Download JEE Main 2025 Question Paper (28 Jan - Shift 2) Mathematics

Please wait! Loading...

 Download as PDF

29 Jan 2025:Shift-1

29 Jan 2025:Shift-2

All the Papers

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. Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$ , is :
   
      
   1. $\frac{1}{3}$
   2. $\frac{4}{15}$
   3. $\frac{2}{3}$
   4. $\frac{2}{5}$
2. Let $A$, $B$, $C$ be three points in $xy-$plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $a\hat{i} + (1– a)\hat{j}$ respectively with respect to the origin $O$. If the distance of the point $C$ from the line bisecting the angle between the vectors $\vec{OA}$ and $\vec{OB}$ is $\frac{9}{\sqrt{2}}$, then the sum of all the possible values of $a$ is :
   
      
   1. 1
   2. 9/2
   3. 0
   4. 2
3. If the components of $\vec{a}$=$\alpha \hat{i}$+$\beta \hat{j}$+$\gamma \hat{k}$ along and perpendicular to $\vec{b}$=$3\hat{i}$+$\hat{j}$$-\hat{k}$ respectively, are $\frac{16}{11}(3\hat{i}+\hat{j}-\hat{k})$ and $\frac{1}{11}(-4\hat{i}-5\hat{j}-17\hat{k})$, then $\alpha^2$+$\beta^2$+$\gamma^2$ is equal to:
   
      
   1. 23
   2. 18
   3. 16
   4. 26
4. If $\alpha+i\beta$ and $\gamma+i\delta$ are the roots of $x^2-(3-2i)x-(2i-2)$=0, $i=\sqrt{-1}$, then $\alpha \gamma$+$\beta \delta$ is equal to :
   
      
   1. 6
   2. 2
   3. -2
   4. -6
5. If the midpoint of a chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}$=1 is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{\alpha}}{3}$, then $\alpha$ is:
   
      
   1. 18
   2. 22
   3. 26
   4. 20
6. Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :
   
      
   1. $\frac{1}{4}$
   2. $\frac{2}{3}$
   3. $\frac{1}{3}$
   4. $\frac{1}{2}$
7. Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)$=$\int \limits_{0}^{x} t f(t)dt$. If $g(x^3)$=$x^6+x^7$, then the value of $\sum \limits_{r=1}^{15}f(r^3)$ is:
   
      
   1. 320
   2. 340
   3. 270
   4. 310
8. The square of the distance of the point $\left(\frac{15}{7}, \frac{32}{7}, 7\right)$ from the line $\frac{x+1}{3}$=$\frac{y+3}{5}$=$\frac{z+5}{7}$ in the direction of the vector $\hat{i}+4\hat{j}+7\hat{k}$ is :
   
      
   1. 54
   2. 41
   3. 66
   4. 44
9. The area of the region bounded by the curves $x(1 + y^2)$ = 1 and $y^2 = 2x$ is :
   
      
   1. $2\left(\frac{\pi}{2}-\frac{1}{3}\right)$
   2. $\frac{\pi}{4}-\frac{1}{3}$
   3. $\frac{\pi}{2}-\frac{1}{3}$
   4. $\frac{1}{2}\left(\frac{\pi}{2}-\frac{1}{3}\right)$
10. Let $\begin{equation*}A=\begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \\ end{bmatrix}\end{equation*}$ and $\begin{equation*} P= \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}\end{equation*}$, $\theta > 0$. If $B$=$PAP^T$, $C$ = $P^TB^{10}P$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $gcd(m, n)$ = 1, then $m + n$ is :
    
       
    1. 65
    2. 127
    3. 258
    4. 2049
11. If $f(x)$=$\int \frac{1}{x^{1/4}(1+x^{1/4})}dx$, $f(0)$=$-6$, then $f(1)$ is equal to :
    
