Download JEE Main 2026 Question Paper (23 Jan - Shift 1) Mathematics

» Hold on briefly for download button.

Wait-starts soon.

 Download Now!

24 Jan 2026:Shift-1

24 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. Let the line $y-x=1$ intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}$=1 at the points $A$ & $B$. Then the angle made by the line segment $AB$ at the centre of the ellipse is
    
   1. $\frac{\pi}{2}-\tan^{-1}\frac{1}{4}$
   2. $\frac{\pi}{2}+2\tan^{-1}\frac{1}{4}$
   3. $\frac{\pi}{2}+\tan^{-1}\frac{1}{4}$
   4. $\pi-\tan^{-1}\frac{1}{4}$
2. Number of solutions of
   $\sqrt{3} \cos 2\theta$+$8\cos \theta$+$3\sqrt{3}$=0 where $\theta \in [-3\pi, 2\pi]$
    
   1. 4
   2. 0
   3. 3
   4. 5
3. Let the direction cosines of two lines satisfy the equation: $4l+m-n=0$ and $2mn+10nl+3lm=0$. Then the cosine of the acute angle between these lines is
    
   1. $\frac{10}{3\sqrt{38}}$
   2. $\frac{20}{3\sqrt{38}}$
   3. $\frac{10}{7\sqrt{38}}$
   4. $\frac{10}{\sqrt{38}}$
4. Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta)$=$4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)$$-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right)$, $\theta \in R$. Then $\alpha+2\beta$ is equal to
    
   1. 4
   2. 3
   3. 5
   4. 6
5. Let
   $f(x)=\left\{ \begin{array} {cc} \frac{ax^2+2ax+3}{4x^2+4x-3}, x \neq -\frac{3}{2}, \frac{1}{2} \\ b, x=-\frac{3}{2}, \frac{1}{2} \end{array}\right.$
   be continuous at $x=-\frac{3}{2}$. If $fof(x)$=$\frac{7}{5}$, then $x$ is equal to
    
   1. 0
   2. 2
   3. 1
   4. 1.4
6. If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation
   $(-2+\sqrt{3})$$(|\sqrt{x}-3|)$+$(x-6\sqrt{x})$+$(9-2\sqrt{3})$=0, $x \geq 0$, then $\sqrt{\frac{\beta}{\alpha}}$+$\sqrt{\alpha \beta}$ is equal to
    
   1. 8
   2. 11
   3. 9
   4. 10
7. The vertices $B$ and $C$ of a triangle $ABC$ lie on the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. The coordinates of $A$ and $B$ are (1, 6, 3) and $(4, 9, \alpha)$ respectively and $C$ is at a distance of 10 units from $B$. The area (in sq. units) of $\Delta ABC$ is
    
   1. $10\sqrt{13}$
   2. $15\sqrt{13}$
   3. $5\sqrt{13}$
   4. $20\sqrt{13}$
8. Among the statements
   $I$: If $\begin{equation*} \begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} \end{equation*}$=$\begin{equation*} \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0 \end{vmatrix} \end{equation*}$, then $\cos^2\alpha$+$\cos^2\beta$+$\cos^2\gamma$=$\frac{3}{2}$, and
   
   II: If $\begin{equation*} \begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} \end{equation*}$=$px+q$, then $p^2=196q^2$,
    
   1. Both are true
   2. Only I is true
   3. Both are false
   4. Only II is true
9. The value of the integral $\int \limits_{\pi/24}^{5\pi/24} \frac{1}{(1+\sqrt{3}{\tan 2x})}dx$ is:
    
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{18}$
   3. $\frac{\pi}{12}$
   4. $\frac{\pi}{3}$
10. Let the domain of the function $f(x)$=$\log_3\log_5\log_7(9x-x^2-13)$ be the interval $(m, n)$. Let the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 have eccentricity $\frac{n}{3}$ & the length of latus rectum is $\frac{8m}{3}$. Then $b^2-a^2$ is equal to:
     
    1. 9
    2. 11
    3. 5
    4. 7
11. Let $f(x)$=$\int e^x \left(\frac{2-x^2}{\sqrt{1+x}(1-x)^{3/2}}\right)dx$. If $f(0)=0$, then $f(1/2)$ is equal to
     
