Download JEE Main 2026 Question Paper (21 Jan - Shift 2) Mathematics

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22 Jan 2026:Shift-1

22 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. Let $A$={2, 3, 5, 7, 9}. Let $R$ be the relation on $A$ defined by $x R y$ if and only if $2x \leq 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l+m$ is equal to:
    
   1. 23
   2. 21
   3. 25
   4. 27
2. Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $34\left|\frac{5z-12}{5iz+16}\right|^2$ is equal to
    
   1. 12
   2. 16
   3. 26
   4. 20
3. Let $\alpha$ and $\beta$ be the roots of the equation $x^2+2ax+(3a+10)$=0 such that $\alpha < 1 < \beta$. Then the set of all possible values of $\alpha$ is
    
   1. $\left(-\infty, \frac{-11}{5}\right) \cup (5, \infty)$
   2. $(-\infty, -3)$
   3. $(-\infty, -2) \cup (5, \infty)$
   4. $\left(-\infty, \frac{-11}{5}\right)$
4. If the line $ax+4y=\sqrt{7}$, where $a \in R$, touches the ellipse $3x^2+4y^2=1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is:
    
   1. $\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{5}}$
   2. $\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{7}}$
   3. $\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{5}}$
   4. $\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{11}}$
5. Let $A$={$x: |x^2-10|\leq 6$} and $B$={$x: |x-2| > 1$}. Then
    
   1. $A-B=[2, 3)$
   2. $A \cap B$=$[-4, -2] \cup [3, 4]$
   3. $B-A$=$(-\infty, -4) \cup (-2, 1) \cup (4, \infty)$
   4. $A \cup B$=$(-\infty, 1] \cup (2, \infty)$
6. Let $f(x)=x^3$+$x^2f'(1)$+$2xf''(2)$+$f'''(3)$, $x \in R$. Then the value of $f'(5)$=
    
   1. $\frac{62}{5}$
   2. $\frac{657}{5}$
   3. $\frac{2}{5}$
   4. $\frac{117}{5}$
7. Let the line $L_1$ be parallel to the vector $-3\hat{i}+2\hat{j}+4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i}+\hat{j}+3\hat{k}$ and pass through the point $(4, 3, 5)$. If the line  $L_3$ is parallel to the vector $-3\hat{i}+5\hat{j}+16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then $|\vec{CD}|^2$ is equal to:
    
   1. 290
   2. 89
   3. 312
   4. 171
8. Let $y=y(x)$ be the solution of the differential equation $\sec x\frac{dy}{dx} -2y$=$2+3\sin x$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0)=-\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :
    
   1. $-\frac{5}{2}$
   2. $-\frac{5}{4}$
   3. $-3\sqrt{2}-7$
   4. $-3\sqrt{3}-7$
9. If the area of the region {$(x, y): 1-2x \leq y \leq 4-x^2$, $x \geq 0, y \geq 0$} is $\frac{\alpha}{\beta}$, $\alpha, \beta \in N$, $gcd (\alpha, \beta)$=1, then the value of $(\alpha+\beta)$ is:
    
   1. 67
   2. 85
   3. 91
   4. 73
10. Let $a_1$, $\frac{a_2}{2}$, $\frac{a_3}{2^2}$, ...., $\frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1$+$a_2$+...+$a_{10}$=62, then $a_1$ is equal to
     
    1. $2-\sqrt{2}$
    2. $2(2-\sqrt{2})$
    3. $\sqrt{2}-1$
    4. $2(\sqrt{2}-1)$
11. Let $f: R \to R$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in R$ and $f'(a-1)=0$, where $a$ is a real number. Let $g(x)=f(\tan^2x-2\tan x +a)$, $0 < x < \frac{\pi}{2}$. Consider the following two statements:
    (I) $g$ is increasing in $\left(0, \frac{\pi}{4}\right)$ (II) $g$ is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
     
    1. Only (II) is true
    2. Only (I) is true
    3. Both (I) and (II) are true
    4. Neither (I) nor (II) is true.
12. For the matrices $A=\begin{equation*} \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \end{equation*}$ and $B=\begin{equation*} \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} \end{equation*}$, if $(A^{15}+B)\begin{equation*} \begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{equation*}$, then among the following which one is true?
     
