Download JEE Main 2025 Question Paper (22 Jan - Shift 2) Mathematics

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23 Jan 2025:Shift-1

23 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. Let $\alpha$, $\beta$, $\gamma$ and $\delta$ be the coefficients of $x^7$, $x^5$, $x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5$+$(x-\sqrt{x^3-1})^5$, $x > 1$. If $u$ and $v$ satisfy the equations
   $\alpha u$+$\beta v$=18,
   $\gamma u$+$\delta v$=20,
   then $u+v$ equals:
   
      
   1. 5
   2. 4
   3. 3
   4. 8
2. In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is :
   
      
   1. 144
   2. 72
   3. 96
   4. 120
3. Let $P(4,4 \sqrt{3})$ be a point on the parabola $y^2$ = $4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to:
   
      
   1. $\frac{263\sqrt{3}}{8}$
   2. $17\sqrt{3}$
   3. $\frac{343\sqrt{3}}{8}$
   4. $\frac{34\sqrt{3}}{3}$
4. For a 3 × 3 matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a 3 × 3 matrix such that $|A|$ = $\frac{1}{2}$ and trace $(A)$ = 3. If $B$ = $adj(adj(2A))$, then the value of $|B|$ + $trace (B)$ equals:
   
      
   1. 156
   2. 132
   3. 174
   4. 280
5. Suppose that the number of terms in an A.P. is $2k$, $k \in N$.  If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then $k$ is equal to
   
      
   1. 5
   2. 8
   3. 6
   4. 4
6. Let a line pass through two distinct points $P(–2, –1, 3)$ and $Q$, and be parallel to the vector $3\hat{i}$+$2\hat{j}$+$2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the square of the area of $\Delta PQR$ is equal to: to
   
      
   1. 136
   2. 140
   3. 144
   4. 148
7. If $\lim \limits_{x \to \infty}\left(\left(\frac{e}{1-e}\right)\left(\frac{1}{e}-\frac{x}{1+x}\right)\right)^x$=$\alpha$, then the value of $\frac{\log_e \alpha}{1+\log_e \alpha}$ equals:
   
      
   1. $e$
   2. $e^{-2}$
   3. $e^2$
   4. $e^{-1}$
8. Let $f(x)$=$\int \limits_{0}^{x^2}\frac{t^2-8t+15}{e^t}dt$, $x \in R$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
   
      
   1. 2 and 3
   2. 3 and 2
   3. 1 and 3
   4. 2 and 2
9. The perpendicular distance, of the line $\frac{x-1}{2}$=$\frac{y+2}{-1}$=$\frac{z+2}{3}$ from the point $P(2, –10,1)$, is:
   
      
   1. 6
   2. $5\sqrt{2}$
   3. $3\sqrt{5}$
   4. $4\sqrt{3}$
10. If $x = f(y)$ is the solution of the differential equation
    $(1+y^2)$+$(x-2e^{\tan^{-1}y})\frac{dy}{dx}$=0, $y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)$=1, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :
    
       
    1. $e^{\pi/4}$
    2. $e^{\pi/12}$
    3. $e^{\pi/3}$
    4. $e^{\pi/6}$
11. If $\int e^{x} \left(\frac{x \sin^{-1}x}{\sqrt{1-x^2}}+\frac{\sin^{-1}x}{(1-x^2)^{3/2}}+\frac{x}{1-x^2}\right)dx$=$g(x)+C$ where $C$ is the constant of integration, then $g\left(\frac{1}{2}\right)$ equal:
    
       
    1. $\frac{\pi}{6}\sqrt{\frac{e}{2}}$
    2. $\frac{\pi}{4}\sqrt{\frac{e}{2}}$
    3. $\frac{\pi}{6}\sqrt{\frac{e}{3}}$
    4. $\frac{\pi}{4}\sqrt{\frac{e}{3}}$
12. Let $\alpha_{\theta}$ and $\beta_{\theta}$ be the distinct roots of $2x^2$ + $(\cos \theta)x$ –1 = 0, $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of
    
       
    1. 24
    2. 25
    3. 27
    4. 17
13. Let $A$ = {1, 2, 3, 4} and $B$ = {1, 4, 9, 16}. Then the number of many-one functions $f : A \to B$ such that $1 \in f(A)$ is equal to :
    
       
    1. 127
    2. 151
    3. 163
    4. 139
14. If the system of linear equations :
    $x + y + 2z$= 6,
    $2x + 3y + az$ = $a + 1$,
    $–x – 3y + bz$ = $2b$,
    where $a, b \in R$, has infinitely many solutions, then $7a + 3b$ is equal to :
    
       
    1. 9
    2. 12
    3. 16
    4. 22
15. Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}$+$2\vec{b}$ and $3\vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in [–1, 3] is :
    
       
    1. 3
    2. 2
    3. 1
    4. 0
16. Let $E$:$\frac{x^2}{a^2}$+$\frac{y^2}{b^2}$=1, $a > b$ and $H$: $\frac{x^2}{A^2}-\frac{y^2}{B^2}$=1. Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a-A$=2, and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$, then the sum of the lengths of their latus rectums is equal to :
    
       
    1. 10
    2. 7
    3. 8
    4. 9
17. If $A$ and $B$ are two events such that $P(A \cap B)$ = 0.1, and $P(A|B)$ and $P(B|A)$ are the roots of the equation $12x^2$ – $7x$ + 1 = 0, then the value of $\frac{P(\bar{A} \cup \bar{B}}{P(\bar{A}\cap \bar{B})}$ is:
    
       
    1. $\frac{5}{3}$
    2. $\frac{4}{3}$
    3. $\frac{9}{4}$
    4. $\frac{7}{4}$
18. The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2\sin^2\theta$ = $\cos2 \theta$ and $2cos^2 \theta$ = $3sin \theta$ is
    
       
    1. $\frac{\pi}{2}$
    2. $4\pi$
    3. $\frac{5\pi}{6}$
    4. $\pi$
19.  Let the curve $z(1 + i)$ + $\bar{z}(1 – i)$= 4, $z \in C$, divide the region $|z – 3| \leq1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha – \beta|$ equals :
    
       
    1. $1+\frac{\pi}{2}$
    2. $1+\frac{\pi}{3}$
    3. $1+\frac{\pi}{4}$
    4. $1+\frac{\pi}{6}$
20. The area of the region enclosed by the curves $y = x^2– 4x$ + 4 and $y^2 = 16 – 8x$ is :
    
       
    1. $\frac{8}{3}$
    2. $\frac{4}{3}$
    3. 5
    4. 8

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. Let $y = f(x)$ be the solution of the differential equation $\frac{dy}{dx}$+$\frac{xy}{x^2-1}$=$\frac{x^6+4x}{\sqrt{1-x^2}}$, $-1 < x < 1$ such that $f(0)$=0. If $6\int \limits_{-1/2}^{1/2}f(x)dx$=$2\pi-\alpha$, then $\alpha^2$ is equal to ................
2. Let $A(6, 8)$, $B(10 cos\alpha, –10 sin\alpha)$ and $C (–10 sin\alpha, 10 cos\alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5a – 3h + 6k + 100 sin2\alpha)$is equal to___________
3. Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while R lies on the other. Then $(QR)^2$ is equal to ________.
4. If $\sum \limits_{r=1}^{30}\frac{r^2({}^30C_r)^2}{{}^{30}C_{r-1}}$=$\alpha$×$10^{29}$, then $\alpha$ is equal to..............
5. Let $A$ = {1, 2, 3}. The number of relations on $A$, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is ______.

22 Jan 2025:Shift-1

22 Jan 2025:Ans-Keys

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