       
    1. $\log_e2+2$
    2. $4(\log_e2-2)$
    3. $2-\log_e2$
    4. $4(\log_e2+2)$
12. Let $f : R \to R$ be a twice differentiable function such that $f(2)$ = 1. If $F(x)$ = $xƒ(x)$ for all $x \in R$, $\int \limits_{0}^{2}xF'(x)dx$=6 and $\int \limits_{0}^{2}x^2F"(x)dx$=40, then $F'(2)+\int \limits_0^2F(x)dx$ is equal to:
    
       
    1. 11
    2. 15
    3. 9
    4. 13
13. For positive integers $n$, if $4a_n$=$(n^2+5n+6)$ and $S_n$=$\sum \limits_{k=1}^n\left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is :
    
       
    1. 540
    2. 1350
    3. 675
    4. 135
14. Let $f : [0, 3] \to A$ be defined by $f(x) = 2x^3– 15x^2 + 36x$ + 7 and $g : [0, \infty) \to B$ be defined by $g(x)$=$\frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and $S$ = {$x \in Z : x \in A or x \in B$}, then $n (S)$ is equal to :
    
       
    1. 30
    2. 36
    3. 29
    4. 31
15. Let $[x]$ denote the greatest integer less than or equal to $x$. Then domain of $f(x)$ = $sec^{–1}(2[x]+1)$ is :
    
       
    1. $(-\infty, -1] \cup [0, \infty)$
    2. $(-\infty, \infty)$
    3. $(-\infty, -1] \cup [1, \infty)$
    4. $(-\infty, \infty)-${0}
16. If $\sum \limits_{r=1}^{13}\left\{\frac{1}{\sin\left(\frac{\pi}{4}+(r-1)\frac{\pi}{6}\right) \sin\left(\frac{\pi}{4}+\frac{r\pi}{6}\right)}\right\}$=$a\sqrt{3}+b$, $a, b \in Z$, then $a^2+b^2$ is equal to :
    
       
    1. 10
    2. 2
    3. 8
    4. 4
17. Two equal sides of an isosceles triangle are along $–x + 2y$ = 4 and $x + y$ = 4. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is :
    
       
    1. $-6$
    2. $12$
    3. $6$
    4. $-2\sqrt{10}$
18. Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a + b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt{4}{3}+\sqrt{3}{4})^{12}$. Then $p + q$ is equal to :
    
       
    1. 283
    2. 295
    3. 287
    4. 299
19.  If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2– 8x$ = 0 and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}$=1 and a point $P$ moves on the line $2x – 3y + 4$ = 0, then the centroid of $\Delta PAB$ lies on the line :
    
       
    1. $4x – 9y = 12$
    2. $x + 9y = 36$
    3. $9x – 9y = 32$
    4. $6x – 9y = 20$
20. Let $ƒ : R – {0} \to (–\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right)$=$f(x)$+$f\left(\frac{1}{x}\right)$. If $ƒ(K) = –2K$, then the sum of squares of all possible values of $K$ is :
    
       
    1. 1
    2. 6
    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. The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is _____.
2. Let $f(x)$=$\lim \limits_{n \to \infty} \sum \limits_{r=0}^{n}\left(\frac{\tan(x/2^{r+1})+\tan^3(x/2^{r+1})}{1-\tan^2(x/2^{r+1})}\right)$. Then $\lim \limits_{x \to 0}\frac{e^x-e^{f(x)}}{(x-f(x))}$ is equal to.............
3. The interior angles of a polygon with $n$ sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then $n$is equal to _____.
4. Let $A$ and $B$ be the two points of intersection of the line $y + 5$ = 0 and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4$ = 0. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\Delta SAB$, where $S$ is the focus of the parabola $y^2$ = $4x$, then the vlaue of $(a + d)$ is ____.
5. If $y = y(x)$ is the solution of the differential equation,
   $\sqrt{4-x^2}\frac{dy}{dx}$=$\left(\left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2-y\right)\sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2)$=$\left(\frac{\pi^2-8}{4}\right)$, then $y^2(0)$ is equal to ...........

28 Jan 2025:Shift-1

28 Jan 2025:Ans-Keys

All the Papers Ans-Keys

Please wait! Loading...

 Download as PDF