Download JEE Main 2025 Question Paper (07 Apr - Shift 2) Mathematics

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08 Apr 2025:Shift-2

02 Apr 2025:Shift-1

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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 (01 – 05) 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 orthocentre of the triangle formed by the lines $y = x + 1$, $y = 4x – 8$ and $y = mx + c$ is at (3, –1), then $m – c$ is :
   
      
   1. 0
   2. -2
   3. 4
   4. 2
2. Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}$=$\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$ is
   
      
   1. $2+4\sqrt{2}$
   2. $1+\sqrt{2}$
   3. $2+\sqrt{2}$
   4. $4+2\sqrt{2}$
3. Let
   $A$ = {$(\alpha, \beta) \in R × R$ : $|\alpha – 1| \leq 4$ and $|\beta – 5| \leq 6$}
   and
   $B$ = {$(\alpha, \beta) \in R × R$ : $16(\alpha – 2)^2$ + $9(\beta – 6)^2\leq 144$}.
   Then
   
      
   1. $B \subset A$
   2. $A \cup B$ = {$(x, y) : – 4 \leq x \leq 4$, $–1 \leq y \leq 11$}
   3. neither $A \subset B$ nor $B \subset A$
   4. $A \subset B$
4. If the range of the function $f(x)$=$\frac{5-x}{x^2-3x+2}$, $x \neq 1, 2$, is $(-\infty, \alpha] \cup [\beta, \infty)$, then $\alpha^2+\beta^2$ is equal to
   
      
   1. 190
   2. 192
   3. 188
   4. 194
5. A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n}$, $gcd(m, n)$ = 1, then $n^2– m^2$is equal to :
   
      
   1. 80
   2. 60
   3. 72
   4. 64
6. Let a random variable $X$ take values 0, 1, 2, 3 with $P(X = 0)$ = $P(X = 1)$ = $p$, $P(X = 2)$ = $P(X = 3)$ and $E(X^2)$ = $2E(X)$. Then the value of $8p – 1$ is :
   
      
   1. 0
   2. 2
   3. 1
   4. 3
7. If the area of the region {$(x, y)$ : $1 + x^2\leq y \leq min \left\{x + 7, 11– 3x\right\}$} is $A$, then $3A$ is equal to
   
      
   1. 50
   2. 49
   3. 46
   4. 47
8. Let $ƒ : R \to R$ be a polynomial function of degree four having extreme values at $x = 4$ and $x = 5$. If $\lim \limits_{x \to 0}\frac{f(x)}{x^2}$=5, then $f(2)$ is equal to:
   
      
   1. 12
   2. 10
   3. 8
   4. 14
9. The number of solutions of the equation $\cos 2\theta \cos \frac{\theta}{2}$+$\cos \frac{5\theta}{2}$=$2\cos^3\frac{5\theta}{2}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is
   
      
   1. 7
   2. 5
   3. 6
   4. 9
10. Let $a_n$ be the $n^{th}$ term of an A. P. If $S_n$ = $a_1$ + $a_2$ + $a_3$ +….+ $a_n$ = 700, $a_6$ = 7 and $S_7$ = 7, then $a_n$ is equal to :
    
       
    1. 56
    2. 65
    3. 64
    4. 70
11. If the locus of $z \in C$, such that $Re\left(\frac{z-1}{2z+i}\right)$+$Re\left(\frac{\bar{z}-1}{2\bar{z}-i}\right)$=2, is a circle of radius $r$ and center $(a, b)$ then $\frac{15ab}{r^2}$ is equal to:
    
       
    1. 24
    2. 12
    3. 18
    4. 16
12. Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 be 10. If its eccentricity is the minimum value of the function $f(t)$ = $t^2$ + $t$ +$\frac{11}{12}$, $t \in R$, then $a^2+b^2$ is equal to:
    
       
    1. 125
    2. 126
    3. 120
    4. 115
13.  Let $y = y(x)$ be the solution of the differential equation $(x^2+1)y'$$–2xy$ = $(x^4 + 2x^2 + 1) \cos x$, $y(0)$ = 1. Then $\int \limits_{-1}^{3}y(x)dx$ is:
    
