Download JEE Main 2026 Question Paper (28 Jan - Shift 2) Mathematics

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21 Jan 2026:Shift-1

21 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. Let $P_1: y = 4x^2$ and $P_2: y = x^2+ 27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ & $P_2$ is six times the area of the bounded region enclosed between the line $y=\alpha x$, $\alpha > 0$ and $P_1$, then $\alpha$ is equal to
    
   1. 8
   2. 15
   3. 6
   4. 12
2. Let $f(x)$=$\int \frac{dx}{x^{2/3}+2x^{\frac{1}{2}}}$ be such that $f(0)=-26+24 \ln 2$. If $f(1)=a+b \ln 3$, where $a, b \in Z$ then $(a+b)$ is equal to
    
   1. -11
   2. -5
   3. -26
   4. -18
3. Given below are two statements
   Statement 1: $25^{13}$+$20^{13}$+$8^{13}$+$3^{13}$ is divisible by 7.
   
   Statement 2: The integral part of $\left(7+4\sqrt{3}\right)^{25}$ is an odd number.
   In the light of the above statements, choose the correct answer from the options given below:
   1. Statement 1 is false and Statement 2 is correct
   2. Statement 1 is correct and Statement 2 is false
   3. Both statements are false
    
   4. Both statements are correct
4. Let the ellipse $E:\frac{x^2}{144}$+$\frac{y^2}{169}$=1 and the hyperbola $H:\frac{x^2}{16}-\frac{y^2}{\lambda^2}$=-1 have same foci. If $e$ and $l$ respectively denote the eccentricity and length of latus rectum of $H$, then the value of $24(e+l)$ is
    
   1. 67
   2. 296
   3. 148
   4. 126
5. Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}$, $a > 2$. If $\alpha$ is such that $a$, $4$, $\alpha$, $b$ are in $A.P.$ then the equation $\alpha x^2-ax+2(\alpha-2b)$ has
    
   1. one root in (1, 4) and another in (-2, 0)
   2. complex roots of magnitude less than 2
   3. Both roots in the interval (-2, 0)
   4. One root in (0, 2) and another in (-4, -2)
6. The sum of the coefficients of $x^{499}$ and $x^{500}$ in the expression :
   $(1+x)^{1000}$+$x(1+x)^{999}$+$x^2(1+x)^{998}$+...+$x^{1000}$ is
    
   1. ${}^{100}C_{501}$
   2. ${}^{1002}C_{500}$
   3. ${}^{1001}C_{501}$
   4. ${}^{1002}C_{501}$
7. Let $y=y(x)$ be the solution of differential equation $x\frac{dy}{dx}-y$=$x^2\cot x$, $x \in (0, \pi)$. If $y\left(\frac{\pi}{2}\right)$=$\frac{\pi}{2}$, then $6y\left(\frac{\pi}{6}\right)-8y\left(\frac{\pi}{4}\right)$ is equal to:
    
   1. $3\pi$
   2. $-3\pi$
   3. $\pi$
   4. $-\pi$
8. An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is
    
   1. $\frac{16}{\sqrt{3}}$
   2. $6$
   3. $4\sqrt{3}$
   4. $6\sqrt{3}$
9. Given below are two statements:
   Statement 1: The function $f: R \to R$ defined by $f(x)=\frac{x}{1+|x|}$ is one.
   
   Statement 2: The function $f: R \to R$ defined by $f(x)=\frac{x^2+4x-30}{x^2-8x+18}$ is many-one.
   
   In the light of the above statements, choose the correct answer from the options given below:
   1. Statement 1 is correct and Statement 2 is false
    
   2. Both statements are correct
   3. Statement 1 is false and Statement 2 is correct
   4. Both statements are false
10. Let $f(x)$=$\lim \limits_{\theta \to 0^{+}} \frac{\cos \pi x-x^{2/\theta} \sin(x-1)}{1-x^{2/\theta}(x-1)}$, $x \in R$. Consider the following two statements:
    Statement 1: $f(x)$ is discontinuous at $x = 1$
    Statement 2: $f(x)$ is continuous at $x = –1$
    
     
    1. Only Statement 1 is true
    2. Neither Statement 1 nor Statement 2 is true
    3. Both Statements are true
    4. Only Statement 2 is true
11. Let $A$ be the focus of the parabola $y^2=8x$. Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$. If the centroid of the triangle is $\left(\frac{7}{3}, \frac{4}{3}\right)$, then $(BC)^2$ is equal to
     
    1. 41
    2. 89
    3. 32
    4. 80
12. Let [•] denotes the greatest integer function. Then $\int \limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{12(3+[x])dx}{3+[\sin x]+[\cos x]}$ is equal to.
     
