Download JEE Main 2026 Question Paper (28 Jan - Shift 1) Mathematics

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21 Jan 2026:Shift-1

21 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 $g(x) = 3x^2 + 2x –3$, $f(0) = –3$, $4g(f(x)) = 3x^2–32x + 72$ then $f(g(2))$ is equal to
    
   1. $-\frac{25}{6}$
   2. $-\frac{7}{2}$
   3. $\frac{25}{6}$
   4. $\frac{7}{2}$
2. Let $y(x)$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x \geq 0$) of diameter 10 passing through origin. Let $C_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_2$, which pass through (2, 3) and is farthest from centre of $C_2$ is $x + ay + b$ = 0,  then $a-b$ is equal to :
    
   1. -2
   2. 10
   3. -6
   4. 6
3. Let $S$={$x^3+ax^2+bx+c: a, b, c \in N$ and $a, b, c \leq 20$} be a set of polynomials. Then the number of polynomials in $S$, which are divisible by $x^2+2$ is
    
   1. 120
   2. 10
   3. 20
   4. 6
4. The mean and variance of 10 observations are 9 and 34.2 respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations, is :
    
   1. 4
   2. 6
   3. 5
   4. 7
5. Let $y=y(x)$ be the solution of the differential equation $x\frac{dy}{dx}-\sin 2y$=$x^3(2-x^3) \cos^2y$, $x \neq 0$. If $y(2)=0$, then find $\tan (y(1))$ is equal to:
    
   1. $3/4$
   2. $-3/4$
   3. $7/4$
   4. $-7/4$
6. A bag contains 10 balls out of which '$k$' are red balls and $(10 – k)$ are black, where $0 \leq k \leq 10$. If 3 balls are drawn at random without replacement and all of them are found to be black, then the probability that bag has 9 black balls & 1 red ball is
    
   1. $\frac{7}{11}$
   2. $\frac{7}{55}$
   3. $\frac{14}{55}$
   4. $\frac{7}{110}$
7. The common difference of the A.P.: $a_1$, $a_2$, ..., $a_m$ is 13 more than the common difference of the A.P.: $b_1$, $b_2$, ..., $b_n$. If $b_{31}=-277$, $b_{43}=-385$ and $a_{78}$=327, then $a_1$ is equal to
    
   1. 16
   2. 19
   3. 24
   4. 21
8. The value of $\sum \limits_{k=1}^{\infty}\frac{(-1)^k.k(k+1)}{k!}$
    
   1. $\frac{1}{e}$
   2. $\frac{2}{e}$
   3. $\sqrt{e}$
   4. $\frac{e}{2}$
9. If the distance of the point $(1, 2, a)$ from the line $L_1: \frac{x-1}{3}$=$\frac{y-2}{4}$=$\frac{z-a}{b}$ and $L_2:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}$ are equal, then $a+b+c$ is equal to
    
   1. 5
   2. 6
   3. 4
   4. 7
10. For three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ satisfying $(\vec{a}-\vec{b})^2$+$(\vec{b}-\vec{c})^2$+$(\vec{c}-\vec{a})^2$ =9 and $|2\vec{a}+k\vec{b}+k\vec{c}|$=3, the positive value of $k$ is
     
    1. 3
    2. 6
    3. 4
    4. 5
11. The value of $\lim \limits_{x \to 0} \frac{\ln (\sec (ex). \sec(e^2x)........\sec(e^{10}x))}{e^2-e^{2\cos x}}$ is equal to
     
    1. $\frac{(e^{10}-1)}{2e^2(e^2-1)}$
    2. $\frac{(e^{20}-1)}{2e^2(e^2-1)}$
    3. $\frac{(e^{10}-1)}{2(e^2-1)}$
    4. $\frac{(e^{20}-1)}{2(e^2-1)}$
12. Let '$z$' a complex number such that $|z - 6|=5$ & $|z + 2 - 6i|=5$. Then the value of $z^3+ 3z^2– 15z + 141$ is equal to
     
    1. 37
    2. 42
    3. 50
    4. 61
13. If $\frac{\tan (A-B)}{\tan A}$+$\frac{\sin^2C}{\sin^2A}$=1 where $A, B, C \in \left(0, \frac{\pi}{2}\right)$ then
     