       
    1. 24
    2. 36
    3. 30
    4. 18
14. If the equation of the line passing through the point $\left(0, -\frac{1}{2}, 0\right)$ and perpendicular to the lines
    $\vec{r}$=$\lambda (\hat{i}+a\hat{j}+b\hat{k})$ and
    $\vec{r}$=$(\hat{i}-\hat{j}-6\hat{k})$+$\mu(-b\hat{i}+a\hat{j}+5\hat{k})$
    is $\frac{x-1}{2}$=$\frac{y+4}{d}$=$\frac{z-c}{-4}$, then $a + b+ c+ d$ is equal to:
    
       
    1. 10
    2. 14
    3. 13
    4. 12
15. Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p + q$ = 126, then the eccentricity of the ellipse $\frac{x^2}{16}$+$\frac{y^2}{n}$=1 is :
    
       
    1. $\frac{3}{4}$
    2. $\frac{1}{2}$
    3. $\frac{\sqrt{7}}{4}$
    4. $\frac{1}{\sqrt{2}}$
16. Consider the lines $L_1$ : $x –1$ = $y–2$ = $z$ and $L_2$ : $x –2$ =$y$=$z–1$. Let the feet of the perpendiculars from the point $P(5,1, –3)$ on the lines $L_1$ and $L_2$ be $Q$ and $R$ respectively. If the area of the triangle $PQR$ is $A$, then $4A^2$is equal to :
    
       
    1. 139
    2. 147
    3. 151
    4. 143
17. The number of real roots of the equation $x|x-2|$+3|x-3|+1=0 is :
    
       
    1. 4
    2. 2
    3. 1
    4. 3
18. Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2}$+$\frac{y^2}{25}$=1 and the hyperbola $\frac{x^2}{16}$+$\frac{y^2}{25}$=1 and the hyperbola $\frac{x^2}{16}-\frac{y^2}{b^2}$=1, respectively. If $b < 5$ and $e_1e_2$ =1, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :
    
       
    1. $\frac{4}{5}$
    2. $\frac{3}{5}$
    3. $\frac{\sqrt{7}}{4}$
    4. $\frac{\sqrt{3}}{2}$
19.  Let the system of equations
    $x$ + $5y –z$ = 1
    $4x$ + $3y –3z$ = 7
    $24x$ + $y + \lambda z$ = $\mu$ $\lambda$, $\mu \in R$, have infinitely many solutions. Then the number of the solutions of this system,
    If $x$, $y$, $z$ are integers and satisfy $7 \leq x+y+z \leq 77$, is
    
       
    1. 3
    2. 6
    3. 5
    4. 4
20. If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :
    
       
    1. 760
    2. 755
    3. 750
    4. 757

SECTION - B
(Numerical Value Type Questions)

This section contains 05 Numerical based questions. The answer to each question is rounded off to the nearest integer.

1. If the function $f(x)$=$\frac{\tan (\tan x)-\sin (\sin x)}{\tan x - \sin x}$ is continuous at $x = 0$, then $f(0)$ is equal to _____.
2. If $\int \left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt{23}{3x^{-24}+x^{-26}}\right)dx$=$-\frac{\alpha}{3(\alpha+1)}$$(3x^{\beta} +x^{\gamma})^{\frac{\alpha+1}{\alpha}}+C$, $x > 0$, $(\alpha, \beta, \gamma \in Z)$, where $C$ is the constant of integration, then $\alpha$ + $\beta$ + $\gamma$ is equal to_____.
3. For $t > –1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left((t+2)^{\frac{1}{7}}-1\right)x^2$+$\left((t+2)^{\frac{1}{6}}-1\right)x$+$\left((t+2)^{\frac{1}{21}}-1\right)$=0. If $\lim \limits_{t \to 1^+} \alpha_t=a$ and $\lim \limits_{t \to 1^+} \beta_t$=$b$, then $72(a+b)^2$ is equal to........
4. Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$, respectively, and one focus and the corresponding directrix of this hyperbola be (–5,0) and $5x + 9$ = 0, respectively. If the product of the focal distances of a point $(\alpha, 2\sqrt{5})$ on the hyperbola is $p$, then $4p$ is equal to ____.
5. The sum of the series 2 × 1 × ${}^{20}C_4$ – 3 × 2 × ${}^{20}C_5$ + 4 × 3 × ${}^{20}C_6$ – 5 × 4 × ${}^{20}C_7$ +…+18 × 17 × ${}^{20}C_{20}$, is equal to

07 Apr 2025:Shift-1

07 Apr 2025:Ans-Keys

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