Download JEE Main 2025 Question Paper (29 Jan - Shift 2) 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. If the set of all $a \in R$, for which the equation $2x^2$+ $(a – 5)x$ + 15 = $3a$ has no real root, is the interval $(\alpha, \beta)$, and $X$ = {$x \in Z : \alpha < x < \beta$}, then $\sum \limits_{x \in X} x^2$ is equal to
   
      
   1. 2109
   2. 2129
   3. 2139
   4. 2119
2. If $\sin x$+$\sin^2x$=1, $x \in \left(0, \frac{\pi}{2}\right)$, then $(cos^{12}x + tan^{12} x)$ + $3(cos^{10}x + tan^{10}x + cos^{8}x + tan^{8}x)$+ $(cos^6x + tan^6x)$ is equal to
   
      
   1. 4
   2. 3
   3. 2
   4. 1
3. Let the area enclosed between the curves $|y|$ = $1 – x^2$and $x^2 + y^2$= 1 be $\alpha$. If $9\alpha$ = $\beta \pi$ + $\gamma$; $\beta$, $\gamma$ are integers, then the value of $|\beta-\gamma|$ equals
   
      
   1. 27
   2. 18
   3. 15
   4. 33
4. If the domain of the function $\log_5(18x-x^2-77)$ is $(\alpha, \beta)$ and the domain of the function $\log_(x-1)\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :
   
      
   1. 195
   2. 174
   3. 186
   4. 179
5. Let the function $f(x)$ = $(x^2– 1)|x^2– ax + 2| + \cos|x|$ be not differentiable at the two points $x = \alpha = 2$and $x = \beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12x + 5y + 10$ = 0 is equal to :
   
      
   1. 3
   2. 4
   3. 2
   4. 5
6. Let a straight line $L$ pass through the point $P(2,–1,3)$ and be perpendicular to the lines $\frac{x-1}{2}$=$\frac{y+1}{1}$=$\frac{z-3}{-2}$ and $\frac{x-3}{1}$=$\frac{y-2}{3}$=$\frac{z+2}{4}$. If the line $L$ intersects the $yz-$plane at the point $Q$, then the distance between the points $P$ and $Q$ is :
   
      
   1. 2
   2. $\sqrt{10}$
   3. 3
   4. $\sqrt{3}$
7. Let $S = N \cup$ {0}. Define a relation $R$ from $S$ to $R$ by :
   $R$=$\left\{(x, y):\log_e y=x\log_e\left(\frac{2}{5}\right), x \in S, y \in R \right\}$
   Then, the sum of all the elements in the range of $R$is equal to
   
      
   1. $\frac{3}{2}$
   2. $\frac{5}{3}$
   3. $\frac{10}{9}$
   4. $\frac{5}{2}$
8. Let the line $x + y$ = 1 meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle $AMN$is inscribed in the triangle $OAB$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$, respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$of the area of the triangle $OAB$ and $AN : NB$ = $\lambda : 1$, then the sum of all possible value(s) of is $\lambda$:
   
      
   1. $\frac{1}{2}$
   2. $\frac{13}{6}$
   3. $\frac{5}{2}$
   4. 2
9. If $\alpha x$+$\beta y$=109 is the equation of the chord of the ellipse $\frac{x^2}{9}$+$\frac{y^2}{4}$=1, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha$+$\beta$ is equal to
   
      
   1. 37
   2. 46
   3. 58
   4. 72
10. If all the words with or without meaning made using all the letters of the word “KANPUR” are arranged as in a dictionary, then the word at $440^{th}$ position in this arrangement, is :
    
       
    1. PRNAKU
    2. PRKANU
    3. PRKAUN
    4. PRNAUK
11. Let $\alpha, \beta (\alpha \neq \beta)$ be the values of m, for which the equations $x + y + z$ = 1; $x + 2y + 4z$ = $m$ and $x + 4y + 10z$ = $m^2$ have infinitely many solutions. Then the value of $\sum \limits_{n=1}^{10}(n^{\alpha}+n^{\beta})$ is equal to:
    
       
    1. 440
    2. 3080
    3. 3410
    4. 560
12. Let $A = [a_{ij}]$ be a matrix of order 3 × 3, with $a_{ij}$=$(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha+\beta \sqrt{2}$, $\alpha, \beta \in Z$, then $\alpha+\beta$ is equal to
    