    1. $13\pi+1$
    2. $12\pi+5$
    3. $11\pi+2$
    4. $15\pi+4$
13. Let $P$ be a point in the plane of the vectors $\vec{AB}$=$3\hat{i}+\hat{j}-\hat{k}$ and $\vec{AC}$=$\hat{i}-\hat{j}+3\hat{k}$ such that $P$ is equidistant from the lines $AB$ and $AC$. If $|\vec{AP}|$=$\frac{\sqrt{5}}{2}$, then the area of the $\Delta APB$ is:
     
    1. $2$
    2. $\frac{3}{2}$
    3. $\frac{\sqrt{26}}{4}$
    4. $\frac{\sqrt{30}}{4}$
14. Let $Q(a, b, c)$ be the image of the point $P(3, 2, 1)$ in the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$. Then the distance of $Q$ from the line $\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2}$ is
     
    1. 8
    2. 7
    3. 6
    4. 5
15. The probability distribution of a random variable $X$ is given below:
                                                   

    X	$4k$	$\frac{30}{7}k$	$\frac{32}{7}k$	$\frac{34}{7}k$	$\frac{36}{7}k$	$\frac{38}{7}k$	$\frac{40}{7}k$	$6k$
    P(X)	$\frac{2}{15}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{1}{5}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{1}{5}$	$\frac{2}{15}$

    
    
    If $E(X)$=$\frac{263}{15}$, then $P(X < 20)$ is equal to:
     
    1. $\frac{3}{5}$
    2. $\frac{14}{15}$
    3. $\frac{8}{15}$
    4. $\frac{11}{15}$
16. Considering the principal values of inverse trignometric functions, the value of the expression $\tan \left[\left(2\sin^{-1}\frac{2}{\sqrt{13}} \right)\right.$$\left.-2\cos^{-1}\left(\frac{3}{\sqrt{10}}\right)\right]$ is equal to:
     
    1. $\frac{33}{56}$
    2. $-\frac{33}{56}$
    3. $\frac{16}{63}$
    4. $-\frac{16}{63}$
17. Let the circle $x^2+ y^2$= 4 intersects $x-$axis at the points $A(a, 0)$, $a > 0$ and $B(b, 0)$. Let $P(2\cos \alpha, 2\sin \alpha)$, $0 < \alpha < \frac{\pi}{2}$ & $Q(2\cos \alpha, 2\sin \alpha)$ be two points such that $\alpha-\beta=\frac{\pi}{2}$. Then the point of intersection of $AQ$ and $BP$ lies on
     
    1. $x^2+y^2-4x-4y-4=0$
    2. $x^2+y^2-4x-4=0$
    3. $x^2+y^2-4y-4=0$
    4. $x^2+y^2-4x-4y=0$
18. Let
    $A$={$z \in C: |z-2| \leq 4$}
    $B$={$z \in C: |z – 2| + |z + 2| = 5$}.
    Then the max {$|z_1-z_2|: z_1 \in A$ and $z_2 \in B$} is
    1. 8
    2. $\frac{15}{2}$
    3. 9
    4. $\frac{17}{2}$
19.   $\frac{6}{3^{26}}$+$10.\frac{1}{3^{25}}$+$10\frac{2}{3^{24}}$+...+$10.\frac{2^{24}}{3^1}$ is equal to:
     
    1. $3^{25}$
    2. $2^{25}$
    3. $3^{26}$
    4. $2^{26}$
20. The sum of all the elements in the range of $f(x) = Sgn(sinx)$ + $Sgn(cosx)$ + $Sgn(tanx)$ + $Sgn(cotx)$, $x \neq \frac{n \pi}{2}, n \in Z$, where $Sgn(t)$=$\left\{ \begin{array} {cc} 1, \text{ if t > 0} \\ -1, \text{if t < 0} \end{array} \right.$, is:
     
    1. 0
    2. 2
    3. -2
    4. 4

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. If $\sum \limits_{r=1}^{25} \frac{r}{r^4+r^2+1}$=$\frac{p}{q}$, where $p$ and $q$ positive integers such that $gcd(p, q)$=1, then $p+q$ is equal to .............
2. Three persons enter in a lift at the ground floor. The lift will go upto $10^{th}$ floor. The number of ways in which the three persons can exit the lift at three different floors, if the lift does not stop at $I^{st}$, $II^{nd}$ and $III^{rd}$ floor is equal to...............
3. If the distance of the point $P(43, \alpha, \beta) < 0$, from the line $\vec{r}$=$4\hat{i}-\hat{k}$+$\mu(2\hat{i}+3\hat{k})$, $\mu \in R$ along a line with direction ratios 3, -1, 0 is $13 \sqrt{10}$, then $\alpha^2+\beta^2$ is equal to .................
4. Let $f$ be a differentiable function satisfying $f(x)=1-2x$+$\int \limits_0^x e^{(x-t)}f(t)dt$, $x \in R$ and let $g(x)=\int \limits_0^x (f(t)+2)^{11}(t+12)^{17}(t-4)^6 dt$, $x \in R$. If $p$ and $q$ are respectively the points of the local minima and the local maxima of $g$, then the value of $|p+q|$ is equal to.............
   
5. Let matrix $A=\begin{equation*} \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \end{equation*}$ and $B$ be two matricessuch that $A^{100}=100B+I$. Then the sum of all the elements in $B^{100}$ is................

28 Jan 2026:Shift-1

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