JEE Main 2026 — 24 Jan Shift 2 (3:00 PM - 6:00 PM) (Previous Year

If the domain of the function $f(x)=\sin^{-1}\left(\frac{1}{x^2-2x-2}\right)$, is $(-\infty, \alpha]\cup[\beta, \gamma]\cup[\delta, \infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to

1. A.5
2. B.2
3. C.4✓
4. D.3

Correct answer: C

Let f be a function such that $3f(x)+2f\left(\frac{m}{19x}\right)=5x$, $x\ne0$, where $m=\sum_{i=1}^{9}(i)^2$. Then $f(5)-f(2)$ is equal to

1. A.18✓
2. B.9
3. C.-9
4. D.36

Correct answer: A

The smallest positive integral value of a, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to

1. A.3
2. B.9
3. C.7✓
4. D.4

Correct answer: C

Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order 3 such that $q_{ij}=2^{(i+j-1)}p_{ij}$ and $\det(Q)=2^{10}$. Then the value of $\det(\operatorname{adj}(\operatorname{adj}P))$ is:

1. A.81
2. B.16✓
3. C.32
4. D.124

Correct answer: B

Let $a_1, a_2, a_3, a_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $a_1+a_2+a_3+a_4=48$ and $a_1a_2a_3a_4+l^4=361$, then the largest term of the A.P. is equal to

1. A.27✓
2. B.23
3. C.24
4. D.21

Correct answer: A

$\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{3^2}+\frac{1}{3}\times\frac{4}{7}+\frac{4^2}{7^2}\right)+\left(\frac{1}{3^3}+\frac{1}{3^2}\times\frac{4}{7}+\frac{1}{3}\times\frac{4^2}{7^2}+\frac{4^3}{7^3}\right)+\ldots$ upto infinite terms, is equal to

1. A.$\frac{7}{4}$
2. B.$\frac{4}{3}$
3. C.$\frac{6}{5}$
4. D.$\frac{5}{2}$✓

Correct answer: D

The largest value of n, for which $40^n$ divides $60!$, is

1. A.13
2. B.11
3. C.14✓
4. D.12

Correct answer: C

The letters of the word \"UDAYPUR\" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word \"UDAYPUR\" is

1. A.1578
2. B.1579
3. C.1580✓
4. D.1581

Correct answer: C

Let $X=\{x\in\mathbb{N}: 1\le x\le19\}$ and for some $a, b\in\mathbb{R}$, $Y=\{ax+b: x\in X\}$. If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of b is

1. A.60✓
2. B.80
3. C.100
4. D.20

Correct answer: A

Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b)$, be 30. If its eccentricity is the maximum value of the function $f(t)=-\frac{3}{4}+2t-t^2$, then $(a^2+b^2)$ is equal to

1. A.276
2. B.516
3. C.256
4. D.496✓

Correct answer: D

Let the image of parabola $x^2=4y$, in the line $x-y=1$ be $(y+a)^2=b(x-c)$, $a, b, c\in\mathbb{N}$. Then $a+b+c$ is equal to

1. A.4
2. B.6✓
3. C.12
4. D.8

Correct answer: B

Let the angles made with the positive $x$-axis by two straight lines drawn from the point P(2, 3) and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1+\theta_2)$ is:

1. A.$\frac{\pi}{6}$
2. B.$\frac{\pi}{2}$✓
3. C.$\frac{\pi}{12}$
4. D.$\frac{\pi}{3}$

Correct answer: B

The sum of all values of $\alpha$, for which the shortest distance between the lines $\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$ and $\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha}$ is $\sqrt{2}$, is

1. A.6
2. B.-6✓
3. C.-8
4. D.8

Correct answer: B

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$, then $|\vec{c}\times\hat{k}|^2$ is equal to

1. A.193
2. B.218✓
3. C.205
4. D.233

Correct answer: B

Let $[t]$ denote the greatest integer less than or equal to t. If the function $f(x)=\begin{cases}b^2\sin\left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x)\cos x\right]\right), & x<0\\ \frac{\sin x-\frac{1}{2}\sin2x}{x^3}, & x>0\\ a, & x=0\end{cases}$ is continuous at $x=0$, then $a^2+b^2$ is equal to

1. A.$\frac{3}{4}$✓
2. B.$\frac{1}{2}$
3. C.$\frac{5}{8}$
4. D.$\frac{9}{16}$

Correct answer: A

Consider the following three statements for the function $f:(0, \infty)\to\mathbb{R}$ defined by $f(x)=|\log_ex|-|x-1|$: (I) f is differentiable at all $x>0$. (II) f is increasing in (0, 1). (III) f is decreasing in $(1, \infty)$. Then.