       
    1. 280
    2. 168
    3. 210
    4. 224
13. Let $P$ be the foot of the perpendicular from the point (1, 2, 2) on the line $L$ : $\frac{x-1}{1}$=$\frac{y+1}{-1}$=$\frac{z-2}{2}$. Let the line $\vec{r}$=$(-\hat{i}+\hat{j}-2\hat{k})$+$\lambda (\hat{i}-\hat{j}+\hat{k})$, $\lambda \in R$, intersect the line $L$ at $Q$. Then $2(PQ)^2$is equal to:
    
       
    1. 27
    2. 25
    3. 29
    4. 19
14. Let a circle $C$ pass through the points (4, 2) and (0, 2), and its centre lie on $3x + 2y$ + 2 = 0. Then the length of the chord, of the circle $C$, whose mid-point is (1, 2), is:
    
       
    1. $\sqrt{3}$
    2. $2\sqrt{3}$
    3. $4\sqrt{2}$
    4. $2\sqrt{2}$
15. Let $A$ = $[a_{ij}]$ be a 2 × 2 matrix such that $a_{ij} \in {0, 1}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:
    
       
    1. $\frac{1}{4}$
    2. $\frac{3}{8}$
    3. $\frac{5}{8}$
    4. $\frac{3}{4}$
16. Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/45, then $n$ is equal to:
    
       
    1. 3
    2. 4
    3. 5
    4. 6
17. The remainder, when $7^{103}$ is divided by 23, is equal to:
    
       
    1. 14
    2. 9
    3. 17
    4. 6
18. Let $f(x)$=$\int \limits_{0}^{x}t(t^2-9t+20)dt$, $1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha + \beta)$ equals:
    
       
    1. 157
    2. 253
    3. 125
    4. 154
19.  Let $\hat{a}$ be a unit vector perpendicular to the vectors $\vec{b}$=$\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}$=$2\hat{i}+3\hat{j}-\hat{k}$, and makes an angle of $\cos^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi}{3}$ with the vector $\hat{i}+\alpha \hat{j}$+$\hat{k}$, then the value of $\alpha$ is:
    
       
    1. $-\sqrt{3}$
    2. $\sqrt{6}$
    3. $-\sqrt{6}$
    4. $\sqrt{3}$
20. If for the solution curve $y = f(x)$ of the differential equation $\frac{dy}{dx}$+$(\tan x)y$=$\frac{2+\sec x}{(1+2\sec x)^2}$, $x \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$, $f\left(\frac{\pi}{3}\right)$=$\frac{\sqrt{3}}{10}$, then $f\left(\frac{\pi}{4}\right)$ is equal to:
    
       
    1. $\frac{9\sqrt{3}+3}{10(4+\sqrt{3})}$
    2. $\frac{\sqrt{3}+1}{10(4+\sqrt{3})}$
    3. $\frac{5-\sqrt{3}}{2\sqrt{2}}$
    4. $\frac{4-\sqrt{2}}{14}$

SECTION - B
(Numerical Answer Type)

This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value.

1. If $24\int \limits_{0}^{\frac{\pi}{4}}\left(\sin \left|4x-\frac{\pi}{12}\right|+2[\sin x]\right)dx$=$2\pi+\alpha$ where $[•]$ denotes the greatest integer function, then $\alpha$ is equal to _________.
2. If $\lim \limits_{t \to 0}\left(\int \limits_0^1(3x+5)^tdx\right)^{\frac{1}{t}}$ =$\frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then $\alpha$ is equal to _______.
3. Let $a_1$, $a_2$, …, $a_{2024}$ be an Arithmetic Progression such that $a_1$ + $(a_5 + a_{10} + a_{15} + … + a_{2020})$ + $a_{2024}$ = 2233. Then $a_1$+ $a_2$+ $a_3$ + … + $a_{2024}$ is equal to ________.
4. Let integers $a, b \in [–3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $\left|\frac{z-a}{z+b}\right|$=1 and $\begin{equation*}\begin{vmatrix}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix}\end{equation*}$ = 1, $z \in C$, where $\omega$ and $\omega^2$are the roots of $x^2+ x + 1$ = 0, is equal to _______.
5. Let $y^2 = 12x$ the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)$ = $\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2$+ $64y^2$– $\alpha x$ – $64 \sqrt{3} y$ = $\beta$, then $\beta – \alpha$ is equal to _________.

29 Jan 2025:Shift-1

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