Download JEE Main 2025 Question Paper (24 Jan - Shift 2) Mathematics

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28 Jan 2025:Shift-1

28 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. The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}$=1, whose mid-point is (3, 1) is:
   
      
   1. $48x + 25y$ = 169
   2. $4x + 122y$ = 134
   3. $25x + 101y$ = 176
   4. $5x + 16y$ = 31
2. The function $ƒ : (–\infty, \infty) \to (–\infty, 1)$, defined by $f(x)$=$\frac{2^x-2^{-x}}{2^x+2^{-x}}$ is:
   
      
   1. One-one but not onto
   2. Onto but not one-one
   3. Both one-one and onto
   4. Neither one-one nor onto
3. If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1}\left\{\beta+\frac{(1+\beta^2)}{\alpha-\beta}\right\}$+$\cot^{-1}\left\{\gamma+\frac{(1+\gamma^2)}{(\beta-\gamma)}\right\}$+$\cot^{-1}\left\{\alpha+\frac{(1+\alpha^2)}{(\gamma-\alpha)}\right\}$ is equal to:
   
      
   1. $\frac{\pi}{2}-(\alpha+\beta+\gamma)$
   2. $3\pi$
   3. 0
   4. $\pi$
4. Let $f : (0, \infty) \to R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2f '(x)$ = $2xf(x)$ + 3, with $f(1)$ = 4. Then $2f(2)$ is equal to:
   
      
   1. 29
   2. 19
   3. 39
   4. 23
5. Let $A$=$\left\{x \in (0, \pi)-\left\{\frac{\pi}{2}\right\} \right.$:$\left.\log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2\right\}$ and $B$=$\left\{x \geq 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\right\}$. Then $n(A \cup B)$ is equal to:
   
      
   1. 4
   2. 2
   3. 8
   4. 6
6. Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r}$, $-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p}+\vec{q}+\vec{r}}{4}$and $\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}$respectively, then $\alpha$+$2\beta$+$3\gamma$ is equal to:
   
      
   1. 3
   2. 1
   3. 6
   4. 4
7. Let $[x]$ denote the gereatest integer function, and let $m$ and $n$ respectively be the numbers of the points, where the function $f(x)$ = $[x] + |x – 2|$, $–2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:
   
      
   1. 6
   2. 9
   3. 8
   4. 7
8. Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y$ = 11, $x + 2y$ = 16 and $2x + 3y$ = 29. Then the product of the smallest and the largest values of $\alpha$ is equal to :
   
      
   1. 22
   2. 44
   3. 33
   4. 55
9. In an arithmetic progression, if $S_{40}$ = 1030 and $S_12$ = 57, then $S_{30} – S_{10}$ is equal to:
   
      
   1. 510
   2. 515
   3. 525
   4. 505
10. If $7+\frac{1}{7}(5+\alpha)$+$\frac{1}{7^2}(5+2\alpha)$+$\frac{1}{7^3}(5+3\alpha)+.....\infty$, then the value of $\alpha$ is:
    
       
    1. 1
    2. $\frac{6}{7}$
    3. 6
    4. $\frac{1}{7}$
11. If the system of equations
    $x + 2y – 3z$ = 2
    $2x + \lambda y + 5z$ = 5
    $14x + 3y + \mu z$ = 33
    has infinitely many solutions, then $\lambda + \mu$ is equal to:
    
       
    1. 13
    2. 10
    3. 11
    4. 12
12. Let (2, 3) be the largest open interval in which the function $f(x)$ = $2 \log_e(x – 2) – x^2 + ax + 1$ is strictly increasing and $(b, c)$ be the largest open interval, in which the function $g(x)$ = $(x – 1)^3(x + 2 – a)^2$ is strictly decreasing. Then $100(a + b – c)$ is equal to:
    
       
    1. 280
    2. 360
    3. 420
    4. 160
13. Suppose $A$ and $B$ are the coefficients of $30^{th}$ and $12^{th}$ terms respectively in the binomial expansion of $(1 + x)^{2n–1}$. If $2A = 5B$, then $n$ is equal to:
    
       
    1. 22
    2. 21
    3. 20
    4. 19
14. Let $\vec{a}$=$3\hat{i}$$-\hat{j}$+$2\hat{k}$, $\vec{b}$=$\vec{a}$×(\hat{i}-2\hat{k})$ and $\vec{c}$=$\vec{b}×\vec{k}$. Then the projection of $\vec{c}-2\hat{j}$ on $\vec{a}$ is:
    
       
    1. $3\sqrt{7}$
    2. $\sqrt{14}$
    3. $2\sqrt{14}$
    4. $2\sqrt{7}$
15. For some $a$, $b$, let
    $f(x)$=$\begin{equation*} \begin{vmatrix} a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 & b+\frac{\sin x}{x}\end{vmatrix} \end{equation*}$, $x \neq 0$, $\lim \limits_{x \to 0} f(x)$=$\lambda$+$\mu a$+ $v b$. Then $(\lambda+\mu+v)^2$ is equal to :
    
       
    1. 25
    2. 9
    3. 36
    4. 16
16. Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:
    
       
    1. 8575
    2. 9100
    3. 8925
    4. 8750
17. The area of the region enclosed by the curves $y = e^x$, $y = |e^x– 1|$ and $y-$axis is:
    
       
    1. $1+\log_e2$
    2. $\log_e2$
    3. $2\log_e2-1$
    4. $1-\log_e2$
18. The number of real solution(s) of the equation $x^2+ 3x + 2$ = $min{|x – 3|, |x + 2|}$ is :
    
       
    1. 2
    2. 0
    3. 3
    4. 1
19.  Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is :
    
       
    1. $\frac{5}{8}$
    2. $\frac{3}{16}$
    3. $\frac{1}{8}$
    4. $\frac{3}{8}$
20. If the equation of the parabola with vertex $V\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y$ = 0 is $\alpha x^2$ + $\beta y^2$ – $\gamma xy$ – $30x$ – $60y$ + 225 = 0, then $\alpha$+$\beta$+$\gamma$ is equal to:
    
       
    1. 6
    2. 8
    3. 7
    4. 9

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. Number of functions $ƒ : {1, 2, …, 100} \to {0, 1}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ______.
2. Let $P$ be the image of the point $Q(7, –2, 5)$ in the line $L$:$\frac{x-1}{2}$=$\frac{y+1}{3}$=$\frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\Delta PQR$ is ____
3. Let $y = y(x)$ be the solution of the differential equation $2\cos x \frac{dy}{dx}$=$\sin 2x-4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)$=0, then $y'\left(\frac{\pi}{4}\right)$+$\left(\frac{\pi}{4}\right)$ is equal to ____.
4. Let $H_1$:$\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 and $H_2$:$-\frac{x^2}{A^2}+\frac{y^2}{B^2}$=1 be two hyperbolas having length of latus rectums $15\sqrt{2}$and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1$=$\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$is equal to ______
5. If $\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}dx$=$x\sqrt{x^2+x+1}$+$\alpha \sqrt{x^2+x+1}$+$\beta \log_e \left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$+C, where C is the constant of integration, then $\alpha + 2\beta$ is equal to ___

24 Jan 2025:Shift-1

24 Jan 2025:Ans-Keys

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