Download JEE Main 2026 Question Paper (24 Jan - Shift 2) Mathematics

» Hold on briefly for download button.

Wait-starts soon.

 Download Now!

28 Jan 2026:Shift-1

28 Jan 2026:Shift-2

All the Papers

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 largest value of $n$ for which $40^n$divides 60! is
    
   1. 13
   2. 11
   3. 14
   4. 12
2. Consider the following three statements $f:(0, \infty) \to R$ defined by $f(x)=|\log _ex|-|x-1|$
   Statement 1: $f$ is differentiable at all $x > 0$
   Statement 2: $f$ is increasing in $(0, 1)$
   Statement 3: $f$ is decreasing in $(1, \infty)$
   Which of the following is correct ?
    
   1. All the statements are correct
   2. Statement -2 & Statement -3 are correct
   3. Statement -1 & Statement -3 are correct
   4. Only Statement -1 is correct
3. Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order 3 such that $q_{ij}$=$2^{i+j-1}p_{ij}$ and $|Q|=2^{10}$ then $|adj(adj(P))|$ is
    
   1. 81
   2. 16
   3. 32
   4. 124
4. Let $X=\left \{x \in N: 1 \leq x \leq 19\right\}$ and for some $a, b \in R$, $Y=\left \{ax+b: x \in X \right\}$. If the mean and variance of the elements of $Y$ are 30 & 750 respectively, then the sum of all possible values of $b$ is
    
   1. 60
   2. 80
   3. 100
   4. 20
5.   Let the angles made with positive $x-$ axis by two straight lines drawn from the point $P(2, 3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. The value of $(\theta_1+\theta_2)$ is
    
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{2}$
   3. $\frac{\pi}{12}$
   4. $\frac{\pi}{3}$
6. Let $a_1$, $a_2$, $a_3$ , $a_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $a_1$ + $a_2$ + $a_3$ + $a_4$= 48 & $a_1a_2a_3a_4$ + $l^4$= 361, then the largest term of the A.P. is equal to
    
   1. 27
   2. 23
   3. 24
   4. 21
7. The letters of the word ‘UDAYPUR’ are written in all possible ways with or without meaning and these words are arranged as in dictionary. Then the rank of the word ‘UDAYPUR’ is :
    
   1. 1578
   2. 1579
   3. 1580
   4. 1581
8. The sum of all the values of $\alpha$, for which the shortest distance between the lines
   $\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$
   and $\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha}$ is $\sqrt{2}$
   is
    
   1. 6
   2. -6
   3. -8
   4. 8
9. If the domain of the function $f(x)=\sin^{-1}\left(\frac{1}{x^2-2x-2}\right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $(\alpha+\beta+\gamma+\delta)$ is equal to
    
   1. 5
   2. 2
   3. 4
   4. 3
10. Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 $(a > b)$ be 30. If its eccentricity is the maximum value of the function $f(t) =\frac{-3}{4}+2t-t^2$ and length of the latus rectum is 30 then $a^2+b^2$ is equal to
     
    1. 276
    2. 516
    3. 256
    4. 496
11. Let $\vec{a}$=$2 \hat{i}-\hat{j}-\hat{k}$, $\vec{b}$=$\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of the vectors $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\frac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
     
    1. $\frac{\sqrt{35}}{2}$
    2. $\frac{\sqrt{21}}{2}$
    3. 7
    4. 13
12. Let $f$ be a function such that $3f(x)+2f\left(\frac{m}{19x}\right)=5x$, $x \neq 0$, where $m=\sum \limits_{i=1}^{9}i^2$. Then $f(5)-f(2)$ ie equal to
     
    1. 18
    2. 9
    3. -9
    4. 36
13. Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0$, $x=1$, $y^2=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha x^2$, then $f(0)+f(1)$ is equal to
     
    1. 12
    2. 9
    3. 7
    4. 14
14. The smallest positive integral value of $a$, for which all the roots of $x^4– ax^2 + 9$ = 0 are real & distinct, is equal to
     
