Download JEE Main 2026 Question Paper (21 Jan - Shift 1) Mathematics

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22 Jan 2026:Shift-1

22 Jan 2023: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. If the coefficient of $x$ in the expansion of $(ax^2+bx+c)(1-2x)^{26}$ is -56 and  the coefficients of $x^2$ and $x^3$ are both zero, then $a+b+c$ is equal to :
    
   1. 1300
   2. 1500
   3. 1403
   4. 1483
2. If $x^2+x+1$=0, then the value of $\left(x+\frac{1}{x}\right)^4$+$\left(x^2+\frac{1}{x^2}\right)^4$+of $\left(x^3+\frac{1}{x^3}\right)^4$+...+$\left(x^{25}+\frac{1}{x^{25}}\right)^4$ is
    
   1. 128
   2. 175
   3. 145
   4. 162
3. Let $f : R \to (0, \infty)$ be twice differentiable function such that $f(3)=18$, $f'(3)=0$ and $f''(3)=4$. Then $\lim \limits_{x \to 1}\left(\log_e\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to
    
   1. 2
   2. 1
   3. 18
   4. 9
4. Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be a point on it. Let the locus of the point $P$, which divides the line segment $OQ$ internally in the ratio 2:3 be the conic $C$. The equation of the chord of $C$, which is bisected atthe point $(1, 2)$ is: whose mid point is (1, 2):
    
   1. $5x – 4y + 3 = 0$
   2. $x – 2y + 3 = 0$
   3. $5x - y - 3 = 0$
   4. $4x - 5y + 6 = 0$
5. The value of integral $\int \limits_{-\pi/6}^{\pi/6}\frac{\pi+4x^{11}}{1-\sin (|x|+\pi/6)}dx$ is
    
   1. $8\pi$
   2. $2\pi$
   3. $6\pi$
   4. $4\pi$
6. The number of relations, defined on the set {$a, b, c, d$}, which are both reflexive and symmetric, is
    
   1. 1024
   2. 64
   3. 16
   4. 256
7. Let $a_1$, $a_2$, $a_3$……. be a G.P. of increasing positive terms such that  $a_2a_3a_4$ and $a_1$+$a_3$+$a_5$= $\frac{813}{7}$. Then $a_3 + a_5 + a_7$ is
    
   1. 3256
   2. 3248
   3. 3244
   4. 3252
8. The number of strictly increasing functions $f$ from the set {1, 2, 3, 4, 5, 6} to the set {1, 2, 3, 4, 5, 6, 7, 8, 9} such that $f(i) \neq i$ for $1 \leq i \leq 6$, is equal to
    
   1. 22
   2. 27
   3. 21
   4. 28
9. Let $\vec{a}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{b}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}×\vec{c}=\vec{b}$. If $\vec{c}•(\hat{i}+\hat{j}+\hat{k})$=4, then $\left|\vec{a}+\vec{c}\right|^2$ is equal to:
    
   1. 33
   2. 35
   3. 27
   4. 30
10. Let $PQ$ and $MN$ be two straight lines touching the circle $x^2 + y^2 – 4x – 6y- 3$ = 0 at the points $A$ & $B$ respectively. Let $O$ be the centre of the circle and $\angle{AOB}=\frac{\pi}{3}$. Then the locus of the point of intersection of the lines $PQ$ and $MN$ is:
     
    1. $x^2+y^2-18x-12y-25=0$
    2. $3(x^2+y^2)-18x-12y+25=0$
    3. $3(x^2+ y^2)– 12x – 18y - 25 = 0$
    4. $x^2+ y^2– 12x – 18y - 25 = 0$
11. The area of the region inside the ellipse $x^2+ 4y^2= 4$ and outside the region bounded by curves $y = |x| – 1$ and $y = 1 – |x|$ :
     
    1. $2\pi -1$
    2. $3(\pi - 1)$
    3. $2(\pi - 1)$
    4. $2\pi - \frac{1}{2}$
12. Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}$=1. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:
     
    1. $24 \sqrt{5}$
    2. 12
    3. 16
    4. $\frac{96}{\sqrt{5}}$
13. The sum of all the roots of the equation $\left(x-1\right)^2-5|x-1|+6=0$ is
     