1. A.All (I), (II) and (III) are TRUE.
2. B.Only (II) and (III) are TRUE.
3. C.Only (I) and (III) are TRUE.✓
4. D.Only (I) is TRUE.

Correct answer: C

Let $f(x)=\int\frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}dx$, $x>0$, $\lim_{x\to0}f(x)=0$ and $f(1)=\frac{1}{4}$. If $A=\begin{bmatrix}0 & 0 & 1\\ \frac{1}{4} & f'(1) & 1\\ \alpha^2 & 4 & 1\end{bmatrix}$ and $B=\operatorname{adj}(\operatorname{adj}A)$ be such that $|B|=81$, then $\alpha^2$ is equal to

1. A.1
2. B.2
3. C.3
4. D.4✓

Correct answer: D

Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0$, $x=1$, $y^2=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha x^2$. Then $(f(0)+f(1))$ is equal to

1. A.12
2. B.9
3. C.7✓
4. D.14

Correct answer: C

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $\lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3$ for each $x>0$. Then $2y(2)$ is equal to

1. A.23✓
2. B.12
3. C.18
4. D.27

Correct answer: A

Let $z=(1+i)(1+2i)(1+3i)\ldots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then n is equal to ______

Correct answer: 5

Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set $P(S)\times P(S)$ such that $A\cap B=\emptyset$. If the probability of the event E is $\frac{3^p}{2^q}$, where $p, q\in\mathbb{N}$, then $p+q$ is equal to ______

Correct answer: 15

Let $(h, k)$ lie on the circle $C: x^2+y^2=4$ and the point $(2h+1, 3k+2)$ lie on an ellipse with eccentricity e. Then the value of $\frac{5}{e^2}$ is equal to ______.

Correct answer: 9

The number of elements in the set $\{x\in[0, 180^\circ]: \tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ)\}$ is ______.

Correct answer: 4

If $f(x)$ satisfies the relation $f(x)=e^x+\int_0^1(y+xe^x)f(y)dy$, then $e+f(0)$ is equal to ______.

Correct answer: 2

In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division = 0.05 mm, then the least count of the vernier callipers is ______ mm.

1. A.0.02
2. B.0.005
3. C.0.002✓
4. D.0.05

Correct answer: C

A thin uniform rod (X) of mass M and length L is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is ______. ($g$ = gravitational acceleration)

1. A.$\sqrt{\frac{3g}{2L}}$
2. B.$\frac{3}{\sqrt{2}}\sqrt{\frac{g}{L}}$
3. C.$\sqrt{\frac{3g}{L}}$✓
4. D.$\frac{1}{\sqrt{2}}\sqrt{\frac{g}{L}}$

Correct answer: C

The velocity (v) $-$ Distance (x) graph is shown in figure. Which graph represents acceleration (a) versus distance (x) variation of this system?

1. A.Straight line from a negative value rising to zero✓
2. B.Straight line from zero going increasingly negative
3. C.Constant (horizontal line)
4. D.Straight line rising from a positive intercept

Correct answer: A

A flexible chain of mass m hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^\circ$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is ______.

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

Correct answer: B

In case of vertical circular motion of a particle by a thread of length r if the tension in the thread is zero at an angle $30^\circ$ shown in figure, the velocity at the bottom point (A) of the circular path is ($g$ = gravitational acceleration)

1. A.$\sqrt{\frac{7}{2}gr}$✓
2. B.$\sqrt{4gr}$
3. C.$\sqrt{5gr}$
4. D.$\sqrt{\frac{5}{2}gr}$

Correct answer: A

A cubical block of density $\rho_b=600$ kg/m$^3$ floats in a liquid of density $\rho_e=900$ kg/m$^3$. If the height of block is $H=8.0$ cm then height of the submerged part is ______ cm.

