Download JEE Main 2025 Question Paper (02 Apr - Shift 2) Mathematics

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03 Apr 2025:Shift-1

03 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. If the image of the point $P(1, 0, 3)$ in the line joining the points $A(4, 7, 1)$ and $B(3, 5, 3)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to
   
      
   1. $\frac{47}{3}$
   2. $\frac{46}{3}$
   3. 18
   4. 13
2. Let $ƒ : [1, \infty) \to [2, \infty)$ be a differentiable function, If $10\int \limits_{1}^{x}f(t)dt$=$5xf(x)-x^5-9$ for all $x \geq 1$, then the value of $ƒ(3)$ is :
   
      
   1. 18
   2. 32
   3. 22
   4. 26
3. The number of terms of an $A.P.$ is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the $A.P.$ is :
   
      
   1. 4
   2. 10
   3. 6
   4. 8
4. Let $A$ = {1, 2, 3,...., 10} and $R$ be a relation on $A$ such that $R$ = {$(a, b) : a = 2b + 1$}. Let $(a1, a2)$, $(a2, a3)$, $(a3, a4)$, ....,$(ak, ak+1)$ be a sequence of $k$elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to :
   
      
   1. 6
   2. 7
   3. 5
   4. 8
5. If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
   
      
   1. $\frac{4}{\sqrt{17}}$
   2. $\frac{\sqrt{3}}{16}$
   3. $\frac{4}{\sqrt{19}}$
   4. $\frac{\sqrt{5}}{7}$
6. The line $L_1$ is parallel to the vector $\vec{a}$=$-3\hat{i}+2\hat{j}+4\hat{k}$ and passes through the point (7, 6, 2) and the line $L_2$ is parallel to the vector $\vec{b}$=$2\hat{i}+\hat{j}+3\hat{k}$ and passes through the point (5, 3, 4). The shortest distance between the lines $L_1$ and $L_2$ is :
   
      
   1. $\frac{23}{\sqrt{38}}$
   2. $\frac{21}{\sqrt{57}}$
   3. $\frac{23}{\sqrt{57}}$
   4. $\frac{21}{\sqrt{38}}$
7. Let $(a, b)$ be the point of intersection of the curve $x^2$ = $2y$ and the straight line $y –2x –6$ = 0 in the second quadrant. Then the integral $I$=$\int \limits_{a}^{b}\frac{9x^2}{1+5^x}dx$ is equal to:
   
      
   1. 24
   2. 27
   3. 18
   4. 21
8. If the system of equation $2x$ + $\lambda y$ + $3z$ = 5
   $3x$ + $2y – z$ = 7
   $4x$ + $5y$ + $\mu z$ = 9
   has infinitely many solutions, then $(\lambda^2 + \mu^2)$ is equal to :
   
      
   1. 22
   2. 18
   3. 26
   4. 30
9. If $\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$, then the number of solutions of $\sqrt{3} cosec^2\theta -2(\sqrt{3}-1) cosec \theta$-4=0, is equal to
   
      
   1. 6
   2. 8
   3. 10
   4. 7
10. Given three indentical bags each containing 10 balls, whose colours are as follows :
       

    	Red	Blue	Green
    Bag I	3	2	5
    Bag II	4	3	3
    Bag III	5	1	4

    
    
    A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag $I$ is $p$ and if the balls is Green, the probability that it is from bag $III$ is $q$, then the value of $\left(\frac{1}{p}+\frac{1}{q}\right)$ is:
    
       
    1. 6
    2. 9
    3. 7
    4. 8
11. If the mean and the variance of 6, 4, $a$, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a + b + ab$ is equal to :
    
       
    1. 105
    2. 103
    3. 100
    4. 106
12. If the domain of the function
    $f(x)$=$\frac{1}{\sqrt{10+3x-x^2}}$+$\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2$+$b^2$ is equal to:
    
