Download JEE Main 2025 Question Paper (04 Apr - Shift 2) 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 $a > 0$. If the function $f(x)$ = $6x^3$$– 45ax^2$ + $108a^2x$ + 1 attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2$ = 54, then $a + x_1 + x_2$ is equal to :
   
      
   1. 15
   2. 18
   3. 24
   4. 13
2. Let $f$ be a differentiable function on $R$ such that $f(2)$=1, $f'(2)$ = 4. Let $\lim \limits_{x \to 0} (f(2+x))^{3/x}$=$e^{\alpha}$. Then the number of times the curve $y$ = $4x^3$$– 4x^2$$– 4(\alpha –7) x$$–\alpha$ meets $x-$axis is :
   
      
   1. 2
   2. 1
   3. 0
   4. 3
3. The sum of the infinite series $\cot^{-1}\left(\frac{7}{4}\right)$+$\cot^{-1}\left(\frac{19}{4}\right)$+$\cot^{-1}\left(\frac{39}{4}\right)$+$\cot^{-1}\left(\frac{67}{4}\right)$+.....
   
      
   1. $\frac{\pi}{2}+\tan^{-1}\left(\frac{1}{2}\right)$
   2. $\frac{\pi}{2}-\cot^{-1}\left(\frac{1}{2}\right)$
   3. $\frac{\pi}{2}+\cot^{-1}\left(\frac{1}{2}\right)$
   4. $\frac{\pi}{2}-\tan^{-1}\left(\frac{1}{2}\right)$
4. Let $A$ ={–3, –2, –1, 0, 1, 2, 3} and $R$ be a relation on $A$ defined by $x R y$ if and only if $2x – y \in {0, 1}$. 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. 18
   2. 17
   3. 15
   4. 16
5. Let the product of $\omega_1$ = $(8 + i)sin \theta$ + $(7 + 4i)cos \theta$and $\omega_2$ = $(1 + 8i)sin \theta$ + $(4 + 7i)cos \theta$ be $\alpha + i \beta$, $i$=$\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha$ + $\beta$ respectively.
   
      
   1. 140
   2. 130
   3. 160
   4. 150
6. Let the values of $p$, for which the shortest distance between the lines $\frac{x+1}{3}$=$\frac{y}{4}$=$\frac{z}{5}$ and $\vec{r}$=$(p\hat{i}+2\hat{j}+\hat{k})$+$\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ is $\frac{1}{\sqrt{6}}$, be $a$, $b$, $(a < b)$. Then the length of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 is
   
      
   1. 9
   2. $\frac{3}{2}$
   3. $\frac{2}{3}$
   4. 18
7. The axis of a parabola is the line $y = x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of $k$ is
   
      
   1. 4
   2. 9
   3. 3
   4. 8
8. Let the domains of the functions $f(x)$ = $log_4log_3log_7(8 – log_2(x^2 + 4x + 5))$ and $g(x)$ = $\sin^{-1}\left(\frac{7x+10}{x-2}\right)$ be $\alpha-\beta$ and $[\gamma, \delta]$, respectively. Then $\alpha^2$+$\beta^2$ + $\gamma^2$ + $\delta^2$is equal to :
   
      
   1. 15
   2. 13
   3. 16
   4. 14
9. A line passing through the point $A(–2, 0)$, touches the parabola $P : y^2 = x – 2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $AB$, parabola $P$ and the $x-$axis, is :
   
      
   1. $\frac{7}{3}$
   2. 2
   3. $\frac{8}{3}$
   4. 3
10. Let the sum of the focal distances of the point $P(4, 3)$on the hyperbola $H$:$\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 be $8\sqrt{\frac{5}{3}}$. If for $H$, the length of the latus rectum is $l$ and the product of the focal distances of the point $P$ is $m$, then $9l^2$+ $6 m$ is equal to
    
       
    1. 184
    2. 186
    3. 185
    4. 187
11. Let the matrix $A$=$\begin{equation*} \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \end{equation*}$ satisfy $A^n$=$A^{n-2}$+$A^2$-$I$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ is:
    
       
    1. 53
    2. 52
    3. 39
    4. 44
12. If the sum of the first 20 terms of the series $\frac{4•1}{4+3•1^2+1^4}$+$\frac{4•2}{4+3•2^2+2^4}$+$\frac{4•3}{4+3•3^2+3^4}$+$\frac{4•4}{4+3•4^2+4^4}$+... is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m + n$ is equal to :
    