1. A.5.3✓
2. B.6.3
3. C.7.3
4. D.4.3

Correct answer: A

10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha$ Joule ($P_1=21.7$ Pa and $P_2=30$ Pa, $C_v=21$ J/K.mol, R = 8.3 J/mol.K). The value of $\alpha$ is ______.

1. A.15
2. B.21✓
3. C.28
4. D.24

Correct answer: B

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is 5/x. The value of x is ______.

1. A.2✓
2. B.3
3. C.4
4. D.1

Correct answer: A

A moving coil galvanometer of resistance 100 $\Omega$ shows a full scale deflection for a current of 1 mA. The value of resistance required to convert this galvanometer into an ammeter, showing full scale deflection for a current of 5 mA, is ______ $\Omega$

1. A.25✓
2. B.2.5
3. C.10
4. D.0.5

Correct answer: A

Two identical circular loops P and Q each of radius r are lying in parallel planes such that they have common axis. The current through P and Q are I and 4I respectively in clockwise direction as seen from O. The net magnetic field at O is:

1. A.$\frac{\mu_0I}{4\sqrt{2}r}$ towards Q
2. B.$\frac{\mu_0I}{4\sqrt{2}r}$ towards P
3. C.$\frac{3\mu_0I}{4\sqrt{2}r}$ towards P
4. D.$\frac{3\mu_0I}{4\sqrt{2}r}$ towards Q✓

Correct answer: D

A regular hexagon is formed by six wires each of resistance r $\Omega$ and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be

1. A.$\frac{4}{5}r$✓
2. B.$\frac{3}{4}r$
3. C.$\frac{3}{5}r$
4. D.$\frac{5}{8}r$

Correct answer: A

Three parallel plate capacitors each with area A and separation d are filled with two dielectric ($k_1$ and $k_2$) in the following fashion. Which of the following is true? $(k_1>k_2)$

1. A.$C_B>C_C>C_A$
2. B.$C_C>C_A>C_B$
3. C.$C_C>C_B>C_A$
4. D.$C_A>C_C>C_B$✓

Correct answer: D

A point source is kept at the center of a spherically enclosed detector. If the volume of the detector increased by 8 times, the intensity will

1. A.increase by 8 times
2. B.increase by 64 times
3. C.decrease by 4 times✓
4. D.decrease by 8 times

Correct answer: C

Five persons P$_1$, P$_2$, P$_3$, P$_4$ and P$_5$ recorded object distance (u) and image distance (v) using same convex lens having power +5D as (25,96), (30,62), (35,37), (45,35) and (50,32) respectively. Identify correct statement

1. A.Readings recorded by P$_4$ and P$_5$ persons are incorrect
2. B.Readings recorded by P$_3$ and P$_2$ persons are incorrect
3. C.Readings recorded by all persons are correct
4. D.Readings recorded by P$_3$ person are incorrect✓

Correct answer: D

In the Young's double slit experiment the intensity produced by each one of the individual slits is $I_0$. The distance between two slits is 2 mm. The distance of screen from slits is 10 m. The wavelength of light is 6000 $\AA$. The intensity of light on the screen in front of one of the slits is ______.

1. A.$I_0$✓
2. B.$2I_0$
3. C.$\frac{I_0}{2}$
4. D.$4I_0$

Correct answer: A

Distance between an object and three times magnified real image is 40 cm. The focal length of the mirror used is ______ cm.

