JEE Main 2026 (Online) 23rd January Morning Shift

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is cm. [Assume that the mass of the pendulum remains same.]

1. 0.75
2. 7.5
3. 15
4. 120

The de Broglie wavelength of an oxygen molecule at 27 is x 10-12 m. The value of x is (take Planck's constant = 6.63 10-34 J s, Boltzmann constant = 1.38 10-23 J/K, mass of oxygen molecule = 5.31 10-26 kg)

1. 24
2. 30
3. 20
4. 26

The strain-stress plot for materials A, B, C and D is shown in the figure. Which material has the largest Young's modulus ? [figure]

1. B
2. A
3. D
4. C

The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at Vz = 5 V and the desired current in load is 5 mA. The unregulated voltage source can supply up to 25 V. Considering the Zener diode can withstand four times the load current, the value of resistor Rs (shown in circuit) should be . [figure]

1. 100
2. 10
3. 1000
4. 4000

A 20 m long uniform copper wire held horizontally is allowed to fall under the gravity (g = 10 m/s2) through a uniform horizontal magnetic field of 0.5 Gauss perpendicular to the length of the wire. The induced EMF across the wire when it travels a vertical distance of 200 m is mV.

1. 200 10
2. 0.2 10
3. 20 10
4. 2 10

Match List-I with List-II. Choose the correct answer from the options given below :

1. A-I, B-IV, C-III, D-II
2. A-II, B-III, C-IV, D-I
3. A-IV, B-I, C-II, D-III
4. A-II, B-III, C-I, D-IV

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is :

1. 43 Lmd2
2. 32 Lmd2
3. 2L5md2
4. 2Lmd2

An object is projected with kinetic energy K from a point A at an angle 60 with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction is : [figure]

1. 3 : 4
2. 1 : 4
3. 2 : 3
4. 1 : 2

In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg, moving in opposite directions with speeds of 10 m/s and 30 m/s, respectively, strike each other and stick together. The rise in temperature (in ), if all the heat produced during the collision is retained by these spheres, is : (specific heat of sphere material 31 cal/(kg C) and 1 cal = 4.2 J)

1. 1.75
2. 1.44
3. 1.95
4. 1.15

In hydrogen atom spectrum, (R Rydberg's constant) Choose the correct answer from the options given below :

1. A, B Only
2. B, D Only
3. A, B and D Only
4. A, B and C Only

Two point charges 2q and q are placed at vertex A and centre of face CDEF of the cube as shown in figure. The electric flux passing through the cube is: [figure]

1. 3q2 0
2. 3q4 0
3. q 0
4. 3q 0

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (R < L) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M):

1. 34MR2 + 16ML2
2. 38MR2 + 712ML2
3. 34MR2 + 712ML2
4. 38MR2 + 16ML2

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Consider a ferromagnetic material : In the light of the above statements, choose the correct answer from the options given below :

1. (A) is true but (R) is false
2. Both (A) and (R) are true and (R) is the correct explanation of (A)
3. (A) is false but (R) is true
4. Both (A) and (R) are true but (R) is not the correct explanation of (A)

A small bob A of mass m is attached to a massless rigid rod of length 1 m pivoted at point P and kept at an angle of 60 with vertical as shown in figure. At distance of 1 m below point P, an identical bob B is kept at rest on a smooth horizontal surface that extends to a circular track of radius R as shown in figure. If bob B just manages to complete the circular path of radius R upto a point Q after being hit elastically by bob A, then radius R is m. [figure]

1. 15
2. 2 - 35
3. 35
4. 2 + 35

A thin prism with angle 5 of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is .

1. 5
2. 4
3. 4.5
4. 6

A wire of uniform resistance /m is bent into a circle of radius r and another piece of wire with length 2r is connected between points A and B (AOB) as shown in figure. The equivalent resistance between points A and B is . [figure]

1. ( + 1)2r
2. 6 r3 + 16
3. 3 r8
4. 2 r

Two blocks with masses 100 g and 200 g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E. The energy stored in B, when spring constants kA, kB of A and B, respectively satisfy the relation 4kA = 3kB, is : [figure]

1. 4E
2. 2E
3. 43E
4. 3E

In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is mm. [Assume least count is 0.01 mm]

