Download JEE Main 2025 Question Paper (23 Jan - Shift 2) Mathematics

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24 Jan 2025:Shift-1

24 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. If in the expansion of $(1 + x )^p$$(1 – x)^q$, the coefficients of $x$ and $x^2$ are 1 and –2, respectively, then $p^2+ q^2$ is equal to :
   
      
   1. 8
   2. 18
   3. 13
   4. 20
2. Let $A$ = {$(x, y) \in R × R : |x + y| \geq 3$} and $B$ = {$(x, y) \in R × R : |x| + |y| \leq 3$}. If $C$ = {$(x, y) \in A \cap B : x = 0 or y = 0$}, then $\sum \limits_{(x, y)\in C}|x+y|$ is:
   
      
   1. 15
   2. 18
   3. 24
   4. 12
3. The system of equations $x + y + z$ = 6,
   $x + 2y + 5z$ = 9,
   $x + 5y + \lambda z = µ$, has no solution if
   
      
   1. $\lambda=17, \mu \neq 18$
   2. $\lambda \neq 17, \mu \neq 18$
   3. $\lambda=15, \mu \neq 17$
   4. $\lambda=17, \mu = 18$
4. Let $\int x^3\sin x dx$=$g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g'\left(\frac{\pi}{2}\right)\right)$=$\alpha \pi^3$+$\beta \pi^2$+$\gamma$, $\alpha$+$\beta$$-\gamma$ equals:
   
      
   1. 55
   2. 47
   3. 48
   4. 62
5. A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x – y + 2$ = 0 and $y + 2$ = 0, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio 2 : 1 is $9(x^2 + \alpha y^2+ \beta xy + \gamma x + 28 y)$ – 76 = 0, then $\alpha – \beta – \gamma$ is equal to :
   
      
   1. 24
   2. 23
   3. 21
   4. 22
6. The distance of the line $\frac{x-2}{2}$=$\frac{y-6}{3}$=$\frac{z-3}{4}$ from the point (1, 4, 0) along the line $\frac{x}{1}$=$\frac{y-2}{2}$=$\frac{z+3}{3}$ is
   
      
   1. $\sqrt{17}$
   2. $\sqrt{14}$
   3. $\sqrt{15}$
   4. $\sqrt{13}$
7. Let the point $A$ divide the line segment joining the points $P(–1, –1, 2)$ and $Q(5, 5,10)$ internally in the ratio $r : 1 (r > 0)$. If $O$ is the origin and $(\vec{OQ}•\vec{OA})$$-\frac{1}{5}|\vec{OP}×\vec{OA}|^2$=10, then the value of $r$ is :
   
      
   1. 14
   2. 3
   3. $\sqrt{7}$
   4. 7
8. If the area of the region {$(x, y) : –1 \leq x \leq 1$, $0 \leq y \leq a$ + $e^{|x|}–e^{–x}, a > 0$} is $\frac{e^2+8e+1}{e}$, then the value of $a$ is :
   
      
   1. 7
   2. 6
   3. 8
   4. 5
9. A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81 cm^3 /min$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} cm/min$. The surface area $(in cm^2)$ of the chocolate ball (without the ice-cream layer) is :
   
      
   1. $225 \pi$
   2. $128 \pi$
   3. $196 \pi$
   4. $256 \pi$
10. A board has 16 squares as shown in the figure :
    
    Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
    
       
    1. $\frac{4}{5}$
    2. $\frac{7}{10}$
    3. $\frac{3}{5}$
    4. $\frac{23}{30}$
11. Let $x = x(y)$ be the solution of the differential equation $y$=$\left(x-y\frac{dx}{dy}\right)\sin \left(\frac{x}{y}\right)$, $y > 0$ and $x(1)$=$\frac{\pi}{2}$. Then $\cos(x(2))$ is equal to:
    
       
    1. $1-2(\log_e2)^2$
    2. $2(\log_e2)^2-1$
    3. $2(\log_e2)-1$
    4. $1-2(\log_e2)$
12. Let the range of the function $f(x)$ = 6 + $16 cosx$•$\cos\left(\frac{\pi}{3}-x\right)$•$\cos\left(\frac{\pi}{3}+x\right)$•$\sin 3x • \cos 6x$, $x \in R$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12$ = 0 is :
    
