Download JEE Main 2025 Question Paper (24 Jan - Shift 1) 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. Let $\vec{a}$=$\hat{i}$+$2\hat{j}$+$3\hat{k}$, $\vec{b}$=$3\hat{i}$+$\hat{j}$$-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a}•\vec{c}$=5, then $|\vec{c}|$ is equal to
   
      
   1. $\frac{1}{3\sqrt{2}}$
   2. 18
   3. 16
   4. $\sqrt{\frac{11}{6}}$
2. In $I(m, n)$=$\int \limits_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, $m, n$ > 0, then $I(9, 14) + I(10, 13)$ is
   
      
   1. $I(9, 1)$
   2. $I(19, 27)$
   3. $I(1, 13)$
   4. $I(9, 13)$
3. Let $f : R – {0} \to R$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)$=$\frac{35}{3x}$$-\frac{5}{2}$. If the $\lim \limits_{x \to 0}\left(\frac{1}{\alpha x}+f(x)\right)$=$\beta$;  $\alpha$, $\beta \in R$, then $\alpha+2\beta$ is equal to
   
      
   1. 3
   2. 5
   3. 4
   4. 6
4. Let $S_n$=$\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{12}$+$\frac{1}{20}$+......upto $n$ terms. If the sum of the first six terms of an A.P. with first term $–p$ and common difference $p$ is $\sqrt{2026S_{2025}}$, then the absolute difference between $20^{th}$ and $15^{th}$ terms of the A.P. is
   
      
   1. 25
   2. 90
   3. 20
   4. 45
5. Let $f(x)$=$\frac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac{1}{15}\right)+f\left(\frac{2}{15}\right)+...+f\left(\frac{59}{15}\right)\right)$ is equal to
   
      
   1. 118
   2. 92
   3. 102
   4. 108
6. If $\alpha$ and $\beta$ are the roots of the equation $2z^2$$-3z$$-2i$=0, where $i$=$\sqrt{-1}$, then 16•$Re\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$•$Im\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to........
   
      
   1. 398
   2. 312
   3. 409
   4. 441
7. $\lim \limits_{x \to 0} cosec x(\sqrt{2\cos^2x+3\cos x}$$-\sqrt{\cos^2x+\sin x+4})$ is
   
      
   1. 0
   2. $\frac{1}{2\sqrt{5}}$
   3. $\frac{1}{\sqrt{15}}$
   4. $-\frac{1}{2\sqrt{5}}$
8. Let in a $\Delta ABC$, the length of the side $AC$ be 6, the vertex $B$ be (1, 2, 3) and the vertices $A$, $C$ lie on the line $\frac{x-6}{3}$=$\frac{y-7}{2}$=$\frac{z-7}{-2}$. Then the area (in sq. units) of $\Delta ABC$ is
   
      
   1. 42
   2. 21
   3. 56
   4. 17
9. Let $y = y(x)$ be the solution of the differential equation $(xy-5x^2\sqrt{1+x^2})dx$+$(1+x^2)dy$=0, $y(0)$=0. Then $y(\sqrt{3})$ is equal to
   
      
   1. $\frac{5\sqrt{3}}{2}$
   2. $\sqrt{\frac{14}{3}}$
   3. $2\sqrt{2}$
   4. $\sqrt{\frac{15}{2}}$
10. Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is
    
       
    1. $\frac{3-2\sqrt{2}}{3\sqrt{2}}$
    2. $\frac{1-\sqrt{2}}{\sqrt{2}}$
    3. $\frac{3-2\sqrt{2}}{2\sqrt{3}}$
    4. $\frac{1-2\sqrt{2}}{\sqrt{3}}$
11. A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is
    
       
    1. $\frac{9}{17}$
    2. $\frac{9}{19}$
    3. $\frac{8}{17}$
    4. $\frac{8}{19}$
12. Consider the region
    $R$=$\left\{(x,y):x \leq y \leq 9-\frac{11}{3}x^2, x \geq 0 \right \}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is :
    