    1. 3
    2. 9
    3. 7
    4. 4
15. Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ & $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}×\vec{c})$+$3(\vec{b}×\vec{c})$=$\vec{0}$ and $(\vec{a}-\vec{b})•\vec{c}$=-97, then $|\vec{c}×\hat{k}|^2$ is equal to
     
    1. 193
    2. 218
    3. 205
    4. 233
16. Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
    $f(x)=\left\{ \begin{array} {ccc} b^2 \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\sin x+\cos x).\cos x \right]\right); x < 0 \\ \frac{\sin x-\frac{\sin 2x}{2}}{x^3}; x > 0 \\ a; x=0 \end{array} \right.$
    is continuous at $x = 0$, then $(a^2+ b^2)$ is equal to
     
    1. $\frac{3}{4}$
    2. $\frac{1}{2}e$
    3. $\frac{5}{8}e$
    4. $\frac{9}{16}$
17. Let $f(x)=\int \frac{(7x^{10}+9x^8)}{(1+x^2+2x^9)^2}dx$, $x > 0$, $\lim \limits_{x \to 0} f(x)=0$ and $f(1)=\frac{1}{4}$. If $A=\begin{equation*} \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix} \end{equation*}$ and $B=adj(adj A)$ be such that $|B|$=81, then $\alpha^2$ is equal to
     
    1. 1
    2. 2
    3. 3
    4. 4
18. $\left(\frac{1}{3}+\frac{4}{7}\right)$+$\left(\left(\frac{1}{3}\right)^2+\left(\frac{1}{3}\right)\left(\frac{4}{7}\right)+\left(\frac{4}{7}\right)^2\right)$+$\left(\left(\frac{1}{3}\right)^3+\left(\frac{1}{3}\right)^2•\frac{4}{7}+\left(\frac{1}{3}\right)\left(\frac{4}{7}\right)^2+\left(\frac{4}{7}\right)^3\right)$+... upto $\infty$ infinite terms, is equal to
     
    1. $\frac{7}{4}$
    2. $\frac{4}{3}$
    3. $\frac{6}{5}$
    4. $\frac{5}{2}$
19. Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$ and $\lim \limits_{t \to x}\frac{t^2y(x)-x^2y(t)}{x-t}$=3 for each $x > 0$. Then $2y(2)$ is equal to
     
    1. 23
    2. 12
    3. 18
    4. 27
20. Let the image of parabola $x^2= 4y$ in the line $x – y = 1$ be $(y + a)^2 = b(x – c)$, $a, b, c \in N$, then $(a + b + c)$ is equal to:
     
    1. 4
    2. 6
    3. 12
    4. 8

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. The number of elements in the set $\left\{x \in [0, 180°]: \tan (x + 100°)\right.$ = $\left. \tan (x + 50°) \tan x \tan (x – 50°) \right\}$ is.............
2. Let $z = (1 + i) (1 + 2i) (1 + 3i)…..(1 + ni)$, where $i=\sqrt{-1}$. If $|z^2|$ = 44200, then $n$ is equal to .............
3. Let $(h, k)$ lie on the circle $x^2 + y^2 = 4$ & the point $(2h + 1, 3k + 3)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$is equal to ..............
4. If $f(x)$ satisfies the relation $f(x)=e^x+\int \limits_0^1(y+xe^x)f(y)dy$, then $f(0)+e$ is equal to ...............
5. Let $S$ be a set of 5 elements and $P(S)$ denotes the power set of $S$. Let $E$ be event of choosing an ordered pair $(A, B)$ from the set $P(S) × P(S)$ such that $A \cap B=\phi$. If the probability of the event $E$ is $\frac{3^p}{2^q}$, where $p, q \in N$, then $(p+q)$ is equal to ................

24 Jan 2026:Shift-2

24 Jan 2026:Ans-Keys

All the Papers Ans-Keys