    1. $\tan A$, $\tan B$, $\tan C$ are in G.P.
    2. $\tan A$, $\tan C$, $\tan B$ are in G.P.
    3. $\tan A$, $\tan B$, $\tan C$ are in A.P.
    4. $\tan A$, $\tan C$, $\tan B$ are in A.P.
14. The area of the region $R$= {$(x, y) : xy \leq 8$; $1 \leq y \leq x^2$, $x \geq 0$} is :
     
    1. $\frac{2}{3}(20\log_e2+9)$
    2. $\frac{2}{3}(40\log_e2+27)$
    3. $\frac{2}{3}(49\log_e2-15)$
    4. $\frac{2}{3}(24\log_e2-7)$
15. If $\alpha$, $\beta$, where $\alpha < \beta$, are the roots of the equation $\lambda x^2-(\lambda+3)x+3=0$ such that $\frac{1}{\alpha}-\frac{1}{\beta}$=$\frac{1}{3}$, then the sum of all the possible values of $\lambda$ is
     
    1. 8
    2. 6
    3. 4
    4. 2
16. Let $S$={1, 2, 3, 4, 5, 6, 7, 8, 9}. Let $x$ be the number of 9 digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9 digit numbers formed using the digits of the set $S$ such that only 2 digits are repeated and each of these is repeated exactly twice. Then
     
    1. $21x=4y$
    2. $45x=7y$
    3. $56x=9y$
    4. $29x=5y$
17. Let $A$, $B$ and $C$ be three 2×2 matrices with real entries such that $B=(I+A)^{-1}$ and $A+C$=$I$. If $BC=\begin{equation*} \begin{bmatrix} 1 & -5 \\ -1 & 2 \end{bmatrix} \end{equation*}$ and $CB \begin{equation*} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}=\begin{bmatrix} 12 \\ -6 \end{bmatrix} \end{equation*}$, then $x_1+x_2$ is
     
    1. 4
    2. 0
    3. -2
    4. 2
18. Let $ABC$ be an equilateral triangle with orthocentre at the origin and the side $BC$ on the line $x+2\sqrt{2}y$=4. If the coordinates of the vertex $A$ are $(a, b)$, then the value of $\left[ \left|a+\sqrt{2}b\right|\right]$, where [•] denotes $G.I.F$, is:
     
    1. 2
    2. 4
    3. 5
    4. 3
19. If $\int \frac{1-5\cos^2x}{\sin^5x \cos^2x}dx$=$f(x)+c$, where $c$ is the constant of the integration, then $f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$ is equal to:
     
    1. $\frac{1}{\sqrt{3}}(26-\sqrt{3})$
    2. $\frac{1}{\sqrt{3}}(26+\sqrt{3})$
    3. $\frac{4}{\sqrt{3}}(8-\sqrt{6})$
    4. $\frac{2}{\sqrt{3}}(4+\sqrt{6})$
20. Let $f$ be a polynomial function such that $f(x^2+1)$=$x^4+5x^2+1$, for all $x \in R$. Then $\int \limits_0^3 f(x)dx$ is equal to:
     
    1. $\frac{5}{3}$
    2. $\frac{27}{2}$
    3. $\frac{33}{2}$
    4. $\frac{41}{3}$

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. In a G.P., if the product of the first 3 terms is 27 and the set of all possible values for the sum of its first three terms is $R–(a,b)$, then $a^2 + b^2$ is equal to ............
2. If $k$= $\tan\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\frac{2}{3}\right)$+$\tan \left(\frac{1}{2}\sin^{-1}\frac{2}{3}\right)$, t the number of solutions of the equation $\sin^{-1}(kx-1)$=$\sin ^{-1}x-\cos^{-1}x$ is..............
3. For some $\theta \in \left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2 – y^2\sec^2\theta$= 8 be $e_1$ & $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2 \sec^2\theta + y^2$ = 6 be $e^2$ & $l_2$, respectively. If $e_1^2$ = $e_2^2(1 + \sec^2 \theta)$ then $\frac{l_1l_2}{e_1e_2} \tan^2 \theta$ is equal to...............
4. The value of $\sum \limits_{r=1}^{20}\left(\left|\sqrt{\pi \int \limits_0^rx|\sin \pi x|dx}\right|\right)$ is ............
5. Let $PQR$ be a triangle such that $\vec{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{PR}$=$a \hat{i}+b \hat{j}-4\hat{k}$, $a, b \in Z$. Let $S$ be the point on $QR$, which is equidistant from the line $PQ$ and $PR$. If $|\vec{PR}|$=9 and $\vec{PS}$=$\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is ..........

28 Jan 2026:Shift-2

28 Jan 2026:Ans-Keys

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