       
    1. 423
    2. 420
    3. 421
    4. 422
13.  If $1^2•({}^{15}C_1$+$2^2•({}^{15}C_2$+$3^2•({}^{15}C_3$+...+$15^2•({}^{15}C_{15}$=$2^m•3^n•5^k$, where $m$, $n$, $k \in N$, then $m + n + k$ is equal to :
    
       
    1. 19
    2. 21
    3. 18
    4. 20
14. Let for two distinct values of p the lines $y$ = $x + p$touch the ellipse $E$ : ${x^2}{4^2}$+$\frac{y^2}{3^2}$= 1 at the points $A$and $B$. Let the line $y = x$ intersect $E$ at the points $C$and $D$. Then the area of the quadrilateral $ABCD$ is equal to
    
       
    1. 36
    2. 24
    3. 48
    4. 20
15. Consider two sets $A$ and $B$, each containing three numbers in $A.P.$ Let the sum and the product of the elements of $A$ be 36 and $p$ respectively and the sum and the product of the elements of $B$ be 36 and $q$ respectively. Let $d$ and $D$ be the common differences of $AP’s$ in $A$ and $B$ respectively such that $D = d + 3$, $d > 0$. If $\frac{p+q}{p-q}$=$\frac{19}{5}$, then $p – q$ is equal to
    
       
    1. 600
    2. 450
    3. 630
    4. 540
16. If a curve $y = y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation $(7x^4 \cot y-e^x cosec y) \frac{dx}{dy}$=$x^5$, $x \geq 1$, then at $x = 2$, the value of $\cos y$ is:
    
       
    1. $\frac{2e^2-e}{64}$
    2. $\frac{2e^2+e}{64}$
    3. $\frac{2e^2-e}{128}$
    4. $\frac{2e^2+e}{128}$
17. The centre of a circle $C$ is at the centre of the ellipse $E$:$\frac{x^2}{a^2}$+$\frac{y^2}{b^2}$ = 1, $a > b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $PF_1F_2$ is 30 and the length of the major axis of $E$ is 17, then the distance between the foci of $E$ is :
    
       
    1. 26
    2. 13
    3. 12
    4. $\frac{13}{2}$
18. Let $f(x)$+$2f\left(\frac{1}{x}\right)$=$x^2+5$ and $g(x)$$-3g\left(\frac{1}{2}\right)$=$x$, $x > 0$. If $\alpha$=$\int \limits_1^2f(x)dx$, and $\beta$=$\int \limits_1^2g(x)dx$, then the value of $9\alpha+\beta$ is:
    
       
    1. 1
    2. 0
    3. 10
    4. 11
19.  Let $A$ be the point of intersection of the lines $L_1$:$\frac{x-7}{1}$=$\frac{y-5}{0}$=$\frac{z-3}{-1}$ and $L_2$:$\frac{x-1}{3}$=$\frac{y+3}{4}$=$\frac{z+7}{5}$. Let $B$ and $C$ be the point on the lines $L_1$ and $L_2$ respectively such that $AB$ = $AC$ = 15. Then the square of the area of the triangle $ABC$ is :
    
       
    1. 54
    2. 63
    3. 57
    4. 60
20. Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, $a$, $b$ be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :
    
       
    1. 1
    2. $\frac{3}{4}$
    3. 2
    4. $\frac{1}{2}$

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 $\alpha$ is a root of the equation $x^2$ + $x$ + 1 = 0 and $\sum \limits_{k=1}^{n}\left(\alpha^k+\frac{1}{\alpha^k}\right)^2$=20, then $n$ is equal to ........
2. If $\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9}dx$=$\frac{1}{m}\left(\left(\sqrt{1+x^2}+x\right)^n\left(n\sqrt{1+x^2}-x\right)\right)$+$C$ where $C$ is the constant of integration and $m, n \in N$, then $m + n$ is equal to
3. A card from a pack of 52 cards is lost. From the remaining 51 cards, $n$ cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}$, the $n$ is equal to
4. Let $m$ and $n$, $(m < n)$ be two 2-digit numbers. Then the total numbers of pairs $(m, n)$, such that $gcd (m, n)$ = 6, is ____
5. Let the three sides of a triangle $ABC$ be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the centroid of the triangle $ABC$. Then $6(|\vec{AG}|^2+|\vec{BG}|^2$+$|\vec{CG}|^2)$ is equal to........

04 Apr 2025:Shift-1

04 Apr 2025:Ans-Keys

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