Download JEE Main 2025 Question Paper (28 Jan - Shift 1) Mathematics

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29 Jan 2025:Shift-1

29 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. The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is
   
      
   1. 4608
   2. 5720
   3. 5719
   4. 4607
2. Let $ABCD$ be a trapezium whose vertices lie on the parabola $y^2$ = $4x$. Let the sides $AD$ and $BC$ of the trapezium be parallel to $y-$axis. If the diagonal $AC$ is of length $\frac{25}{4}$ and it passes through the point (1, 0), then the area of $ABCD$ is :
   
      
   1. $\frac{75}{4}$
   2. $\frac{25}{2}$
   3. $\frac{125}{8}$
   4. $\frac{75}{8}$
3. Two number $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1}+i^{k_2}$, $(i=\sqrt{-1})$ is non-zero, equals
   
      
   1. $\frac{1}{2}$
   2. $\frac{1}{4}$
   3. $\frac{3}{4}$
   4. $\frac{2}{3}$
4. If $f(x)$=$\frac{2^x}{2^x+\sqrt{2}}$, $x \in R$, then $\sum \limits_{k=1}^{81}f\left(\frac{k}{82}\right)$ is equal to:
   
      
   1. 41
   2. $\frac{81}{2}$
   3. 82
   4. $81\sqrt{2}$
5. Let $f : R \to R$ be a function defined by $f(x)$ = $(2 + 3a)x^2$+$\left(\frac{a+2}{a-1}\right)x+b$, $a \neq 1$. If $f(x + y)$ = $f(x)$ + $f(y)$ + 1 – $\frac{2}{7}xy$, then the value of $28\sum \limits_{i=1}^{5}|f(i)|$ is:
   
      
   1. 715
   2. 735
   3. 545
   4. 675
6. Let $A(x, y, z)$ be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let $B$ = (1, 4, –1) and $C$ = (2, 0, –2). Then among the statements
   (S1) : $\Delta ABC$ is an isosceles right angled triangle and
   (S2) : the area of $\Delta ABC$ is
   
      
   1. both are true
   2. only (S1) is true
   3. only (S2) is true
   4. both are false
7. The relation $R$={$(x, y) : x, y \in z$ and $x + y$ is even} is :
   
      
   1. reflexive and transitive but not symmetric
   2. reflexive and symmetric but not transitive
   3. an equivalence relation
   4. symmetric and transitive but not reflexive
8. Let the equation of the circle, which touches $x-$axis at the point $(a, 0), a > 0$ and cuts off an intercept of length $b$ on $y-$axis be $x^2$+ $y^2$– $\alpha x$ + $\beta y$ + $\gamma$ = 0. If the circle lies below $x-$axis, then the ordered pair $(2a, b^2)$ is equal to :
   
      
   1. $(\alpha, \beta^2+\gamma)$
   2. $(\gamma, \beta^2-4\alpha)$
   3. $(\gamma, \beta^2+4\alpha)$
   4. $(\alpha, \beta^2-\gamma)$
9. Let $< a_n >$ be a sequence such that $a_0$ = 0, $a_1$ = $\frac{1}{2}$ and $2a_{n+2}$= $5a_{n+1}$ – $3a_n$, $n$ = 0, 1, 2, 3, …… Then $\sum \limits_{k=1}^{100}a_k$ is equal to:
   
      
   1. $3a_{99}-100$
   2. $3a_{100}-100$
   3. $3a_{100}+100$
   4. $3a_{99}+100$
10. $\cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right)$ is equal to:
    
       
    1. 1
    2. 0
    3. $\frac{33}{65}$
    4. $\frac{32}{65}$
11. Let $T_r$ be the $r^{th}$ term of an A.P. If for some $m$, $T_m$=$\frac{1}{25}$, $T_{25}$=$\frac{1}{20}$ and $20\sum \limits_{r=1}^{25}T_r$=13, then $5m\sum \limits_{r=m}^{2m}T_r$ is equal to:
    
       
    1. 112
    2. 126
    3. 98
    4. 142
12. If the image of the point (4, 4, 3) in the line
    $\frac{x-1}{2}$=$\frac{y-2}{1}$=$\frac{z-1}{3}$ is $(\alpha, \beta, \gamma)$, then $\alpha$+$\beta$+$\gamma$ is equal to
    
       
    1. 9
    2. 12
    3. 8
    4. 7
13. If $\int \limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{96x^2\cos^2x}{(1+e^x)}dx$=$\pi(\alpha \pi^2+\beta)$, $\alpha$, $\beta \in Z$, then $(\alpha+\beta)^2$ equals :
    