       
    1. 26
    2. 29
    3. 25
    4. 30
13.  $4\int \limits_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)dx$$-3\log_e(\sqrt{3})$ is equal to:
    
       
    1. 2+$\sqrt{2}$+$\log_e(1+\sqrt{2})$
    2. 2-$\sqrt{2}$-$\log_e(1+\sqrt{2})$
    3. 2+$\sqrt{2}$-$\log_e(1+\sqrt{2})$
    4. 2-$\sqrt{2}$+$\log_e(1+\sqrt{2})$
14. If $\lim \limits_{x \to 0} \frac{\cos (2x)+a \cos (4x)-b}{x^4}$ is infinite, then $(a+b)$ is equal to:
    
       
    1. $\frac{1}{2}$
    2. 0
    3. $\frac{3}{4}$
    4. $-1$
15. If $\sum \limits_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right)•{}^{11}C_{r+1}$=$\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to:
    
       
    1. 15
    2. 11
    3. 24
    4. 20
16. The number of ways, in which the letters $A$, $B$, $C$, $D$, $E$ can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :
    
    
       
    1. 5880
    2. 960
    3. 840
    4. 5760
17. Let the point $P$ of the focal chord $PQ$ of the parabola $y^2 = 16x$ be (1, –4). If the focus of the parabola divides the chord $PQ$ in the ratio $m : n$, $gcd(m, n)$ = 1, then $m^2 + n^2$is equal to :
    
       
    1. 17
    2. 10
    3. 37
    4. 26
18. Let $\vec{a}$=$2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}$=$3\hat{i}$+$2\hat{j}$+$5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a}-\vec{c})×\vec{b}$=$-18\hat{i}-3\hat{j}+12\hat{k}$ and $\vec{a}•\vec{c}$=3. If $\vec{b}×\vec{c}$=$\vec{d}$, then $|\vec{a}•\vec{d}|$ is equal to :
    
       
    1. 18
    2. 12
    3. 9
    4. 15
19.  Let the area of the triangle formed by a straight Line $L$ : $x + by + c$ = 0 with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of 45° with the positive $x-$axis, then the value of $b^2+c^2$is:
    
       
    1. 90
    2. 93
    3. 97
    4. 83
20. Let $A$ be a 3 × 3 real matrix such that $A^2(A – 2I) –4(A – I)$ = $O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5$= $\alpha A^2$+ $\beta A$ + $\gamma I$, where $\alpha$, $\beta$ and $\gamma$ are real constants, then $\alpha$ + $\beta$ + $\gamma$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 $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx}$+$2y \sec^2x$=$2\sec^2x$+$3\tan x•\sec^2x$ such that $y(0)$=$\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-e^{-2}\right)$ is equal to.........
2. If the sum of the first 10 terms of the series $\frac{4•1}{1+4•1^4}$+$\frac{4•2}{1+4•2^4}$+$\frac{4•3}{1+4•3^4}$+.... is $\frac{m}{n}$, where $gcd(m, n)$=1, then $m+n$ is equal to...........
3. If $y$=$\cos \left(\frac{\pi}{3}+\cos^{-1}\frac{x}{2}\right)$, then $(x-y)^2+3y^2$ is equal to.........
4. Let $A(4, –2)$, $B(1, 1)$ and $C(9, –3)$ be the vertices of a triangle $ABC$. Then the maximum area of the parallelogram $AFDE$, formed with vertices $D$, $E$ and $F$ on the sides $BC$, $CA$ and $AB$ of the triangle $ABC$ respectively, is __________.
5. If the set of all $a \in R – {1}$, for which the roots of the equation $(1 – a)x^2$ + $2(a – 3)x$ + 9 = 0 are positive is $(–\alpha, –\beta] \cup [\beta, \gamma)$, then $2\alpha + \beta + \gamma$ is equal to _________.

02 Apr 2025:Shift-1

02 Apr 2025:Ans-Keys

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