Download JEE Main 2026 Question Paper (23 Jan - Shift 2) Mathematics

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24 Jan 2023:Shift-1

24 Jan 2026: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 (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. The system of linear equations
   $x + y + z$ = 6
   $2x + 5y + az$ = 36
   $x + 2y + 3z = b$
   has
    
   1. Infinitely many solutions for $a = 8$ & $b = 14$
   2. Infinitely many solutions for $a = 8$ & $b = 16$
   3. Unique solution for $a = 8$ & $b = 16$
   4. Unique solution for $a = 8$ & $b = 14$
2. If the mean and the variance of the data
    

   Class	4-8	8-12	12-16	16-20
   Frequency	3	$\lambda$	4	7

   
   
   
   are $\mu$ & 19 respectively, then the value of $(\lambda+\mu)$ is
    
   1. 21
   2. 18
   3. 19
   4. 20
3. Let $I(x)$=$\int \frac{3dx}{(4x+6)\sqrt{4x^2+8x+3}}$ and $I(0)=\frac{\sqrt{3}}{4}+20$. If $I\left(\frac{1}{2}\right)$=$\frac{a\sqrt{2}}{b}+c$, where $a, b, c \in N$, $gcd(a, b)$=1, then $a+b+c$ is equal to
    
   1. 29
   2. 28
   3. 30
   4. 31
4. An equilateral triangle $OAB$ is inscribed in the parabola $y = 4x^2$ with the vertex $O$ at the vertex of the parabola. Then the minimum distance of the circle having $AB$ as diameter from the origin is
    
   1. $4(6+\sqrt{3})$
   2. $4(3-\sqrt{3})$
   3. $2(8-3\sqrt{3})$
   4. $2(3+\sqrt{3})$
5. The sum of all the real solutions of the equation
   $\log_{x+3}(6x^2+28x+30)$=$5-2\log_{6x+10}(x^2+6x+9)$, are:
    
   1. 1
   2. 0
   3. 2
   4. 4
6. The least value of $\cos^2\theta$-$6\sin \theta \cos \theta$+$3\sin^2\theta$+2 is:
    
   1. -1
   2. $4+\sqrt{10}$
   3. $4-\sqrt{10}$
   4. 1
7. Let $A$ = {1, 2, 3, …, 9}. Let $R$ be a relation on $A$ defined by $x R y$, if and only if $|(x – y)|$ is multiple of 3.
   $S_1$: Number of elements in $R$ is 36 $S_2$: $R$ is an equivalence relation.
    
   1. $S_1$ & $S_2$ both correct
   2. $S_1$ & $S_2$ both incorrect
   3. $S_2$ is correct, but $S_1$ is not correct
   4. $S_1$ is correct, but $S_2$ is not correct
8. The area of the region enclosed between the circles $x^2 + y^2= 4$ & $x^2+ (y–2)^2$ = 4 is
    
   1. $\frac{4}{3}(2\pi-3\sqrt{3})$ Sq. units
   2. $\frac{2}{3}(4\pi-3\sqrt{3})$ Sq. units
   3. $\frac{4}{3}(2\pi-\sqrt{3})$ Sq. units
   4. $\frac{2}{3}(2\pi-3\sqrt{3})$ Sq. units
9. Bag A contains 9 White & 8 Black balls and bag B contains 6 White & 4 Black balls. A ball is randomly picked up from bag B and mixed up with the balls in the bag $A$. If probability that drawn ball is White, is $\frac{p}{q}$ (where $p$ and $q$ are coprime), then $(p+q)$ is
    
   1. 23
   2. 22
   3. 21
   4. 24
10. If the points of the intersection of the ellipses $x^2+ 2y^2– 6x – 12y$ + 23 = 0 and $4x^2 + 2y^2– 20x – 12y$ + 35 = 0 lie on a circle of radius r and centre $(a, b)$, then the value of $ab + 18r^2$ is
     
    1. 53
    2. 51
    3. 55
    4. 52
11. If $f(x)$=$\left\{ \begin{array} {cc} \frac{a|x|+2x^2-2\sin|x|\cos|x|}{x}; x \neq 0 \\ b; x=0 \end{array} \right.$
    is continuous at $x=0$, then $ a + b$ is equal to
     