1. A.$-15/2$
2. B.-10
3. C.-20
4. D.-15✓

Correct answer: D

When a light of a given wavelength falls on a metallic surface the stopping potential for photoelectrons is 3.2 V. If a second light having wavelength twice of first light is used, the stopping potential drops to 0.7 V. The wavelength of first light is ______ m. ($h=6.63\times10^{-34}$ J.s, $e=1.6\times10^{-19}$ C, $c=3\times10^8$ m/s)

1. A.$2.2\times10^{-8}$
2. B.$3.1\times10^{-7}$
3. C.$2.5\times10^{-7}$✓
4. D.$2.9\times10^{-8}$

Correct answer: C

The binding energy for the following nuclear reactions are expressed in MeV. $_2$He$^3$ + $_0$n$^1$ $\to$ $_2$He$^4$ + 20 MeV; $_2$He$^4$ + $_0$n$^1$ $\to$ $_2$He$^5$ $-$ 0.9 MeV. If X$_3$, X$_4$, X$_5$ denote the stability of $_2$He$^3$, $_2$He$^4$ and $_2$He$^5$, respectively, then the correct order is :

1. A.X$_4$ > X$_5$ > X$_3$✓
2. B.X$_4$ = X$_5$ = X$_3$
3. C.X$_4$ > X$_5$ < X$_3$
4. D.X$_4$ < X$_5$ < X$_3$

Correct answer: A

Identify the correct truth table of the given logic circuit.

1. A.A B Y: 0 0 0; 0 1 0; 1 0 1; 1 1 0✓
2. B.0 0 0; 0 1 1; 1 0 1; 1 1 0
3. C.0 0 1; 0 1 0; 1 0 1; 1 1 0
4. D.0 0 1; 0 1 1; 1 0 1; 1 1 0

Correct answer: A

The reading of the ammeter (A) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is ______ A.

1. A.2
2. B.1/2
3. C.0
4. D.1✓

Correct answer: D

A point charge $q=1$ $\mu$C is located at a distance 2 cm from one end of a thin insulating wire of length 10 cm having a charge $Q=24$ $\mu$C, distributed uniformly along its length, as shown in figure. Force between q and wire is ______ N. (Use : $\frac{1}{4\pi\epsilon_0}=9\times10^9$ N.m$^2$/C$^2$)

Correct answer: 90

A soap bubble of surface tension 0.04 N/m is blown to a diameter of 7 cm. If $(15000-x)$ $\mu$J of work is done in blowing it further to make its diameter 14 cm, then the value of x is ______. ($\pi=22/7$)

Correct answer: 11304

A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length L/2 and radius R/3 carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is ______.

Correct answer: 162

In a meter bridge experiment to determine the value of unknown resistance, first the resistances 2 $\Omega$ and 3 $\Omega$ are connected in the left and right gaps of the bridge and the null point is obtained at a distance $l$ cm from the left. Now when an unknown resistance x $\Omega$ is connected in parallel to 3 $\Omega$ resistance, the null point is shifted by 10 cm to the right of wire. The value of unknown resistance x is ______ $\Omega$.

Correct answer: 6

One mole of Cl$_2$(g) was passed into 2 L of cold 2M KOH solution. After the reaction, the concentrations of Cl$^-$, ClO$^-$ and OH$^-$ are respectively (assume volume remains constant)

1. A.1M, 1M, 1M
2. B.0.5M, 0.5M, 0.5M
3. C.0.5M, 0.5M, 1M✓
4. D.0.75M, 0.75M, 1M

Correct answer: C

The wavelength of spectral line obtained in the spectrum of Li$^{2+}$ ion, when the transition takes place between two levels whose sum is 4 and difference is 2, is

1. A.$1.14\times10^{-7}$ cm
2. B.$2.28\times10^{-7}$ cm
3. C.$2.28\times10^{-6}$ cm
4. D.$1.14\times10^{-6}$ cm✓

Correct answer: D

Pair of species among the following having same bond order as well as paramagnetic character will be-

1. A.O$_2^-$, N$_2^-$
2. B.O$_2^+$, N$_2^{2-}$
3. C.O$_2^-$, N$_2^+$
4. D.O$_2^+$, N$_2^-$✓

Correct answer: D

The heat of atomisation of methane and ethane are 'x' kJ mol$^{-1}$ and 'y' kJ mol$^{-1}$ respectively. The longest wavelength ($\lambda$) of light capable of breaking the C$-$C bond can be expressed in SI unit as:

1. A.$\frac{hc}{1000}\left(\frac{y-6x}{4}\right)^{-1}$
2. B.$\frac{N_Ahc}{250(y-6x)}$
3. C.$N_Ahc\left(y-\frac{6x}{4}\right)^{-1}$
4. D.$\frac{N_Ahc}{250(4y-6x)}$✓

