Download JEE Main 2026 Question Paper (22 Jan - Shift 1) Mathematics

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23 Jan 2026:Shift-1

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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. Two distinct numbers $a$ and $b$ are selected at random from 1, 2, 3, ..., 50. The probability, that their product $ab$ is divisible by 3, is
    
   1. $\frac{8}{25}$
   2. $\frac{561}{1225}$
   3. $\frac{664}{1225}$
   4. $\frac{272}{1225}$
2. If a random variable $x$ has the probability distribution
   

   X	0	1	2	3	4	5	6	7
   P(X)	0	$2k$	$k$	$3k$	$2k^2$	$2k$	$k^2+k$	$7k^2$

   
   
   
   then $P(3 < x \leq 6)$ is equal to
    
   1. 0.22
   2. 0.33
   3. 0.34
   4. 0.64
3. Let $f: (1, \infty) \to R$ be a differentiable function. If $6\int \limits_1^xf(t)dt$=$3xf(x)$+$x^3-4$ for all $x \geq 1$, then the value of $f(2)-f(3)$ is
    
   1. 3
   2. -4
   3. -3
   4. 4
4. If the image of the point $P(1, 2, \alpha)$ in the line $\frac{x-6}{3}$=$\frac{y-7}{2}$=$\frac{7-x}{2}$ is $Q(5, b, c)$, then $a^2+b^2+c^2$ is equal to
    
   1. 293
   2. 298
   3. 264
   4. 283
5. If the chord joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2-y_1y_2$ is equal to
    
   1. 292
   2. 288
   3. 284
   4. 280
6. If the domain of the function $f(x)=\sin^{-1}\left(\frac{5-x}{3+2x}\right)$+$\frac{1}{\log_e(10-x)}$ is $(-\infty, \alpha)\cup[\beta, \gamma)-\left\{\delta\right\}$, then $6(\alpha+\beta+\gamma+\delta)$ is equal to
    
   1. 68
   2. 66
   3. 70
   4. 67
7. Let $P(\alpha, \beta, \gamma)$ be the point on the line $\frac{x-1}{2}$=$\frac{y+1}{-3}$=$z$ at a distance $4\sqrt{14}$ from the point $(1, -1, 0)$ and nearer to the origin. Then the shortest distance between the lines $\frac{x-\alpha}{1}$=$\frac{y-\beta}{2}$=$\frac{z-\gamma}{3}$ and $\frac{x+5}{2}$=$\frac{y-10}{1}$=$\frac{z-3}{1}$, is equal to
    
   1. $7\sqrt{\frac{5}{4}}$
   2. $4\sqrt{\frac{5}{7}}$
   3. $2\sqrt{\frac{7}{4}}$
   4. $4\sqrt{\frac{7}{5}}$
8. If $A=\begin{equation*} \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix} \end{equation*}$, then the determinant of the matrix $(A^{2025}-3A^{2024}+A^{2023})$ is
    
   1. 28
   2. 16
   3. 24
   4. 12
9. Let the relation $R$ on the set $M$={1, 2, 3, ..., 16} be given by $R$={$(x, y): 4y=5x-3$, $x, y \in M$}. Then the minimum number of elements required to be added in $R$, in order to make the relation symmetric, is equal to
    
   1. 3
   2. 4
   3. 2
   4. 1
10. Let the set of all values of $r$, for which the circles $(x+1)^2$+ $(y + 4)^2$= $r^2$ and $x^2$+ $y^2$ – $4x$ – $2y$ – 4 =0 intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha \beta$ is equal to:
     
    1. 25
    2. 21
    3. 24
    4. 20
11. Let the solution curve of the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx$, $x > 0$, $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to
     
    1. 4
    2. 2
    3. 1
    4. 6
12. Let the line $x=-1$ divide the area of region {$(x, y):1+x^2 \leq y \leq 3-x$} in the ratio $m:n$, $gcd(m, n)$=1. Then $m+n$ is equal to
     
    1. 27
    2. 26
    3. 25
    4. 28
13. The no. of solutions of $\tan^{-1}4x$+$\tan^{-1}6x$=$\frac{\pi}{6}$, where $-\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}}$, is equal to:
     