    1. 0
    2. 1
    3. 4
    4. 2
12. Let $\vec{a}$, $\vec{b}$, $\vec{c}$ are three vectors such that $\vec{a}×\vec{b}$=$2(\vec{a}×\vec{c})$. If $|\vec{a}|$=1, $|\vec{b}|=4$, $|\vec{c}|=2$ and angle between $\vec{b}$ and $\vec{c}$ is 60°, then $|\vec{a}•\vec{c}|$ is:
     
    1. 4
    2. 2
    3. 0
    4. 1
13. Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{v}$=$\vec{a}×\vec{b}$. If $\vec{v}•\vec{c}$=11 and the length of the projection of $\vec{b}$ on $\vec{c}$ is $p$, then $9p^2$ is equal to
     
    1. $12$
    2. $4$
    3. $9$
    4. $6$
14. Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ perpendicular to the $x-$ axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of $\Delta OPQ$ is
     
    1. $\frac{11}{5}$
    2. $\frac{9}{5}$
    3. $\frac{8}{5}\sqrt{3}$
    4. $2\sqrt{3}$
15. Let $\frac{\pi}{2} < \theta < \pi$ and $\cot \theta$=$-\frac{1}{2\sqrt{2}}$. Then value of $\sin \left(\frac{15\theta}{2}\right)(\sin 8\theta+\cos 8\theta)$+$\cos \left(\frac{15\theta}{2}\right)(\cos 8\theta-\sin 8\theta)$ is:
     
    1. $\frac{\sqrt{2}-1}{\sqrt{3}}$
    2. $-\frac{\sqrt{2}}{\sqrt{3}}$
    3. $\frac{1-\sqrt{2}}{\sqrt{3}}$
    4. $\frac{\sqrt{2}}{\sqrt{3}}$
16. If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}$, $i=\sqrt{-1}$, then $(z^{201}-1)^8$ is:
     
    1. 0
    2. 256
    3. -1
    4. 1
17. Consider two sets $A$: {$x \in Z: |(|x-3|)-3| \leq 1$} and
    set $B$: $\left\{x \in R - {1, 2}: \right.$ $\left. \frac{(x-2)(x-4)}{(x-1)} \log_e(|x-2|)=0\right\}$.
    Then the number of onto functions $f: A \to B$ is equal to.
     
    1. 32
    2. 62
    3. 81
    4. 79
18. Let $A(1,2)$, $C(–3,–6)$ be two diagonally opposite vertices of a rhombus whose sides $AD$ & $BC$ are parallel to the line $7x – y = 14$. If $B(\alpha, \beta)$ & $D(\gamma, \delta)$ are the other two vertices, then $|\alpha+ \beta + \gamma + \delta|$ is equal to
     
    1. 6
    2. 1
    3. 9
    4. 3
19. Let $\sum \limits_{k=1}^{n}a_k=\alpha n^2+\beta n$ & $a_{10}=59$ and $a_6$=$7a_1$, then $\alpha+\beta$ is equal to:
     
    1. 3
    2. 5
    3. 7
    4. 12
20. The number of ways, in which 16 oranges can be distributed to 4 children such that each child gets atleast one oranges is
     
    1. 403
    2. 384
    3. 429
    4. 455

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. Let $S$ denote the set of 4 digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the number of 5 digit numbers having product of its digits is equal to 20, then $n(S) + n(P)$ is equal to..........
2. If the image of the point $P(a, 2, a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is $Q$ and the image of $Q$ in the line $\frac{x-2b}{2}$=$\frac{y-a}{1}$=$\frac{z+2b}{-5}$ is P, then $a+b$ is equal to................
3. Let $A$=$\begin{equation*} \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \end{equation*}$ and $B$ be a matrix such that $B(I-A)=I+A$. Then the sum of the diagonal elements of $B^TB$ is equal to.......
4. The number of elements in the set $S$=$\left\{x: x \in [0, 100]\right.$ and $\left. \int \limits_0^x t^2 \sin(x-t)dt=x^2\right\}$ is..............
5. If the solution curve $y = f(x)$ of the differential equation $(x^2– 4)y' – 2xy + 2x(4 – x^2)^2$ = 0, $x > 2$, passes through the point $(3, 15)$ then the local maximum value of $f$ is

23 Jan 2026:Shift-1

23 Jan 2026:Ans-Keys

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