Download JEE Main 2025 Question Paper (29 Jan - Shift 1) Mathematics

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22 Jan 2025:Shift-1

22 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 the line $x + y$ = 1 meet the circle $x^2 + y^2$ = 4 at the points $A$ and $B$. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects is the circle at $C$ and $D$, then the area of the quadrilateral $ADBC$ is equal to
   
      
   1. $3\sqrt{7}$
   2. $2\sqrt{14}$
   3. $5\sqrt{7}$
   4. $\sqrt{14}$
2. Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x)$=$\begin{equation*} \begin{vmatrix} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2x & 1+\cos^2 x & 4\sin 4x \\ \sin^2x & \cos^2 x & 1+4\sin 4x \end{vmatrix} \end{equation*}$, $x \in R$. Then $M^4-m^4$ is equal to:
   
      
   1. 1280
   2. 1295
   3. 1040
   4. 1215
3. Two parabolas have the same focus (4,3) and their directrices are the $x-$axis and the $y-$axis, respectively. If these parabolas intersects at the points $A$ and $B$, then $(AB)^2$ is equal to
   
      
   1. 192
   2. 384
   3. 96
   4. 392
4. Let ABC be a triangle formed by the lines $7x – 6y$ + 3 = 0, $x + 2y$ –31 = 0 and $9x –2y$–19 = 0, Let the point $(h,k)$ be the image of the centroid of $\Delta ABC$ in the line $3x + 6y$ –53 = 0. Then $h^2 + k^2+ hk$ is equal to
   
      
   1. 37
   2. 47
   3. 40
   4. 36
5. Let $\vec{a}$=$2\hat{i}$$-\hat{j}$+$3\hat{k}$, $\vec{b}$=$3\hat{i}$$-5\hat{j}$+$\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}×\vec{c}$=$\vec{c}×\vec{b}$ and $(\vec{a}+\vec{c})•(\vec{b}+\vec{c})$=168. Then the maximum value of $|\vec{c}|^2$ is :
   
      
   1. 77
   2. 462
   3. 308
   4. 154
6. Let $P$ be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in $P$ are formed by using the digits 1,2 and 3 only, then the number of elements in the set $P$ is :
   
      
   1. 158
   2. 173
   3. 164
   4. 161
7. Let the area of the region{$(x,y) : 2y \leq x^2+3, y+|x|\leq3, y \geq |x –1|$} be $A$. Then $6A$ is equal to:
   
      
   1. 16
   2. 12
   3. 18
   4. 14
8. The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt{3}{7}+\sqrt{12}{11})^n$ is 183, is:
   
      
   1. 2184
   2. 2148
   3. 2172
   4. 2196
9. The number of solutions of the equation $\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)$$\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)$=0 is:
   
      
   1. 2
   2. 4
   3. 1
   4. 3
10. Let $y = y(x)$ be the solution of the differential equation $\cos x(log_e (\cos x))^2dy$ + $(\sin x –3y\sin x log_e(\cos x))dx$ = 0, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right)$=$\frac{-1}{\log_e2}$, then $y\left(\frac{\pi}{6}\right)$ is:
    
       
    1. $\frac{2}{\log_e(3)-\log_e(4)}$
    2. $\frac{1}{\log_e(4)-\log_e(3)}$
    3. $-\frac{1}{\log_e(4)}$
    4. $\frac{1}{\log_e(3)-\log_e(4)}$
11. Define a relation $R$ on the interval $\left[0, \frac{\pi}{2}\right)$ by $x R y$ if and only if $sec^2x – tan^2y$ = 1. Then $R$ is :
    
       
    1. an equivalence relation
    2. both reflexive and transitive but not symmetric
    3. both reflexive and symmetric but not transitive
    4. reflexive but neither symmetric not transitive
12. Let the ellipse, $E_1$:$\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1, $a > b$ and $E_2$:$\frac{X^2}{A^2}+\frac{Y^2}{B^2}$=1, $A < B$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$meet at $A$, $B$, $C$ and $D$, then the area of the quadrilateral $ABCD$ equals:
    
       
    1. $6\sqrt{6}$
    2. $\frac{18\sqrt{6}}{5}$
    3. $\frac{12\sqrt{6}}{5}$
    4. $\frac{24\sqrt{6}}{5}$
13. Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{th}$ term is :
    