Correct answer: D

At 298 K, the mole percentage of N$_2$(g) in air is 80%. Water is in equilibrium with air at a pressure of 10 atm. What is the mole fraction of N$_2$(g) in water at 298 K? ($K_H$ for N$_2$ is $6.5\times10^7$ mm Hg)

1. A.$9.35\times10^{-5}$✓
2. B.$1.17\times10^{-4}$
3. C.$9.35\times10^{5}$
4. D.$1.23\times10^{-7}$

Correct answer: A

Two liquids A and B form an ideal solution at temperature T K. At T K, the vapour pressures of pure A and B are 55 and 15 kN m$^{-2}$ respectively. What is the mole fraction of A in solution of A and B in equilibrium with a vapour in which the mole fraction of A is 0.8?

1. A.0.340
2. B.0.663
3. C.0.480
4. D.0.5217✓

Correct answer: D

Consider the following gaseous equilibrium in a closed container of volume 'V' at T(K). P$_2$(g) + Q$_2$(g) $\rightleftharpoons$ 2PQ(g). 2 moles each of P$_2$(g), Q$_2$(g) and PQ(g) are present at equilibrium. Now one mole each of 'P$_2$' and 'Q$_2$' are added to the equilibrium keeping the temperature at T(K). The number of moles of P$_2$, Q$_2$ and PQ at the new equilibrium, respectively, are

1. A.1.21, 2.24, 1.56
2. B.2.67, 2.67, 2.67✓
3. C.1.66, 1.66, 1.66
4. D.2.56, 1.62, 2.24

Correct answer: B

The correct order of C, N, O and F in terms of second ionisation potential is

1. A.C < N < F < O✓
2. B.F < N < C < O
3. C.C < O < N < F
4. D.C < F < N < O

Correct answer: A

Choose the INCORRECT statement

1. A.Carbon exhibits negative oxidation states along with +4 and +2.
2. B.CO$_2$ is the most acidic oxide among the dioxides of group of 14 elements.
3. C.Among the isotopes of carbon, $^{13}$C is a radioactive isotope.✓
4. D.Carbon cannot exceed its covalency more than four.

Correct answer: C

\"X\" is an oxoanion of the lightest element of group 7 (in the periodic table). The metal is in +6 oxidation state in \"X\". The color of the potassium salt of X is

1. A.purple
2. B.green✓
3. C.orange
4. D.yellow

Correct answer: B

The wavelength of light absorbed for the following complexes are in the order [Co(NH$_3$)$_6$]$^{3+}$ (I); [Co(H$_2$O)$_6$]$^{3+}$ (II); [Co(CN)$_6$]$^{3-}$ (III); [Co(NH$_3$)$_5$(H$_2$O)]$^{3+}$ (IV); [CoF$_6$]$^{3-}$ (V)

1. A.III < I < IV < II < V✓
2. B.III < I < II < IV < V
3. C.III < IV < I < II < V
4. D.III < I < IV < V < II

Correct answer: A

Find out the statements which are **not** true. A. Resonating structures with more number of covalent bonds and lesser charge separation are more stable. B. In electromeric effect, an unsaturated system shows +E effect with nucleophile and $-$E effect with electrophile. C. Inductive effect is responsible for high melting point, boiling point and dipole moment of polar compounds. D. The greater the number of alkyl groups attached to the doubly bonded carbon atoms, higher is the heat of hydrogenation. E. Stability of carbanion increases with the increase in s $-$ character of the carbon carrying the negative charge. Choose the correct answer from the options given below:

1. A.B, D & E only
2. B.A, D & E only
3. C.B & D only✓
4. D.A, C & D only

Correct answer: C

Given below are two statements: Statement I: There are several conformers for n$-$butane. Out of those conformers, (X) [Newman projection with the two CH$_3$ groups eclipsed] is the least stable and most stable conformer is (Y) [Newman projection with the two CH$_3$ groups anti, 180 degrees apart]. Statement II: As the dihedral angle increases, torsional strain decreases from (X) to (Y). In the light of the above statements, choose the correct answer from the options given below