Download JEE Main 2026 Question Paper (24 Jan - Shift 1) Mathematics

» Hold on briefly for download button.

Wait-starts soon.

 Download Now!

28 Jan 2026:Shift-1

28 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 729, 81, 9, 1, …… be a sequence and $P_n$ denotes the product of $I^{st}$ $n$ terms of this sequence. If $2\sum \limits_{n=1}^{40}(P_n)^{\frac{1}{n}}$=$\frac{3^{\alpha}-1}{3^{\beta}}$ and $gcd(\alpha, \beta)$=1, then $(\alpha+\beta)$ is equal to:
    
   1. 73
   2. 75
   3. 76
   4. 74
2. The value of $\frac{\sqrt{3} cosec20°-\sec20°}{\cos20°\cos40°\cos60°\cos80°}$ is equal to:
    
   1. 32
   2. 64
   3. 12
   4. 16
3. Let the lines $L_1: \vec{r}=(\hat{i}+2\hat{j}+3\hat{k})$+$\lambda(2\hat{i}+3\hat{j}+4\hat{k})$, $\lambda \in R$ and $L_2:\vec{r}=(4\hat{i}+\hat{j})+\mu(5\hat{i}+2\hat{j}+\hat{k})$, $\mu \in R$ intersect at the point $R$. Let $P$ and $Q$ be the points lying on the line $L_1$ & $L_2$ respectively, such that $|\vec{PR}|=\sqrt{29}$ and $|\vec{PQ}|$=$\sqrt{\frac{47}{3}}$. If the point $P$ lies is the first octant then $27(QR)^2$ is equal to:
    
   1. 340
   2. 348
   3. 360
   4. 320
4. The number of the real solutions of the equation $x |x–3| + |x–1| +3$ = 0 is:
    
   1. 5
   2. 4
   3. 2
   4. 3
5. Let $A_1$ be the bounded area enclosed by $y = x^2 + 2$, $x + y = 8$, $y-$axis that lies in the $I^{st}$ quadrant. Let $A_2$ be the bounded area enclosed by the curves $y = x^2 + 2$, $y^2 = x$, $x = 2$ and $y-$ axis that lies in the $1^{st}$ quadrant. Then $(A_1– A_2)$ is equal to:
    
   1. $\frac{2}{3}\left(4\sqrt{2}+1\right)$
   2. $\frac{2}{3}\left(3\sqrt{2}+1\right)$
   3. $\frac{2}{3}\left(2\sqrt{2}+1\right)$
   4. $\frac{2}{3}\left(\sqrt{2}+1\right)$
6. Let $R$ be a relation defined on the set {1, 2, 3, 4} × {1, 2, 3, 4} by $R$={$((a, b), (c, d)): 2a+3b=3c+4d$}. Then the number of elements in $R$ is
    
   1. 18
   2. 12
   3. 6
   4. 15
7. Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=\hat{i}+\hat{j}$, $\vec{c}=\vec{a}×\vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|$=$\sqrt{11}$, $|\vec{c}×\vec{d}|$=3 & the angle between $\vec{c}$ and $\vec{d}$ is $\frac{\pi}{4}$. Then $\vec{a}•\vec{d}$ is
    
   1. 1
   2. 3
   3. 11
   4. 0
8. Let each of the two ellipse $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a > b)$ and $E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$, $(A < B)$ have eccentricity $\frac{4}{5}$. Let the length of latus recta of $E_1$ and $E_2$ be $l_1$  and $l_2$, respectively, such that $(2l_1^2=9l_2)$. If the distance between the focii of $E_1$ is 8, then the distance between the focii of $E_2$ is
    
   1. $\frac{16}{5}$
   2. $\frac{96}{5}$
   3. $\frac{8}{5}$
   4. $\frac{32}{5}$
9. Let $S$=$\left\{z \in C: \left|\frac{z-6i}{z-2i}\right|=1\right.$ & $\left. \left|\frac{z-8+2i}{z+2i}\right|=\frac{3}{5} \right\}$
   then $\sum \limits_{z \in S} |z|^2$ is equal to
    
   1. 398
   2. 385
   3. 423
   4. 413
10. Let $f(t)=\int \left( \frac{1-\sin(ln t)}{1-\cos(ln t)}\right)dt$, $t > 1$. If $f(e^{\frac{\pi}{2}})$=$-e^{\frac{\pi}{2}}$ and , then $f\left(e^{\frac{\pi}{4}}\right)$=$\alpha e^{\frac{\pi}{4}}$, then $\alpha$ equals
     
    1. $-1+\sqrt{2}$
    2. $-1-2\sqrt{2}$
    3. $-1-\sqrt{2}$
    4. $1+\sqrt{2}$
11. Let $S$=$\frac{1}{25!}$+$\frac{1}{3!23!}$+$\frac{1}{5!21!}$+...+ upto 13 terms. If $13S=\frac{2^k}{n!}$, $k \in N$, then $n+k$ ie equal to
     
