={$(x, y) \in R×R$: $x^2 + y2 = 25$}, $B$ = {$(x, y) \in R×R$ : $x^2 + 9y^2 = 144$}, $C$ = {$(x, y) \in Z × Z$ : $x^2 + y^2 \leq 4$}, and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
   
      
   1. 15120
   2. 19320
   3. 17160
   4. 18290
3. Let $A$ = {1, 6, 11, 16, …} and $B$ = {9, 16, 23, 30, …} be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n (A \cup B)$ is
   
      
   1. 3814
   2. 4027
   3. 3761
   4. 4003
4. For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n–3}$is 16, then the distance of the point $P(2n –1, n^2– 4n)$from the line $x + y = 8$ is:
   
      
   1. 16923
   2. 3763
   3. 33845
   4. 18817
5. The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:
   
      
   1. $\frac{129}{182}$
   2. $\frac{103}{182}$
   3. $\frac{17}{26}$
   4. $\frac{29}{26}$
6. Let the shortest distance between the lines $\frac{x-3}{3}$=$\frac{y-\alpha}{-1}$=$\frac{z-3}{1}$ and $\frac{x+3}{-3}$=$\frac{y+7}{2}$=$\frac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha+\beta$ is
   
      
   1. 42
   2. 46
   3. 48
   4. 40
7. If $\lim \limits_{x \to 1^+}\frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin(1-x)}{(x-1)^3}$=$-1$, where $\lambda, \mu \in R$, then $\lambda+\mu$ is equal to
   
      
   1. 18
   2. 20
   3. 19
   4. 17
8. Let $f : [0, \infty) \to R$ be differentiable function such that $f(x) = 1 – 2x$+$\int \limits_{0}^{x}e^{x-t}f(t)dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is
   
      
   1. $\sqrt{5}$
   2. $\frac{1}{2}$
   3. $\sqrt{2}$
   4. 2
9. Let $A$ and $B$ be two distinct points on the line $L$:$\frac{x-6}{3}$=$\frac{y-7}{2}$=$\frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point (1, 2, 3) on the line $L$. If $O$ is the origin, then $\vec{OA}$•$\vec{OB}$ is equal to:
   
      
   1. 49
   2. 47
   3. 21
   4. 62
10. Let $f : R \to R$ be a continuous function satisfying $f(0)$ = 1 and $f(2x) – f(x)$ = $x$ for all $x \in R$ . If $\lim \limits_{n \to \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}$=$G(x)$, then $\sum \limits_{r=1}^{10}G(r^2)$ is equal to
    
       
    1. 540
    2. 385
    3. 420
    4. 215
11. 1 + 3 + $5^2$ + 7 + $9^2$ + … upto 40 terms is equal to
    
       
    1. 43890
    2. 41880
    3. 33980
    4. 40870
12. In the expansion of $\left(\sqrt{3}{2}+\frac{1}{\sqrt{3}{3}}\right)^n$, $n \in N$, if the ratio of $15^{th}$ term from the beginning to the $15^{th}$term from the end is $\frac{1}{6}$, then the value of ${}^nC_3$ is:
    
       
    1. 4060
    2. 1040
    3. 2300
    4. 4960
13. Considering the principal values of the inverse trigonometric functions, $\sin^{-1}\left(\frac{\sqrt{3}}{2}x +\frac{1}{2}\sqrt{1-x^2}\right)$, $-\frac{1}{2} < x < \frac{1}{\sqrt{2}}$, is equal to
    
       
    1. $\frac{\pi}{4}+\sin^{-1}x$
    2. $\frac{\pi}{6}+\sin^{-1}x$
    3. $\frac{-5\pi}{6}-\sin^{-1}x$
    4. $\frac{5\pi}{6}-\sin^{-1}x$
14. Consider two vectors $\vec{u}$=$3\hat{i}-\hat{j}$ and $\vec{v}$=$2\hat{i}$+$\hat{j}$$-\lambda\hat{k}$, $\lambda > 0$. The angle between them is given by $\cos^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v}$=$\vec{v_1}$+$\vec{v_2}$, where $\vec{v_1}$ is parallel to $\vec{u}$ and $\vec{v_2}$ is perpendicular to $\vec{u}$. Then the value $|\vec{v_1}|^2$+$|\vec{v_2}|^2$ is equal to
    
       
    1. $\frac{23}{2}$
    2. 14
    3. $\frac{25}{2}$
    4. 10
15. Let the three sides of a triangle are on the lines $4x – 7y$ + 10 = 0, $x + y$ = 5 and $7x + 4y$ = 15. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x$ = 0, $y$ = 0 and $x + y$ = 1 is
    
       
    1. 5
    2. $\sqrt{5}$
    3. $\sqrt{20}$
    4. 20
16. The value of $\int \limits_{-1}^{1}\frac{(1+\sqrt{|x|-x})e^x+(\sqrt{|x|-x})e^{-x}}{e^x+e^{-x}}dx$ is equal to
    
       
    1. $3-\frac{2\sqrt{2}}{3}$
    2. $2+\frac{2\sqrt{2}}{3}$
    3. $1-\frac{2\sqrt{2}}{3}$
    4. $1+\frac{2\sqrt{2}}{3}$
17. The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, –3) and eccentricity is $\frac{4}{5}$, is
    
       
    1. $\frac{6}{5}$
    2. $\frac{50}{3}$
    3. $\frac{10}{3}$
    4. $\frac{18}{5}$
18. Consider the equation $x^2$ + $4x – n$ = 0, where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to
    
       
    1. 7
    2. 8
    3. 6
    4. 5
19.  A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is
    
       
    1. $\frac{11}{15}$
    2. $\frac{28}{75}$
    3. $\frac{2}{15}$
    4. $\frac{3}{5}$
20. If $10 sin^4 \theta$ + $15 cos^4\theta$ = 6, then the value of $\frac{27 cosec^6\theta+8\sec^6\theta}{16\sec^8\theta}$ is
    
       
    1. $\frac{2}{5}$
    2. $\frac{3}{4}$
    3. $\frac{3}{5}$
    4. $\frac{1}{5}$

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. If the area of the region {$(x, y) : |x–5| \leq y \leq4 \sqrt{x}$} is $A$, then $3A$ is equal to ____.
2. Let $A$=$\begin{equation*} \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix} \end{equation*}$. If for some $\theta \in (0, \pi)$, $A^2$=$A^T$, then the sum of the diagonal elements of the matrix $(A+I)^3$+$(A-I)^3$$-6A$ is equal to........
3. Let $A$ = {$z \in C$ : $|z – 2 – i|$ = 3}, $B$ = {$z \in C : Re (z – iz)$ = 2} and $S$ = $A \cap B$. Then
4. Let $C$ be the circle $x^2$+ $(y – 1)^2$ = 2, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line $x + y$ = 3 touch the curves $C$, $E_1$ and $E_2$ at $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$respectively. Given that $P$ is the mid-point of the line segment $QR$ and $PQ=\frac{2\sqrt{2}}{3}$, the value of $9(x_1y_1 + x_2y_2 + x_3y_3)$ is equal to _____.
5. Let m and n be the number of points at which the function $ƒ(x)$ = max{$x,x^3,x^5,…..,x^{21}$}, $x \in R$, is not differentiable and not continuous, respectively. Then $m + n$ is equal to _____.

04 Apr 2025:Shift-2

04 Apr 2025:Ans-Keys

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