    1. 5
    2. $3$
    3. $4$
    4. $1$
14. Let $(\alpha, \beta, \gamma)$ be the coordinate of the foot of the perpendicular drawn from the point (5, 4, 2) on the line $\vec{r}$=$\left(-\hat{i}+3\hat{j}+\hat{k}\right)$+$\lambda\left(2\hat{i}+3\hat{j}-\hat{k}\right)$. Then the length of the projection of the vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ on the vector $6\hat{i}+2\hat{j}+3\hat{k}$ is:
     
    1. 3
    2. $\frac{15}{7}$
    3. $\frac{18}{7}$
    4. $4$
15. Let a point $A$ lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle $ABC$, where the points $B$ and $C$ lie on the lines $L_1$ and $L_2$ respectively, is
     
    1. $21\sqrt{3}$
    2. $15\sqrt{6}$
    3. $27 $
    4. $12 \sqrt{2}$
16. The value of $cosec 10° - \sqrt{3} \sec 10°$ is equal to:
     
    1. 8
    2. 2
    3. 6
    4. 4
17. If the domain of the function $f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)$+$\sin^{-1}(2x^2-3x+1)$ is the interval $[\alpha, \beta]$, then $(\alpha+2\beta)$ is equal to:
     
    1. 3
    2. 5
    3. 1
    4. 2
18. Let $y = y(x)$ be the solution curve of the differential equation $(1 + x^2) dy$ + $(y – tan^{–1}x) dx$ = 0, $y(0)$ = 1. Then the value of $y(1)$ is :
    1. $\frac{4}{e^{\frac{\pi}{4}}}-$$\frac{\pi}{2}-1$
    2. $\frac{2}{e^{\frac{\pi}{4}}}$+$\frac{\pi}{4}-1$
    3. $\frac{2}{e^{\frac{\pi}{4}}}-$$\frac{\pi}{4}-1$
    4. $\frac{4}{e^{\frac{\pi}{4}}}$+$\frac{\pi}{2}-1$
19. Let $\vec{c}$ and $\vec{d}$ be vectors such that $\left|\vec{c}+\vec{d}\right|$=$\sqrt{29}$ and $\vec{c}×\left(2\hat{i}+3\hat{j}+4\hat{k}\right)$=$\left(2\hat{i}+3\hat{j}+4\hat{k}\right)×\vec{d}$. If $\lambda_1, \lambda_2(\lambda_1 > \lambda_2)$ are the possible values of $(\vec{c}+\vec{d})(-7\hat{i}+2\hat{j}+3\hat{k})$, then the equation $K^2x^2$+$(K^2-5K+\lambda_1)xy$+$\left(3K+\frac{\lambda_2}{2}\right)^2$$-8x+12y+\lambda_2$=0 represents a circle, for $K$ equal to
     
    1. 2
    2. -1
    3. 1
    4. 4
20. Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x>y, be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x-4, y, 5} one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is:
     
    1. $\frac{4}{5}$
    2. $\frac{3}{5}$
    3. $\frac{2}{5}$
    4. $\frac{1}{3}$

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 $f: R \to R$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f'(x)m+f''(x)=0$ in $m$, has two equal roots for every $x \in R$. If $f(0)=1$, $f'(0)=2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_ex-x)$ is increasing, then $\alpha+\beta$ is equal to........
2. Let $S$=$\left\{(m, n): m, n \in \left\{1, 2, 3, ..., 50 \right\}\right\}$. If the number of elements $(m, n)$ in $S$ such that $6^m+9^n$ is a multiple of 5 is $p$ and the number of elements $(m, n)$ in $S$ such that $m+n$ is a square of a prime number is $q$, then $p+q$ is equal to.......
3. For some $\alpha, \beta \in R$, let $A=\begin{equation*} \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix} \end{equation*}$ and $B=\begin{equation*} \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix} \end{equation*}$ and $A^2-4A+2I$=$B^2-3B+1=O$ then $\left(det (adj (A^3-B^3))\right)^2$ is equal to
4. Let $a_1=1$ and for $n \geq 1$, $a_{n+1}=\frac{1}{2}a_n+\frac{n^2-2n-1}{n^2(n+1)^2}$. Then $\left|\sum \limits_{n=1}^{\infty}\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to...........
5. $6\int \limits_0^{\pi}\left|\sin 3x+\sin 2x+\sin x \right|dx$ is equal to.........

21 Jan 2026:Shift-2

21 Jan 2026:Ans-Keys

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