Download JEE Main 2026 Question Paper (22 Jan - Shift 2) Mathematics

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23 Jan 2026:Shift-1

23 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. Among the statements
   Statement 1 : If $A(5, –1)$ & $B(–2, 3)$ are two vertices of a triangle whose orthocentre is (0, 0) then its third vertex is (–4, –7).
   Statement 2: If positive numbers $2a$, $b$, $c$ are three consecutive terms of an A.P., then the lines $ax$ + $by$ + $c$ = 0 are concurrent at point (2,–2).
    
   1. Both statement-1 and statement-2 are correct
   2. Statement-1 is incorrect and statement-2 is correct
   3. Statement-1 is correct and statement-2 is incorrect
   4. Both statement-1 and statement-2 are incorrect
2. Let $n$ be the number obtained on rolling a fair die. If the probability that the system
   $x – ny + z$ = 6
   $x – (n–2)y + (n+1)z$ = 8
   $(n–1)y + z$ = 1
   has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is
    
   1. 21
   2. 24
   3. 20
   4. 22
3. Let the domain of the function
   $f(x)$=$\log_3(\log_5(7-\log_2(x^2-10x+85)))$+$\sin^{-1}\left(\left|\frac{3x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$ then value of $\alpha+\beta$ is equal to :
    
   1. 9
   2. 12
   3. 8
   4. 10
4. Let [•] denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2}x, x^2 \right\}$. Let $S$={$x \in (-2, 2)$: the function $g(x)=|x|\left[x^2 \right]$ is discontinuous at $x$}. Then $\sum \limits_{x \in S}f(x)$ equals
    
   1. $2-\sqrt{2}$
   2. $1-\sqrt{2}$
   3. $2\sqrt{6}-3\sqrt{2}$
   4. $\sqrt{6}-2\sqrt{2}$
5. The mean deviation about the median of the numbers $k$, $2k$, $3k$, …, $1000k$ is 500, then $k^2$ is equal to
    
   1. 4
   2. 16
   3. 1
   4. 9
6. Let $P(10, 2\sqrt{15})$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 whose focii are $S$ and $S'$. If the length of its latus rectum is 8, then the square of the area of $\Delta PS_1S_2$ is equal to
    
   1. 4200
   2. 900
   3. 1462
   4. 2700
7. The area of the region $A$={$(x, y)$: $4x^2+y^2 \leq 8$ and $y^2 \leq 4x$} (in sq. units) is
    
   1. $\frac{\pi}{2}+2$
   2. $\pi+4$
   3. $\pi+\frac{2}{3}$
   4. $\frac{\pi}{2}+\frac{1}{3}$
8. Let l the locus of the mid point of the chord through the origin $O$ of the parabola $y^2 = 4x$ be the curve $S$. Let $P$ be a point on $S$. Then the locus of the point, which internally divides $OP$ in the ratio 3:1 is:
    
   1. $3y^2=2x$
   2. $3x^2=2y$
   3. $2y^2=3x$
   4. $2x^2=3y$
9. If $X=\begin{equation*} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{equation*}$ is a solution of system of equations $AX=B$ where $Adj A=\begin{equation*} \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix} \end{equation*}$ & $B=\begin{equation*} \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix} \end{equation*}$, then $|x+y+z|$ is
    
   1. 1
   2. $\frac{3}{2}$
   3. 3
   4. 2
10. Let $\alpha$, $\beta$ be the roots of the quadratic equation $12x^2-20x+3\lambda$=0, $\lambda \in Z$. If $\frac{1}{2} \leq |\beta-\alpha|\leq \frac{3}{2}$, then the sum of all the possible values of $\lambda$ is:
     
    1. 1
    2. 6
    3. 4
    4. 3
11. Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1+x)^n$, $n \in N$, $0 \leq r \leq n$. If $P_n=C_0-C_1+\frac{2^2}{3}C_2-\frac{2^3}{4}C_3$+...+$\frac{(-2)^n}{(n+1)}C_n$, then find the value of $\sum \limits_{n=1}^{25}P_{2n}$ equals
     
    1. 650
    2. 525
    3. 675
    4. 580
12. The number of elements in the relation $R$ ={$(x, y): 4x^2+y^2$ < 52, $x, y \in Z$} is
     