       
    1. 11
    2. 8
    3. 10
    4. 9
13. Let the shortest distance from $(a, 0), a > 0$, to the parabola $y^2 = 4x$ be 4. Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is:
    
       
    1. $x^2$+$y^2$$-6x$+5=0
    2. $x^2$+$y^2$$-4x$+5=0
    3. $x^2$+$y^2$$-10x$+5=0
    4. $x^2$+$y^2$$-8x$+5=0
14. Let $X = R × R$. Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2)$ <=> $b_1 = b_2$.
    Statement-I: $R$ is an equivalence relation.
    Statement-II: For some $(a, b) \in X$, the set $S$ = {$(x, y) \in X : (x, y) R (a, b)$} represents a line parallel to $y = x$. In the light of the above statements, choose the correct answer from the options given below:
    
       
    1. Both Statement-I and Statement-II are false.
    2. Statement-I is true but Statement-II is false.
    3. Both Statement-I and Statement-II are true.
    4. Statement-I is false but Statement-II is true
15. The length of the chord of the ellipse $\frac{x^2}{4}$+$\frac{y^2}{2}$=1, whose mid-point is $\left(1, \frac{1}{2}\right)$, is:
    
       
    1. $\frac{2}{3}\sqrt{15}$
    2. $\frac{5}{3}\sqrt{15}$
    3. $\frac{1}{3}\sqrt{15}$
    4. $\sqrt{15}$
16. Let $A$ = $[a_{ij}]$ be a 3 × 3 matrix such that $\begin{equation*} A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{equation*}$ and $\begin{equation*} A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \end{equation*}$ and $\begin{equation*} A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \end{equation*}$, then $a_{23}$ equals:
    
       
    1. -1
    2. 0
    3. 2
    4. 1
17. The number of complex numbers $z$, satisfying $|z|$ = 1 and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|$=1, is:
    
       
    1. 6
    2. 4
    3. 10
    4. 8
18. If the square of the shortest distance between the lines $\frac{x-2}{1}$=$\frac{y-1}{2}$=$\frac{z+3}{-3}$ and $\frac{x+1}{2}$=$\frac{y+3}{4}$=$\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m + n$ is equal to:
    
       
    1. 6
    2. 9
    3. 21
    4. 14
19.  If $I$=$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin^{\frac{3}{2}}x}{\sin^{\frac{3}{2}}x +\cos^{\frac{3}{2}}x}dx$, then $\int \limits_{0}^{21} \frac{x \sin x \cos x}{\sin^4x +\cos^4x} dx$ equals:
    
       
    1. $\frac{\pi^2}{16}$
    2. $\frac{\pi^2}{4}$
    3. $\frac{\pi^2}{8}$
    4. $\frac{\pi^2}{12}$
20. $\lim \limits_{x \to \infty}\frac{(2x^2-3x+5)(3x-1)^{\frac{x}{2}}}{(3x^2+5x+4)\sqrt{(3x+2)^x}}$ is equal to:
    
       
    1. $\frac{2}{\sqrt{3e}}$
    2. $\frac{2e}{\sqrt{3}}$
    3. $\frac{2e}{3}$
    4. $\frac{2}{3\sqrt{e}}$

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. The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is ________.
2. Let $\alpha$, $\beta$ and $x^2-ax-b$=0, with $Im(\alpha)$  < $Im(\beta)$.  Let $P_a$=$\alpha^n-\beta^n$. If $P_3$=$-5\sqrt{7}i$, $P_4$=$-3\sqrt{7}i$ and $P_5$=$11\sqrt{7}i$ and $P_6$=$45\sqrt{7}i$, then $|\alpha^4+\beta^4|$ is equal to............
3. The focus of the parabola $y^2$ = $4x + 16$ is the centre of the circle $C$ of radius 5. If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3x – y$ = 0 and $x + \lambda y$ = 4, are $\lambda_1$ and $\lambda_2$, $\lambda_1$ < $\lambda_2$, then $12\lambda_1$ + $29\lambda_2$is equal to _______.
4. The variance of the numbers 8, 21, 34, 47, …, 320, is ________.
5. The roots of the quadratic equation $3x^2– px + q$ = 0 are $10^{th}$ and $11^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$.If the sum of the first 11 terms of this arithmetic progression is 88, then $q – 2q$ is equal to _________.

23 Jan 2025:Shift-1

23 Jan 2025:Ans-Keys

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