       
    1. $\frac{625}{111}$
    2. $\frac{730}{119}$
    3. $\frac{567}{121}$
    4. $\frac{821}{123}$
13. The area of the region {$(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|$} is equal to
    
       
    1. 7
    2. 24/5
    3. 20/3
    4. 5
14. For a statistical data $x_1$, $x_2$, …, $x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum \limits_{i=1}^{10}x_i^2$=371. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
    
       
    1. 7
    2. 4
    3. 9
    4. 5
15. Let circle $C$ be the image of $x^2$ + $y^2$– $2x + 4y$ – 4 = 0 in the line $2x – 3y$ + 5 = 0 and $A$ be the point on $C$ such that $OA$ is parallel to $x-$axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta-\sqrt{3}\alpha$ is equal to
    
       
    1. 3
    2. $3+\sqrt{3}$
    3. $4-\sqrt{3}$
    4. 4
16. For some $n \neq 10$, let the coefficients of the $5^{th}$, $6^{th}$ and $7^{th}$ terms in the binomial expansion of $(1 + x)^{n+4}$be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is :
    
       
    1. 70
    2. 35
    3. 20
    4. 10
17. The product of all the rational roots of the equation $(x^2 – 9x + 11)^2$$– (x – 4) (x – 5)$ = 3, is equal to :
    
       
    1. 14
    2. 7
    3. 28
    4. 21
18. Let the line passing through the points (–1, 2, 1) and parallel to the line $\frac{x-1}{2}$=$\frac{y+1}{3}$=$\frac{z}{4}$ intersect the line $\frac{x+2}{3}$=$\frac{y-3}{2}$=$\frac{z-1}{4}$ at the point $P$. Then the distance of $P$ from the point $Q(4, – 5, 1)$ is :
    
       
    1. 5
    2. 10
    3. $5\sqrt{6}$
    4. $5\sqrt{5}$
19.  Let the lines $3x – 4y – \alpha$ = 0, $8x – 11y – 33$ = 0, and $2x – 3y + \lambda$= 0 be concurrent. If the image of the point (1, 2) in the line $2x – 3y + \lambda$ = 0 is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha \lambda|$ is equal to :
    
       
    1. 84
    2. 91
    3. 113
    4. 101
20. If the system of equations
    $2x – y + z$= 4
    $5x + \lambda y + 3z$ = 12
    $100 x – 47 y + µz$ = 212,
    has infinitely many solutions, then $µ – 2\lambda$ is equal to
    
    1. 56
    2. 59
    3. 55
       
    4. 57

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 $f$ be a differentiable function such that $2(x+2}^2f(x)$$-3(x+2)^2$=10$\int \limits_{0}^{x}(t+2)f(t)dt$, $x \geq 0$. Then $f(2)$ is equal to _____.
2. If for some $\alpha$, $\beta$; $\alpha \leq \beta$, $\alpha + \beta$ = 8 and $\sec^2(tan^{–1}\alpha)$ + $cosec^2(cot^{–1}\beta)$ = 36, then $\alpha^2 + \beta$ is ______.
3. The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is
4. Let be a 3 × 3 matrix such that $X^TAX$ = O for all nonzero 3 × 1 matrices $X$ = $\begin{equation*} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{equation*}$. If $\begin{equation*}A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=\begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}\end{equation*}$, $\begin{equation*} A\begin{bmatrix} 1 \\ 2 \\ 1\end{bmatrix}=\begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \end{equation*}$ and $det (adj(2(A + I)))$ = $2^{\alpha}3^{\beta}5^{\gamma}$, $\alpha$, $\beta$, $\gamma \in N$, then $\alpha^2$+$\beta^2$+$\gamma^2$ is
5. Let $S$ = {$p_1$, $p_2$……, $p_{10}$} be the set of first ten prime numbers. Let $A$ = $S \cup P$, where $P$ is the set of all possible products of distinct element of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \in A$, such that $x$ divides $y$ , is ______.

24 Jan 2025:Shift-2

24 Jan 2025:Ans-Keys

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