    1. 1
    2. 2
    3. 0
    4. 3
14. Let $\vec{AB}$=$2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{AD}$=$\hat{i}+2\hat{j}+\lambda \hat{k}$, $\lambda \in R$. Let the projection of the vector $\vec{v}$=$\hat{i}+\hat{j}+\hat{k}$ on the diagonal $\vec{AC}$ of the parallelogram $ABCD$ be of length one unit. If $\alpha, \beta$ where $\alpha > \beta$, be the roots of the equation $\lambda^2x^2-6\lambda x+5$=0, then $2\alpha-\beta$ is equal to
     
    1. 4
    2. 6
    3. 3
    4. 1
15. The value of $\int \limits_{-\pi/2}^{\pi/2} \left(\frac{1}{[x]+4}\right)dx$, where [•] denotes greatest integer function, is:
     
    1. $\frac{1}{60}(\pi-7)$
    2. $\frac{1}{60}(21\pi-1)$
    3. $\frac{7}{60}(3\pi-1)$
    4. $\frac{1}{60}(\pi-3)$
16. Let $f(x)=x^{2025}-x^{2000}$, $x \in [0, 1]$, then minimum value of $f(x)$ in the interval [0, 1] be $(80)^{80}(n)^{-81}$. Then $n$ is equal to:
     
    1. -40
    2. -81
    3. -80
    4. -41
17. If the sum of the first 4 terms of an AP is 6 and sum of its first 6 terms is 4, then sum of its first 12 terms of AP is :
     
    1. -22
    2. -20
    3. -26
    4. -24
18. The coefficient of $x^{48}$ in
    $(1+x)+2(1+x)^2+3(1+x)^3$+...+$100(1+x)^{100}$ is:
     
    1. $100 \cdot{ }^{101} \mathrm{C}_{49}-{ }^{101} \mathrm{C}_{50}$
    2. $100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{48}$
    3. $100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{50}$
    4. ${ }^{100} \mathrm{C}_{50}+{ }^{101} \mathrm{C}_{49}$
19. The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is
     
    1. 2
    2. 0
    3. 3
    4. 1
20. If the line $\alpha x + 2y = 1$, where $\alpha \in R$ does not meet the hyperbola $x^2– 9y^2= 9$ then a possible value of $\alpha$ is :
     
    1. 0.5
    2. 0.6
    3. 0.8
    4. 0.7

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 $A$ be a 3×3 matrix such that $A+A^T=O$. If $A \begin{equation*} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix} \end{equation*}$, $A^2 \begin{equation*} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \end{equation*}$=$\begin{equation*} \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix} \end{equation*}$ and $det(adj(2 adj (A+I)))$=$(2)^{\alpha}•(3)^{\beta}•(11)^{\gamma}$, $\alpha$, $\beta$, $\gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to............
2. Let $\alpha=\frac{-1+i\sqrt{3}}{2}$ & $\beta=\frac{-1-i\sqrt{3}}{2}$, $i=\sqrt{-1}$. If $(7-7\alpha+9\beta)^{20})$+$(9\alpha+7\beta-7)^{20}$+$(-7+9\alpha+7\beta)^{20}$+$(14+7\alpha+7\beta)^{20}$ = $m^{10}$, then $m$ is.............
3. If $\int (\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}}dx$=$-\frac{p_1}{q_1}(cot x)^{\frac{9}{2}}$$-\frac{p_2}{q_2}(cot x)^{\frac{5}{2}}$$-\frac{p_3}{q_3}(cot x)^{\frac{1}{2}}$+$\frac{p_4}{q_4}(cot x)^{\frac{-3}{2}}$+C, where $p_i$ and $q_i$ are positive integers with $gcd(p_i, q_i)$ =1 for $i$ =1, 2, 3, 4 and C is the constant of integration, then $\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to..............
4. If $\frac{\cos^248°-\sin^212°}{\sin^224°-\sin^6°}$ = $\frac{\alpha+\beta \sqrt{5}}{2}$, where $\alpha, \beta \in N$, then $\alpha+\beta$ is equal to............
5. Let $ABC$ be a triangle. Consider four points $p_1$, $p_2$, $p_3$, $p_4$ on the side $AB$, five points $p_5$, $p_6$, $p_7$, $p_8$, $p_9$ on the side $BC$, and four points $p_{10}$, $p_{11}$, $p_{12}$, $p_{13}$ on the side $AC$. None of these points is a vertex of the triangle $ABC$. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1$, $p_2$, ..., $p_{13}$, is..............

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