JEE Main 2026 — 21 Jan Shift 1 (9:00 AM - 12:00 PM) (Previous Year

The number of relations, defined on the set $\{a, b, c, d\}$, which are both reflexive and symmetric, is equal to :

1. A.1024
2. B.64✓
3. C.16
4. D.256

Correct answer: B

If the domain of the function $f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1)$ is the interval $[\alpha, \beta]$, then $\alpha+2\beta$ is equal to :

1. A.3✓
2. B.5
3. C.1
4. D.2

Correct answer: A

The sum of all the roots of the equation $(x-1)^2-5|x-1|+6=0$, is :

1. A.5
2. B.3
3. C.4✓
4. D.1

Correct answer: C

If $x^2+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)^4+\left(x^2+\frac{1}{x^2}\right)^4+\left(x^3+\frac{1}{x^3}\right)^4+\ldots+\left(x^{25}+\frac{1}{x^{25}}\right)^4$ is :

1. A.128
2. B.175
3. C.145✓
4. D.162

Correct answer: C

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2 \cdot a_3 \cdot a_4=64$ and $a_1+a_3+a_5=\frac{813}{7}$. Then $a_3+a_5+a_7$ is equal to :

1. A.3256
2. B.3248
3. C.3244
4. D.3252✓

Correct answer: D

The number of strictly increasing functions $f$ from the set $\{1, 2, 3, 4, 5, 6\}$ to the set $\{1, 2, 3, \ldots, 9\}$ such that $f(i) \neq i$ for $1 \le i \le 6$, is equal to :

1. A.22
2. B.27
3. C.21
4. D.28✓

Correct answer: D

If the coefficient of $x$ in the expansion of $(ax^2+bx+c)(1-2x)^{26}$ is $-56$ and the coefficients of $x^2$ and $x^3$ are both zero, then $a+b+c$ is equal to :

1. A.1300
2. B.1500
3. C.1403✓
4. D.1483

Correct answer: C

Let the mean and variance of 7 observations $2, 4, 10, x, 12, 14, y$, $x>y$, be 8 and 16 respectively. Two numbers are chosen from $\{1, 2, 3, x-4, y, 5\}$ one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is :

1. A.$\frac{4}{5}$✓
2. B.$\frac{3}{5}$
3. C.$\frac{2}{5}$
4. D.$\frac{1}{3}$

Correct answer: A

Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

1. A.$24\sqrt{5}$
2. B.12
3. C.16
4. D.$\frac{96}{\sqrt{5}}$✓

Correct answer: D

Let a point A lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines $L_1$ and $L_2$, respectively, is :

1. A.$21\sqrt{3}$✓
2. B.$15\sqrt{6}$
3. C.27
4. D.$12\sqrt{2}$

Correct answer: A

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4x-6y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle AOB=\frac{\pi}{3}$. Then the locus of the point of intersection of the lines PQ and MN is :

1. A.$x^2+y^2-18x-12y-25=0$
2. B.$3(x^2+y^2)-18x-12y+25=0$
3. C.$3(x^2+y^2)-12x-18y-25=0$✓
4. D.$x^2+y^2-12x-18y-25=0$

Correct answer: C

Let O be the vertex of the parabola $x^2=4y$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio $2:3$ be the conic C. Then the equation of the chord of C, which is bisected at the point $(1, 2)$, is :

1. A.$5x-4y+3=0$✓
2. B.$x-2y+3=0$
3. C.$5x-y-3=0$
4. D.$4x-5y+6=0$

Correct answer: A

The value of $\operatorname{cosec}10^\circ-\sqrt{3}\sec 10^\circ$ is equal to :

1. A.8
2. B.2
3. C.6
4. D.4✓

Correct answer: D

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}-\hat{k})$. Then the length of the projection of the vector $\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ on the vector $6\hat{i}+2\hat{j}+3\hat{k}$ is :