    1. 67
    2. 89
    3. 86
    4. 77
13. Let $f(x)=[x]^2-[x+3]-3$, $x \in R$, where [•] denotes $G.I.F.$ then
    1. $f(x) > 0$ only for $x \in [4, \infty)$
    2. $f(x) < 0$ only for $x \in [–1, 3)$
    3. $\int \limits_0^2 f(x) dx=-6$
     
    4. $f(x) = 0$ for finitely many values of $x$
14. Let $S$={$z \in C$: $4z^2+\bar{z}$=0}. Then $\sum \limits_{z \in S}^{n}|z|^2$ is equal to
     
    1. $\frac{1}{16}$
    2. $\frac{3}{16}$
    3. $\frac{5}{64}$
    4. $\frac{7}{64}$
15. Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2\hat{k}$, $\lambda \in Z$ be two vectors. Let $\vec{c}$=$\vec{a}×\vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in $yz-$ plane. If $|\vec{c}|$=$\sqrt{53}$, then the maximum possible value of $\left(\vec{c}•\vec{d}\right)$ is equal to:
     
    1. 26
    2. 208
    3. 104
    4. 52
16. If $y=y(x)$ satisfies the differential equation
    $16\left(\sqrt{x+9\sqrt{x}}\right)$$\left(4+\sqrt{9+\sqrt{x}}\right)\cos y dy$=$(1+2\sin y)dx$, $x > 0$ and $y(256)=\frac{\pi}{2}$, $y(49)=\alpha$, then $2\sin \alpha$ is equal to
     
    1. $2\sqrt{2}-1$
    2. $\sqrt{2}-1$
    3. $2\sqrt{2}-1$
    4. $3(\sqrt{2}-1)$
17. If $\lim \limits_{x \to 0} \frac{e^{(a-1)x}+2\cos bx+e^{-x}(c-2)}{x \cos x-\ln (1+x)}$=2, then find $a^2+b^2+c^2$
     
    1. 3
    2. 7
    3. 5
    4. 9
18. Let $f$ and $g$ be functions satisfying $f(x+y)$ = $f(x).f(y)$, $f(1)$ = 7 and $g(x+y)$ = $g(xy)$, $g(1)$ = 1, for all $x, y \in N$. If $\sum \limits_{x=1}^{n}\frac{f(x)}{g(x)}$=19607. Then the find the value of $n$ is
     
    1. 5
    2. 4
    3. 6
    4. 7
19. Let $L$ be the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$ and $S$ be the set of all the points $(a, b, c)$ on $L$, whose the distance from the line $\frac{x+1}{2}=\frac{y+1}{3}$=$\frac{z-9}{0}$ along the line is 7, then $\sum \limits_{(a, b, c) \in S} (a+b+c)$ is equal to:
     
    1. 34
    2. 40
    3. 6
    4. 28
20. Let $S$ and $S'$ be the focii of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}$=1 and $P(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $PS^2+PS'^2-PS.PS'$=37, then the value of $(\alpha^2+\beta^2)$ is
     
    1. 17
    2. 13
    3. 15
    4. 11

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. Suppose $a$, $b$, $c$ are in A.P. and $a^2$, $2b^2$, $c^2$ are in G.P. If $a < b < c$ and $a + b + c$ = 1 then $9(a^2+ b^2+ c^2)$ is equal to.........
2. Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A$={$B \subseteq S: n(B) \geq 2$ and the product of all elements of $B$ is even} is...........
3. Let $\cos(\alpha+\beta)$=$\frac{-1}{10}$ and $\sin(\alpha-\beta)$=$\frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha$=$\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$, $r, s \in N$, then $r+s$ is equal to..........
4. Let  [•] denotes greatest integer function. If $\alpha$=$\int \limits_0^{64}\left(x^{\frac{1}{3}}+\left[x^{\frac{1}{3}}\right]\right]dx$, then $\frac{1}{\pi} \int \limits_0^{\alpha \pi}\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}d\theta$ is equal to..........
5. Let a vector $\vec{a}=\sqrt{2}\hat{i}+\hat{j}+\lambda \hat{k}$, $\lambda > 0$, make an obtuse angle with the vector $\vec{b}=-\lambda^2 \hat{i}-4\sqrt{2}\hat{j}$+$4\sqrt{2}\hat{k}$ and an angle $\theta$, $\frac{\pi}{6} < \theta < \frac{\pi}{2}$, with the positive $z-$ axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta)$$ - \left\{\gamma\right\}$, then the value of $\alpha+\beta+\gamma$ is equal to...........

22 Jan 2026:Shift-1

22 Jan 2026:Ans-Keys

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