       
    1. 84
    2. 122
    3. 90
    4. 108
14. Let $\vec{a}$=$\hat{i}$+$2\hat{j}$+$\hat{k}$ and $\vec{b}$=$2\hat{i}$+$7\hat{j}$+$3\hat{k}$. Let
    $L_1$:$\vec{r}$=$(-\hat{i}+2\hat{j}+\hat{k})$+$\lambda \vec{a}$, $\lambda \in R$ and
    $L_2$:$\vec{r}$=$(\hat{j}+\hat{k})$+$\mu \vec{b}$, $\mu \in R$ be two lines. If the line $L_3$ passes through the point of intersection of $L_1$and $L_2$, and is parallel to $\vec{a}$+$\vec{b}$, then $L_3$ passes through the point:
    
       
    1. (8, 26, 12)
    2. (2,8, 5)
    3. (–1, –1, 1)
    4. (5, 17, 4)
15. The value of $\lim \limits_{n \to \infty}\left(\sum \limits_{K=1}^{n}\frac{k^3+6k^2+11k+5}{(k+3)!}\right)$ is:
    
       
    1. $\frac{4}{3}$
    2. 2
    3. $\frac{7}{3}$
    4. $\frac{5}{3}$
16. The integral 80$\int \limits_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16\sin 2\theta}\right)d\theta$ is equal to :
    
       
    1. $3 log_e4$
    2. $6 log_e4$
    3. $4 log_e3$
    4. $2 log_e3$
17. Let $L_1$:$\frac{x-1}{1}$=$\frac{y-2}{-1}$=$\frac{z-1}{2}$ and $L_2$:$\frac{x+1}{-1}$=$\frac{y-2}{2}$=$\frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$ , then $|5\alpha–11\beta–8\gamma|$ equals :
    
       
    1. 18
    2. 16
    3. 25
    4. 20
18. Let $x_1$, $x_2$,......$x_{10}$ be ten observations such that
    $\sum \limits_{i=1}^{10}(x_i-2)$=30, $\sum \limits_{i=1}^{10}(x_i-\beta)^2$=98, $\beta$ > 2 and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1–1)$ + $4\beta$, $2(x_2 –1)$ + $4\beta$, ....., $2(x_{10}–1)+ 4\beta$, then $\frac{\beta \mu}{\sigma^2}$ is equal to:
    
       
    1. 100
    2. 110
    3. 120
    4. 90
19.  Let $|z_1-8-2i| \leq 1$ and $|z_2-2+6i|\leq 2$and $z_1, z_2 \in C$. Then the minimum value of $|z_1-z_2|$ is :
    
       
    1. 3
    2. 7
    3. 13
    4. 10
20. Let $A$=$[a_{ij}]$=$\begin{equation*}\begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \end{equation*}$. If $A_{ij}$ is the cofactor of $a_{ij}$, $C_{ij}$=$\sum \limits_{k=1}^{2}a_{ik}A_{jk}$, $1 \leq i, j \leq 2$, and $C$=$[C_{ij}]$, then $8|C|$ is equal to :
    
    1. 262
    2. 288
    3. 242
       
    4. 222

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 $f : (0, \infty) \to R$ be a twice differentiable function. If for some $a \neq 0$, $\int \limits_{0}^{1}f(\lambda x)d \lambda$=$af(x)$, $f(1)$=1 and $f(16)$=$\frac{1}{8}$, then $16-f'\left(\frac{1}{16}\right)$ is equal to.........
2. Let $S$=$\left\{m \in Z :A^{m^2}+A^m=3I-A^{-6}\right\}$, where $A$=$\begin{equation*}\begin{bmatrix} 2 & -1 \\ 1 & 0\end{bmatrix}\end{equation*}$. Then $n(S)$ is equal to ____.
3. Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in N$for which $\lim \limits_{x \to 0^+}\left(x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+...+\left[\frac{p}{x}\right]\right)\right.$$-x^2\left.\left(\left[\frac{1}{x^2}\right]+\left[\frac{2^2}{x^2}\right]+...+\left[\frac{9^2}{x^2}\right]\right)\right)\geq 1$ is equal to.........
4. The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is 4 ____.
5. Let $S$ = $\left\{x:\cos^{-1}x=\pi+\sin^{-1}x+\sin^{-1}(2x+1)\right\}$. Then $\sum \limits_{x \in S}(2x-1)^2$ is equal to...........

29 Jan 2025:Shift-2

29 Jan 2025:Ans-Keys

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