1. A.3
2. B.$\frac{15}{7}$
3. C.$\frac{18}{7}$✓
4. D.4

Correct answer: C

Let $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c}+\vec{d}|=\sqrt{29}$ and $\vec{c}\times(2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times\vec{d}$. If $\lambda_1, \lambda_2$ $(\lambda_1>\lambda_2)$ are the possible values of $(\vec{c}+\vec{d})\cdot(-7\hat{i}+2\hat{j}+3\hat{k})$, then the equation $K^2x^2+(K^2-5K+\lambda_1)xy+\left(3K+\frac{\lambda_2}{2}\right)y^2-8x+12y+\lambda_2=0$ represents a circle, for K equal to :

1. A.2
2. B.-1
3. C.1✓
4. D.4

Correct answer: C

Let $\vec{a}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{b}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times\vec{c}=\vec{b}$. If $\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=4$, then $|\vec{a}+\vec{c}|^2$ is equal to :

1. A.33
2. B.35
3. C.27✓
4. D.30

Correct answer: C

The value of $\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin\left(|x|+\frac{\pi}{6}\right)}\right)dx$ is equal to :

1. A.$8\pi$
2. B.$2\pi$
3. C.$6\pi$
4. D.$4\pi$✓

Correct answer: D

The area of the region, inside the ellipse $x^2+4y^2=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is :

1. A.$2\pi-1$
2. B.$3(\pi-1)$
3. C.$2(\pi-1)$✓
4. D.$2\pi-\frac{1}{2}$

Correct answer: C

Let $y=y(x)$ be the solution curve of the differential equation $(1+x^2)dy+(y-\tan^{-1}x)\,dx=0$, $y(0)=1$. Then the value of $y(1)$ is :

1. A.$\frac{4}{e^{\pi/4}}-\frac{\pi}{2}-1$
2. B.$\frac{2}{e^{\pi/4}}+\frac{\pi}{4}-1$✓
3. C.$\frac{2}{e^{\pi/4}}-\frac{\pi}{4}-1$
4. D.$\frac{4}{e^{\pi/4}}+\frac{\pi}{2}-1$

Correct answer: B

Let $f:\mathbb{R}\to(0,\infty)$ be a twice differentiable function such that $f(3)=18$, $f'(3)=0$ and $f''(3)=4$. Then $\lim_{x\to1}\left(\log_e\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to :

1. A.2✓
2. B.1
3. C.18
4. D.9

Correct answer: A

For some $\alpha, \beta\in\mathbb{R}$, let $A=\begin{bmatrix}\alpha & 2\\ 1 & 2\end{bmatrix}$ and $B=\begin{bmatrix}1 & 1\\ 1 & \beta\end{bmatrix}$ be such that $A^2-4A+2I=B^2-3B+I=O$. Then $(\det(\operatorname{adj}(A^3-B^3)))^2$ is equal to ______.

Correct answer: 225

Let $a_1=1$ and for $n\ge1$, $a_{n+1}=\frac{1}{2}a_n+\frac{n^2-2n-1}{n^2(n+1)^2}$. Then $\left|\sum_{n=1}^{\infty}\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to ______.

Correct answer: 2

Let $S=\{(m, n): m, n\in\{1, 2, 3, \ldots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m+9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m+n$ is a square of a prime number is q, then $p+q$ is equal to ______.

Correct answer: 1333

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f'(x)m+f''(x)=0$ in $m$, has two equal roots for every $x\in\mathbb{R}$. If $f(0)=1$, $f'(0)=2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x-x)$ is increasing, then $\alpha+\beta$ is equal to ______.

Correct answer: 1

$6\int_0^{\pi}|(\sin3x+\sin2x+\sin x)|\,dx$ is equal to ______.

Correct answer: 17

Consider a modified Bernoulli equation. $\left(P+\frac{A}{Bt^2}\right)+\rho g(h+Bt)+\frac{1}{2}\rho V^2=\text{constant}$. If $t$ has the dimension of time then the dimensions of $A$ and $B$ are ______, ______ respectively.