    1. $\sqrt{2e}-1$
    2. $\sqrt{3e}-1$
    3. $\sqrt{3e}+1$
    4. $\sqrt{2e}+1$
12. A building construction work can be completed by two masons $A$ and $B$ together in 22.5 days. Mason $A$ alone can complete the construction work in 24 days less than mason $B$ alone. Then mason $A$ alone will complete the construction work in
     
    1. 42 days
    2. 24 days
    3. 36 days
    4. 30 days
13. Let $y=y(x)$ be the solution of the differential equation $x^4dy+(4x^3y+2\sin x)dx$=0, $x > 0$, $y\left(\frac{\pi}{2}\right)$=0 then $\pi^4y\left(\frac{\pi}{3}\right)$ is equal to
     
    1. 92
    2. 81
    3. 72
    4. 64
14. A rectangle is formed by lines $x = 0, y = 0, x = 3$ and $y = 4$. Let the line $L$ be perpendicular to $3x + 4y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2}, -5 \right)$ from the line $L$ is equal to:
     
    1. $2\sqrt{5}$
    2. $2\sqrt{10}$
    3. $3\sqrt{10}$
    4. $\sqrt{10}$
15. Let the mean and variance of 8 numbers $-10$, $-7$, $-1$, $x$, $y$, $2$, $9$, $16$ be $\frac{7}{2}$ and $\frac{293}{4}$ respectively. Then the mean of 4 numbers $(1+x+y)$, $x$, $y$, $|y-x|$:
     
    1. 12
    2. 10
    3. 9
    4. 11
16. Let $A$ = {–2, –1, 0, 1, 2, 3, 4}. Let $R$ be a relation defined on $A$ defined by $xRy$ if and only if $2x + y \leq 2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be minimum number of elements required to be added in $R$ to make it reflexive and symmetric relation. Then $(l + m + n)$ is equal to:
     
    1. 34
    2. 35
    3. 32
    4. 33
17. Let $\vec{a}=-\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$, $\vec{c}=\vec{a}×\vec{b}$ and $\vec{d}=\vec{c}×\vec{b}$. Then $(\vec{a}-\vec{b})•\vec{d}$ is equal to:
     
    1. -4
    2. 4
    3. -2
    4. 2
18. Let $S$={$Z: 3 \leq |2Z-3(1+i)| \leq 7$ be a set of complex numbers. Then $\operatorname{Min}_{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to
     
    1. 2
    2. $\frac{3}{2}$
    3. $\frac{1}{2}$
    4. $\frac{5}{2}$
19. The sum of all the possible values of $n \in N$, so that the coefficient of $x$, $x^2$ & $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$, are in arithmetic progression is:
     
    1. 9
    2. 3
    3. 7
    4. 12
20. The value of $\frac{{ }^{100} C_{50}}{51}$+$\frac{{ }^{100} C_{51}}{52}$+$\ldots$+$\frac{{ }^{100} C_{100}}{101}$ is:
     
    1. $\frac{2^{100}}{100}$
    2. $\frac{2^{101}}{100}$
    3. $\frac{2^{101}}{101}$
    4. $\frac{2^{100}}{101}$

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. Find the area of the region bounded by $y$ = max {$sin x, cos x$}, lines $x=0$, $x=\frac{3\pi}{2}$, and the $x-axis$ be $A$. Then, $A+A^2$ is equal to...........:
2. The number of 4 letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is...........
3. Let $f$ be a twice differentiable function such that $(f(x))^2$=25+$\int \limits_0^x(f(t))^2+(f'(t))^2dt$. Then the mean of $f(ln 1)$+$f(ln 2)$+...+$f(ln 625)$ is equal to........
4. From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a-b \geq 10$ is $\frac{m}{n}$, $gcd(m, n)=1$, then $m+n$ is equal to..............
5. Let $|A|$=6, where $A$ is 3×3 matrix. If $\left| adj \left( \left(A^2 adj(2A)\right)\right)\right|$=$2^m•3^n$, $m, n \in N$, then $m + n$ is equal to............

23 Jan 2026:Shift-2

23 Jan 2026:Ans-Keys

All the Papers Ans-Keys