       
    1. 144
    2. 196
    3. 100
    4. 64
14. The sum of all local minimum values of the function $f(x)$=$\left\{ \begin{array} {cc} 1- 2x &, x < -1 \\ \frac{1}{3}(7+2|x|) &, -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5) &, x > 2 \end{array} \right.$
    
       
    1. $\frac{171}{72}$
    2. $\frac{131}{72}$
    3. $\frac{157}{72}$
    4. $\frac{167}{72}$
15. The sum, of the squares of all the roots of the equation $x^2$+ $|2x – 3|$ – 4 = 0, is :
    
       
    1. $3(3-\sqrt{2})$
    2. $6(3-\sqrt{2})$
    3. $6(2-\sqrt{2})$
    4. $3(2-\sqrt{2})$
16. Let for some function $y = f(x)$, $\int \limits_{0}^{x}tf(t)dt$=$x^2f(x)$, $x > 0$ and $f(2)$ = 3. Then $f(6)$ is equal to :
    
       
    1. 1
    2. 2
    3. 6
    4. 3
17. Let ${}^nC_{r–1}$ = 28, ${}^nC_r$ = 56 and ${}^nC_{r+1}$ = 70. Let $A(4\cos t, 4\sin t)$, $B(2\sin t, –2\cos t)$ and $C(3r – n, r^ 2 – n – 1)$be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x – 1)^2$ + $(3y)^2$= $\alpha$, is the locus of the centroid of triangle $ABC$, then $\alpha$ equals :
    
       
    1. 20
    2. 8
    3. 6
    4. 18
18. Let $O$ be the origin, the point $A$ be $z_1$=$\sqrt{3}$+$2\sqrt{2i}$, the point $B(z_2)$ be such that $\sqrt{3}|z_2|$=$|z_1|$ and $arg(z_2)$=$arg(z_1)$+$\frac{\pi}{6}$. Then
    
       
    1. area of triangle $ABO$ is $\frac{11}{\sqrt{3}}$.
    2. $ABO$ is a scalene triangle
    3. area of triangle $ABO$ is $\frac{11}{4}$
    4. ABO is an obtuse angled isosceles triangle
19.  Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of x is :
    
       
    1. 28/75
    2. 14/25
    3. 26/75
    4. 18/25
20. The area (in sq. units) of the region {$(x, y): 0 \leq y \leq2|x| + 1, 0 \leq y \leq x^2+ 1$, $|x| \leq 3$} is
    
    1. $\frac{80}{3}$
    2. $\frac{64}{3}$
    3. $\frac{17}{3}$
       
    4. $\frac{32}{3}$

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 $M$ denote the set of all real matrices of order 3 × 3 and let $S$ = {–3, –2, –1, 1, 2}. Let
   $S_1$ = {$A = [a_{ij}]\in M$ : $A = A^T$and $a_{ij} \in S$, $\forall i, j$} $S_2$ = {$A = [a_{ij}]\in M : A = –A^T$and $a_{ij} \in S$, $\forall i, j$} $S3$ = {$A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33}$ = 0 and $a_{ij} \in S$,$\forall i,j$} If $n(S_1 \cup S_2 \cup S_3)$ = $125 \alpha$, then $\alpha$ equals.
2. If $\alpha$=1+$\sum \limits_{r=1}^{6}(-3)^{r-1}{}^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ form the line $\alpha x – 3y$ + 1 = 0 is ………
3. Let $\vec{a}$=$\hat{i}$+$\hat{j}$+$\hat{k}$, $\vec{b}$=$2\hat{i}$+$2\hat{j}$+$\hat{k}$ and $\vec{d}$=$\vec{a}×\vec{b}$. If $\vec{c}$ is a vector such that $\vec{a}•\vec{c}$=$|\vec{c}|$, $|\vec{c}-2\vec{a}|^2$=8 and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $|10-3\vec{b}•\vec{c}|$+$|\vec{d}×\vec{c}|^2$ is equal to.............
4. Let
   $f(x)$=$\left\{\begin{array}{cc} 3x &, x < 0\\ min{1+x+[x], x+2[x]}, 0 \leq x \leq 2 \\ 5 &, x > 2 \end{array} \right.$
   where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha$ + $\beta$ equals…….
5. Let $E_1$ = $\frac{x^2}{9}+\frac{y^2}{4}$=1 be an ellipse. Ellipses $E_i$'s are constructed such that their centres and eccentricities are same as that of $E_1$, and the length of minor axis of $E_i$ is the length of major axis of $E_{i+1} (i \geq 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi}\left(\sum \limits_{i=1}^{\infty}A_i\right)$, is equal to...........

28 Jan 2025:Shift-2

28 Jan 2025:Ans-Keys

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