1. A.$[ML^0T^{-1}]$ and $[M^0LT]$
2. B.$[ML^0T^{-2}]$ and $[M^0LT^{-1}]$
3. C.$[ML^0T^{-2}]$ and $[M^0LT^{-2}]$
4. D.$[ML^0T^{-1}]$ and $[M^0LT^{-1}]$✓

Correct answer: D

Potential energy $(V)$ versus distance $(x)$ is given by the graph. Rank various regions as per the magnitudes of the force $(F)$ acting on a particle from high to low.

1. A.$F_{CD}>F_{AB}>F_{BC}>F_{DE}$
2. B.$F_{CD}>F_{DE}>F_{AB}>F_{BC}$
3. C.$F_{BC}>F_{AB}>F_{DE}>F_{CD}$✓
4. D.$F_{BC}>F_{CD}>F_{DE}>F_{AB}$

Correct answer: C

A 4 kg mass moves under the influence of a force $\vec{F}=(4t^3\hat{i}-3t\hat{j})$ N where $t$ is the time in second. If mass starts from origin at $t=0$, the velocity and position after $t=2$ s will be :

1. A.$\vec{v}=3\hat{i}+\frac{3}{2}\hat{j}$, $\vec{r}=\frac{6}{5}\hat{i}+\hat{j}$
2. B.$\vec{v}=4\hat{i}-\frac{3}{2}\hat{j}$, $\vec{r}=\frac{6}{5}\hat{i}-\hat{j}$
3. C.$\vec{v}=4\hat{i}+\frac{5}{2}\hat{j}$, $\vec{r}=\frac{8}{5}\hat{i}+2\hat{j}$
4. D.$\vec{v}=4\hat{i}-\frac{3}{2}\hat{j}$, $\vec{r}=\frac{8}{5}\hat{i}-\hat{j}$✓

Correct answer: D

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^\circ$C. The coefficient of linear expansion of aluminium and steel are $24\times10^{-6}/^\circ$C and $1.2\times10^{-5}/^\circ$C, respectively. The length of this composite rod when its temperature is raised to $100^\circ$C, is ______ cm.

1. A.120.20
2. B.120.03
3. C.120.15✓
4. D.120.06

Correct answer: C

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at A and B are 6 cm$^2$ and 3 cm$^2$ respectively. The rate of flow will be ______ cm$^3$/s. (take $g=10$ m/s$^2$)

1. A.$200\sqrt{6}$
2. B.$100\sqrt{3}$
3. C.$\frac{200}{\sqrt{3}}$
4. D.$200\sqrt{3}$✓

Correct answer: D

The electric field in a plane electromagnetic wave is given by : $E_y=69\sin[0.6\times10^3x-1.8\times10^{11}t]$ V/m. The expression for magnetic field associated with this electromagnetic wave is ______ T.

1. A.$B_z=2.3\times10^{-7}\sin[0.6\times10^3x-1.8\times10^{11}t]$✓
2. B.$B_y=69\sin[0.6\times10^3x+1.8\times10^{11}t]$
3. C.$B_z=2.3\times10^{-7}\sin[0.6\times10^3x+1.8\times10^{11}t]$
4. D.$B_y=2.3\times10^{-7}\sin[0.6\times10^3x-1.8\times10^{11}t]$

Correct answer: A

Two strings $(A, B)$ having linear densities $\mu_A=2\times10^{-4}$ kg/m and $\mu_B=4\times10^{-4}$ kg/m and lengths $L_A=2.5$ m and $L_B=1.5$ m respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $t_1$ and $t_2$, respectively, to reach the joint. The ratio $t_1/t_2$ is :

1. A.1.08
2. B.1.90
3. C.1.18✓
4. D.1.67

Correct answer: C

Initially a satellite of 100 kg is in a circular orbit of radius $1.5R_E$. This satellite can be moved to a circular orbit of radius $3R_E$ by supplying $\alpha\times10^6$ J of energy. The value of $\alpha$ is ______. (Take Radius of Earth $R_E=6\times10^6$ m and $g=10$ m/s$^2$)

1. A.1000✓
2. B.150
3. C.100
4. D.500

Correct answer: A

A uniform rod of mass $m$ and length $l$ suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is ______. ($g$ acceleration due to gravity)