    1. 52
    2. 51
    3. 49
    4. 50
12. Let $\alpha, \beta \in R$ be such that the function $f(x)$=$\left \{ \begin{array} {cc} 2\alpha(x^2-2)+2\beta x, x < 1 \\ (\alpha+3)x+(\alpha-\beta), x \geq 1 \end{array} \right.$ be differentiable at all $x \in R$. Then $34(\alpha+\beta)$ is equal to
     
    1. 48
    2. 84
    3. 24
    4. 36
13. The mean & variance of a data of 10 observations are 10 & 2 respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then new mean & variance become 10.1 & 1.99. Then $(\alpha+ \beta)$ equals
     
    1. 10
    2. 15
    3. 20
    4. 5
14. If the function $f(x)$= $\frac{e^x(e^{\tan x-x}-1)+\ln (\sec x+\tan x)-x}{(\tan x-x)}$ is continuous at $x=0$, then the value of $f(0)$ is equal to
     
    1. $\frac{2}{3}$
    2. $\frac{3}{2}$
    3. $2$
    4. $\frac{1}{2}$
15. From a lot containing 10 defective & 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is
     
    1. $\left(\frac{67}{10^8}\right)$
    2. $\left(\frac{73}{10^8}\right)$
    3. $\left(\frac{7}{10^7}\right)$
    4. $\left(\frac{81}{10^8}\right)$
16. Consider an A.P.: $a_1$, $a_2$, ..., $a_n$; $a_1 > 0$, $a_2-a_1=\frac{-3}{4}$, $a_n=\frac{a_1}{4}$, $\sum \limits_{i=1}^{a}a_i=\frac{525}{2}$ then $\sum \limits_{i=1}^{17}a_i$ is equal to:
     
    1. 136
    2. 476
    3. 238
    4. 952
17. Let a circle of radius 4 pass through the origin $O$, the points $A(-\sqrt{3}a, 0)$ & $B= (0, -\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab \neq 0$. Then the locus of the centroid of $\Delta OAB$ is a circle of radius
     
    1. $\frac{8}{3}$
    2. $\frac{5}{3}$
    3. $\frac{11}{3}$
    4. $\frac{7}{3}$
18. Let $A(1, 0)$, $B(2, –1)$ and $C\left(\frac{7}{3}, \frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle $ABC$ is $\alpha x + \beta y$ = 5 then value of $(\alpha^2 + \beta^2)$ is:
     
    1. 10
    2. 8
    3. 13
    4. 5
19. If the domain of the function $f(x)=\log_{(10x^2-17x+7)}(18x^2-11x+1)$ is $(-\infty, a) \cup (b, c) \cup (d, \infty)$ - {e}, then $90(a+b+c+d+e)$ equals:
     
    1. 177
    2. 170
    3. 307
    4. 316
20. If $\cot x=\frac{5}{12}$ for some $x \in \left(\pi, \frac{3\pi}{2}\right)$ then $\sin 7x\left(\cos \frac{13x}{2}+\sin \frac{13x}{2}\right)$+$\cos 7x\left(\cos \frac{13x}{2}-\sin \frac{13x}{2}\right)$ is equal to:
     
    1. $\frac{1}{\sqrt{13}}$
    2. $\frac{4}{\sqrt{26}}$
    3. $-\frac{5}{\sqrt{13}}$
    4. $\frac{6}{\sqrt{26}}$

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. The number of 3×2 matrices $A$, which can be formed using the elements of the set {-2, -1, 0, 1, 2} such that the sum of all the diagonal elements of $A^TA$ is 5, is ..............
2. Let a line $L$ passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4}$=$\frac{y-1}{1}$=$\frac{z-1}{1}$ and $\frac{x-17}{1}$=$\frac{y-71}{1}$=$\frac{z}{0}$. Let the line $L$ intersect the $yz-$ plane at the point $Q$. Another line parallel to $L$ and passing through the point $S(1, 0, -1)$ intersects the $yz-$ plane at the point $R$. Then the square of the area of the parallelogram $PQRS$ is equal to ................
3. Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t)$=$\left|\frac{t+1}{t^2}\right|$; $(t < 0)$ is strictly increasing. Then the local maximum value of the function $g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha$, $x >2$ is .................
4. Let a differentiable function $f$ satisfy the equation $\int \limits_0^{36}f\left(\frac{tx}{36}\right)dt$=$4\alpha f(x)$. If $y=f(x)$ is standard parabola passing through (2, 1) and $(–4, \beta)$. Then value of $\beta^{\alpha}$ is ........
5. The number of numbers greater then 5000, less than 9000 and divisible by 3 that can be formed by using the digits 0, 1, 2, 5, 9, it the repetition is allowed, is..........

24 Jan 2026:Shift-2

24 Jan 2026:Ans-Keys

All the Papers Ans-Keys