Download JEE Main 2025 Question Paper (03 Apr - Shift 1) 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 $A$ be a matrix of order 3 x 3 and $|A|$=5. If $|2 adj(3A adj(2A)|$=$2^{\alpha}•3^{\beta}•5^{\gamma}$. $\alpha$, $\beta$, $\gamma \in N$ then $\alpha$+$\beta$+$\gamma$ is equal to
   
      
   1. 25
   2. 26
   3. 27
   4. 28
2. Let a line passing through the point (4,1,0) intersect the line $L_1$:$\frac{x-1}{2}$=$\frac{y-2}{3}$=$\frac{z-3}{4}$ at the point $A(\alpha, \beta, \gamma)$ and the line $L_2$:$x-6$=$y$=$-z+4$ at the point $B(a, b, c)$. Then $\begin{equation*} \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \end{equation*}$ is equal to
   
      
   1. 8
   2. 16
   3. 12
   4. 6
3. Let $\alpha$ and $\beta$ be the roots of $x^2$+$\sqrt{3}x$$-16$=0, and $\gamma$ and $\delta$ be the roots of $x^2$+$3x$$-1$=0. If $P_n$=$\alpha^n$+$\beta^n$ and $Q_n$=$\gamma^n+\delta^n$, then $\frac{P_{25}+\sqrt{3P_{24}}}{2P_{23}}$+$\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to
   
      
   1. 3
   2. 4
   3. 5
   4. 7
4. The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is
   
      
   1. 16923
   2. 3763
   3. 33845
   4. 18817
5. Let $A$ = {$-3, -2, -1, 0, 1, 2, 3$}. Let $R$ be a relation on $A$ defined by $x R y$ if and only if $0 \leq x^2+2y \leq 4$. Let $l$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added in $R$ to make it a reflexive relation. then $l + m$ is equal to
   
      
   1. 19
   2. 20
   3. 17
   4. 18
6. A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36}+\frac{y^2}{25}$=1 at $A$ and $B$ such that $(PA)•(PB)$ is maximum. Then $5(PA^2+PB^2)$ is equal to:
   
      
   1. 218
   2. 377
   3. 290
   4. 338
7. The sum 1 + 3 + 11 + 25 + 45 + 71 + .. upto 20 terms, is equal to
   
      
   1. 7240
   2. 7130
   3. 6982
   4. 8124
8. If the domain of the function $f(x)$=$\log_e\left(\frac{2x-3}{5+4x}\right)$+$\sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$, then $\alpha^2+4\beta$ is equal to
   
      
   1. 5
   2. 4
   3. 3
   4. 7
9. If $\sum \limits_{r=1}^{9}\left(\frac{r+3}{2^r}\right)•{}^9C_r$=$\alpha\left(\frac{3}{2}\right)^9-\beta$, $\alpha, \beta \in N$, then $(\alpha+\beta)^2$ is equal to
   
      
   1. 27
   2. 9
   3. 81
   4. 18
10. The number of solutions of the equation $2x+3\tan x$=$\pi$, $x \in [-2\pi, 2\pi]$$-\left\{±\frac{\pi}{2}, ±\frac{3\pi}{2}\right\}$ is
    
       
    1. 6
    2. 5
    3. 4
    4. 3
11. If $y(x)$=$\begin{equation*}\begin{vmatrix} \sin x & \cos x & \sin x+\cos x +1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix} \end{equation*}$, $x \in R$, then $\frac{d^2y}{dx^2}+y$ is equal to
    
       
    1. -1
    2. 28
    3. 27
    4. 1
12. Let $g$ be a differentiable function such that $\int \limits_0^xg(t)dt$=$x-\int \limits_0^xtg(t)dt$, $x \geq 0$ and let $y$=$y(x)$ satisfy the differential equation $\frac{dy}{dx}-y\tan x$=$2(x+1)\sec x g(x)$, $x \in \left[0, \frac{\pi}{2}\right)$. If $y(0)$=0, then $y\left(\frac{\pi}{3}\right)$ is equal to
    
       
    1. $\frac{2\pi}{3\sqrt{3}}$
    2. $\frac{4\pi}{3}$
    3. $\frac{2\pi}{3}$
    4. $\frac{4\pi}{3\sqrt{3}}$
13.  A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $L_1$ : $2x + y + 6$ = 0 and $L_2$ : $4x + 2y – p$ = 0, $p > 0$, at the points $A$ and $B$, respectively. If $AB$=$\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$on the line $L_2$ is $M$, then $\frac{AM}{BM}$ is equal to
    
       
    1. 5
    2. 4
    3. 2
    4. 3
14. Let $z \in C$ be such that $\frac{z^2+3i}{z-2+i}$=$2+3i$. Then the sum of all possible values of $z^2$is
    
       
    1. $19-2i$
    2. $-19-2i$
    3. $19+2i$
    4. $-19+2i$
15. Let $f(x)$=$\int x^3\sqrt{3-x^2}dx$. If $5f(\sqrt{2})$=-4, then $f(1)$ is equal to
    
       
    1. $-\frac{2\sqrt{2}}{5}$
    2. $-\frac{8\sqrt{2}}{5}$
    3. $-\frac{4\sqrt{2}}{5}$
    4. $-\frac{6\sqrt{2}}{5}$
16. Let $a_1$, $a_2$, $a_3$,… be a G. P. of increasing positive numbers. If $a_3a_5$ = 729 and $a_2 + a_4$=$\frac{111}{4}$, then $24(a_1 + a_2 + a_3)$ is equal to
    
       
    1. 131
    2. 130
    3. 129
    4. 128
17. Let the domain of the function $ƒ(x)$ = $log_2 log_4 log_6 (3 + 4x – x^2)$ be $(a, b)$. If $\int \limits_0^{b-a}[x^2]dx$=$p-\sqrt{q}-\sqrt{r}$, $p, q, r \in N$, $gcd(p, q, r)$=1, where $[•]$ is the greatest integer function, then $p+q+r$ is equal to
    
       
    1. 10
    2. 8
    3. 11
    4. 9
18. The radius of the smallest circle which touches the parabolas $y = x^2+ 2$ and $x = y^2 + 2$ 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-2

03 Apr 2025:Ans-Keys

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