1. A.$mg/3$
2. B.$mg/2$
3. C.$mg/4$✓
4. D.$mg$

Correct answer: C

A parallel plate capacitor has capacitance C, when there is vacuum within the parallel plates. A sheet having thickness $\left(\frac{1}{3}\right)^{rd}$ of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is :

1. A.$\frac{3KC}{2K+1}$✓
2. B.$\frac{CK}{2+K}$
3. C.$\frac{3CK^2}{(2K+1)^2}$
4. D.$\frac{4KC}{3K-1}$

Correct answer: A

A current carrying solenoid is placed vertically and a particle of mass $m$ with charge $Q$ is released from rest. The particle moves along the axis of solenoid. If $g$ is acceleration due to gravity then the acceleration $(a)$ of the charged particle will satisfy :

1. A.$0<a<g$
2. B.$a>g$
3. C.$a=0$
4. D.$a=g$✓

Correct answer: D

A conducting circular loop of area 1.0 m$^2$ is placed perpendicular to a magnetic field which varies as $B=\sin(100t)$ Tesla. If the resistance of the loop is 100 $\Omega$, then the average thermal energy dissipated in the loop in one period is ______ J.

1. A.$2\pi$
2. B.$\pi$✓
3. C.$\pi^2$
4. D.$\frac{\pi}{2}$

Correct answer: B

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T. If the resistance of the total circuit is 2 $\Omega$ then the force needed to move the rod towards right with constant speed $(v)$ of 1.5 m/s is ______ N.

1. A.$5.7\times10^{-2}$
2. B.$7.5\times10^{-3}$✓
3. C.$7.5\times10^{-2}$
4. D.$5.7\times10^{-3}$

Correct answer: B

In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness $t$ and refractive index $n(=1.5)$, the central fringe shifts by 0.2 cm. The value of $t$ is ______ cm.

1. A.$8\times10^{-4}$✓
2. B.$6.0\times10^{-3}$
3. C.$5.0\times10^{-3}$
4. D.$5.6\times10^{-4}$

Correct answer: A

A light wave described by $E=60[\sin(3\times10^{15})t+\sin(12\times10^{15})t]$ (in SI units) falls on a metal surface of work function 2.8 eV. The maximum kinetic energy of ejected photoelectron is (approximately) ______ eV. $(h=6.6\times10^{-34}$ J.s. and $e=1.6\times10^{-19}$ C)

1. A.3.8
2. B.5.1✓
3. C.6.0
4. D.7.8

Correct answer: B

If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is ______ m. (Atomic number of gold $=79$ and $\frac{1}{4\pi\epsilon_0}=9\times10^9$ in SI units)

1. A.$2.95\times10^{-16}$
2. B.$3.85\times10^{-14}$
3. C.$2.95\times10^{-14}$✓
4. D.$3.85\times10^{-16}$

Correct answer: C

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from $27^\circ$C to $87^\circ$C. The rate of consumption of the gas is ______ g/s. (Take heat of combustion of gas $=5.0\times10^4$ J/g) specific heat capacity of water $=4200$ J/kg$^\circ$C

1. A.0.21
2. B.2.1
3. C.0.42✓
4. D.4.2

Correct answer: C

The given circuit works as :

1. A.NOR gate
2. B.OR gate
3. C.AND gate
4. D.NAND gate✓

Correct answer: D

A point charge of $10^{-8}$ C is placed at origin. The work done in moving a point charge 2 $\mu$C from point $A(4, 4, 2)$ m to $B(2, 2, 1)$ m is ______ J. ($\frac{1}{4\pi\epsilon_0}=9\times10^9$ in SI units)

1. A.$45\times10^{-6}$
2. B.0
3. C.$30\times10^{-6}$✓
4. D.$15\times10^{-6}$

Correct answer: C

In an experiment the values of two spring constants were measured as $k_1=(10\pm0.2)$ N/m and $k_2=(20\pm0.3)$ N/m. If these springs are connected in parallel, then the percentage error in equivalent spring constant is :

1. A.1.33%
2. B.1.67%✓
3. C.2.33%
4. D.2.67%

Correct answer: B

Two identical thin rods of mass $M$ kg and length $L$ m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{x}{12}ML^2$ kg m$^2$. The value of $x$ is ______.