    1. $x=16, y=3$
    2. $x=18, y=11$
    3. $x=5, y=7$
    4. $x=11, y=2$
13. Let one end of a focal chord of the parabola $y^2 = 16x$ be (16,16). If $P(\alpha, \beta)$ divides this focal chord internally in ratio 5 : 2 then minimum value of $(\alpha+\beta)$ is
     
    1. 5
    2. 7
    3. 16
    4. 22
14. Let $y^2 = 12x$ be the parabola with its vertex at $O$. Let $P$ be a point on the parabola and $A$ be a point on the $x-axis$ such that $\angle{OPA}$ =90°. Then the locus of the centroid of such triangle $OPA$ is
     
    1. $y^2-4x+8=0$
    2. $y^2-6x+4=0$
    3. $y^2-9x+6=0$
    4. $y^2-2x+8=0$
15. The positive integer $n$, for which the solutions of the equation $x(x+2)$+$(x+2)(x+4)$+...+$(x+2n-2)(x+2n)$=$\frac{8n}{3}$ are two consecutive even integers, is:
     
    1. 9
    2. 3
    3. 12
    4. 6
16. A random variable $X$ takes values 0, 1, 2, 3 with probabilities $\frac{2a+1}{30}$, $\frac{8a-1}{30}$, $\frac{4a+1}{30}$, $b$ respectively, where $a, b \in R$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2+mu^2$=2. Then $\frac{a}{b}$ is equal to:
     
    1. 12
    2. 3
    3. 60
    4. 30
17. Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angle with the positive coordinate axes. If distance of $L$ from the point $(-2, r, 1)$ is $\sqrt{\frac{14}{3}}$, then sum of all possible values of $r$ is
     
    1. 16
    2. 12
    3. 10
    4. 6
18. The largest $n \in N$, for which $7^n$ divides 101!, is
     
    1. 15
    2. 19
    3. 16
    4. 18
19. For a triangle $ABC$, let $\vec{p}=\vec{BC}$, $\vec{q}=\vec{CA}$ and $\vec{r}=\vec{BA}$. If $\left|\vec{p}\right|$=$2\sqrt{3}$, $\left|\vec{q}\right|$=2 and $\cos \theta$=$\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p}×(\vec{q}-3\vec{r})|^2$+$3|r^2|$ is equal to
     
    1. 340
    2. 220
    3. 200
    4. 410
20. If the system of equations
    $3x+y+4z=3$
    $2x+\alpha y - z=-3$
    $x+2y+z=4$
    has no solution, then the value of $\alpha$ is equal to:
     
    1. 19
    2. 13
    3. 4
    4. 23

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 the maximum value of $(\sin^{-1}x)^2$+$(\cos^{-1}x)^2$ for $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n}\pi^2$, where $gcd(m, n)$=1. Then $m+n$ is equal to
2. If $\left(\frac{1}{{ }^{15} \mathrm{C}_0}+\frac{1}{{ }^{15} \mathrm{C}_1}\right)$ $\left(\frac{1}{{ }^{15} \mathrm{C}_1}+\frac{1}{{ }^{15} \mathrm{C}_2}\right)$ ... $\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)$=$\frac{\alpha^{13}}{{ }^{14} C_0{ }^{14} C_1 \cdot \cdot{ }^{14} C_{12}}$, then $30 \alpha$ is equal to.....
3. Let [•] represents greatest integer function and $f(x)=\lim \limits_{n \to \infty}\left(\frac{1}{n^3}\sum \limits_{k=1}^{n}\left[\frac{k^2}{3^x}\right]\right)$ then $12 \sum \limits_{j=1}^{\infty}f(j)$ is equal to
4. If $P$ be a point on the circle $x^2+y^2=4$, $Q$ is a point on the straight line $5x+2y+2=0$ and $x-y+1=0$ is the perpendicular bisector of $PQ$, then 13 times the sum of abscissa of all such points $P$ is........
5. If $\int \limits_0^1 4\cot^{-1}(1-2x+4x^2)dx$=$a \tan^{-1}2-$$b \log_e 5$, where $a, b \in N$, then $(2a+b)$ is equal to

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