Correct answer: 17

A collimated beam of light of diameter 2 mm is propagating along $x$-axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. If first lens has focal length 40 mm, then the focal length of second lens is ______ mm.

Correct answer: 280

In a microscope the objective is having focal length $f_o=2$ cm and eye-piece is having focal length $f_e=4$ cm. The tube length is 32 cm. The magnification produced by this microscope for normal adjustment is ______.

Correct answer: 100

The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1 $\Omega$ is connected across these points is ______ J.

Correct answer: 1080

10 mole of oxygen is heated at constant volume from $30^\circ$C to $40^\circ$C. The change in the internal energy of the gas is ______ cal. (The molecular specific heat of oxygen at constant pressure, $C_p=7$ cal/mol$^\circ$C and $R=2$ cal/mol$^\circ$C.)

Correct answer: 500

80 mL of a hydrocarbon on mixing with 264 mL of oxygen in a closed U-tube undergoes complete combustion. The residual gases after cooling to 273 K occupy 224 mL. When the system is treated with KOH solution, the volume decreases to 64 mL. The formula of the hydrocarbon is :

1. A.C$_2$H$_4$
2. B.C$_2$H$_6$
3. C.C$_2$H$_2$✓
4. D.C$_4$H$_{10}$

Correct answer: C

14.0 g of calcium metal is allowed to react with excess HCl at 1.0 atm pressure and 273 K. Which of the following statements is **incorrect** ? [Given : Molar mass in g mol$^{-1}$ of Ca $-$ 40, Cl $-$ 35.5, H $-$ 1]

1. A.0.35 mol of H$_2$ gas is evolved.
2. B.7.84 L of H$_2$ gas is evolved.
3. C.The limiting reagent is calcium metal.
4. D.33.3 g of CaCl$_2$ is produced.✓

Correct answer: D

Given below are two statements : Statement I : The number of species among SF$_4$, NH$_4^+$, [NiCl$_4$]$^{2-}$, XeF$_4$, [PtCl$_4$]$^{2-}$, SeF$_4$ and [Ni(CN)$_4$]$^{2-}$, that have tetrahedral geometry is 3. Statement II : In the set [NO$_2$, BeH$_2$, BF$_3$, AlCl$_3$], all the molecules have incomplete octet around central atom. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is false but Statement II is true✓
4. D.Statement I is true but Statement II is false

Correct answer: C

Given below are two statements : Statement I : When an electric discharge is passed through gaseous hydrogen, the hydrogen molecules dissociate and the energetically excited hydrogen atoms produce electromagnetic radiation of discrete frequencies. Statement II : The frequency of second line of Balmer series obtained from He$^+$ is equal to that of first line of Lyman series obtained from hydrogen atom. In the light of the above statements, choose the correct answer from the options given below :

1. A.Statement I is true but Statement II is false
2. B.Both Statement I and Statement II are true✓
3. C.Statement I is false but Statement II is true
4. D.Both Statement I and Statement II are false

Correct answer: B

For the reaction, N$_2$O$_4$ $\rightleftharpoons$ 2NO$_2$, graph is plotted as shown below. Identify **correct** statements. A. Standard free energy change for the reaction is $-5.40$ kJ mol$^{-1}$. B. As $\Delta G^\ominus$ in graph is positive, N$_2$O$_4$ will not dissociate into NO$_2$ at all. C. Reverse reaction will go to completion. D. When 1 mole of N$_2$O$_4$ changes into equilibrium mixture, value of $\Delta G^\ominus=-0.84$ kJ mol$^{-1}$. E. When 2 mole of NO$_2$ changes into equilibrium mixture, $\Delta G^\ominus$ for equilibrium mixture is $-6.24$ kJ mol$^{-1}$. Choose the correct answer from the options given below :