1. 1.54
2. 1.50
3. 1.51
4. 1.48

Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are 2.4 108 m/s and 2.7 108 m/s, respectively, then the value of critical angle is :

1. -1 ( 89 )
2. -1 ( 98 )
3. -1 ( 3 13 )
4. -1 ( 8 17 )

Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is :

1. 0.18
2. 0.24
3. 0.06
4. 0.08

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of 60 with vertical, then maximum induced EMF between the point of suspension and point of oscillation is mV. (Take g = 10 m/s2)

The space between the plates of a parallel plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants K1 = 2, K2 = 3 and K3 = 5 (as shown in figure). If new capacitance is n3C then the value of n is . [figure]

The equation of the electric field of an electromagnetic wave propagating through free space is given by : E = 377 (6.27 103 t - 2.09 10-5 x) N/C The average intensity of the electromagnetic wave is ( 1 ) W/m2. The value of α is . (Take 0 0 = 377 in SI units )

In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are 2 : 1 and 1 : 2 respectively. The corresponding ratio of the distances between the slits and the respective screens (D1/D2) is .

Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for V = 5 (100t). The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is . [figure]

Consider the general reaction given below at 400 K x A(g) y B(g). The values of Kp and Kc are studied under the same condition of temperature but variation in x and y. (i) Kp = 85.87 and Kc = 2.586 appropriate units (ii) Kp = 0.862 and Kc = 28.62 appropriate units The values of x and y in (i) and (ii) respectively are :

1. array|c|c| (i) & (ii) \ 1,2 & 2,1 \ array
2. array|c|c| (i) & (ii) \ 1,3 & 2,1 \ array
3. array|c|c| (i) & (ii) \ 3,1 & 3,1 \ array
4. array|c|c| (i) & (ii) \ 4,1 & 4,1 \ array

Which one of the following graphs accurately represents the plot of partial pressure of CS2 vs its mole fraction in a mixture of acetone and CS2 at constant temperature?

1. [figure]
2. [figure]
3. [figure]
4. [figure]

In the given electrochemical cell, Ag(s)|AgCl(s)|FeCl2(aq), FeCl3(aq) | Pt(s) at 298 K, the cell potential ( Ecell) will increase when : Choose the correct answer from the options given below :

1. C, D and E Only
2. A and B Only
3. B Only
4. A and E Only

Match List - I with List - II.

1. A-IV, B-I, C-II, D-III
2. A-III, B-IV, C-I, D-II
3. A-III, B-IV, C-II, D-I
4. A-IV, B-III, C-II, D-I

'x' is the product which is obtained from propanenitrile and stannous chloride in the presence of hydrochloric acid followed by hydrolysis. 'y' is the product which is obtained from the but-2-ene by the ozonolysis followed by hydrolysis. From the following, which product is not obtained when one mole of 'x' and one mole of 'y' react with each other in the presence of alkali followed by heating?

1. 2-Methylpent-2-enal
2. 2-Methylbut-2-enal
3. 3-Methylbut-2-enal
4. Pent-2-enal

A cup of water at 5 C (system) is placed in a microwave oven and the oven is turned on for one minute during which the water begins to boil. Which of the following options is true?

1. q = +ve, w = -ve, U = +ve
2. q = +ve, w = 0, U = -ve
3. q = +ve, w = -ve, U = -ve
4. q = -ve, w = -ve, U = -ve

Given, (A) n = 5, ml = -1 (B) n = 3, l = 2, ml = -1, ms = + 12 The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) respectively are :

1. 4 and 1
2. 2 and 4
3. 26 and 1
4. 8 and 1

The correct trend in the first ionization enthalpies of the elements in the 3rd period of periodic table is :

1. Si < S < Al < P < Cl
2. Al < Si < S < P < Cl
3. S < Si < Al < P < Cl
4. Al < S < P < Si < Cl

The correct statements from the following are : Choose the correct answer from the options given below :

1. B and D Only
2. C and D Only
3. A and D Only
4. A and C Only

Identify the molecule (X) with maximum number of lone pairs of electrons (obtained using Lewis dot structure) among HNO3, H2SO4, NF3 and O3. Choose the correct bond angle made by the central atom of the molecule (X).

1. 107
2. 102
3. 120
4. 116

From the given following (A to D) cyclic structures, those which will not react with Tollens' reagent are: [figure]

1. A and D
2. A and B
3. B and D
4. B and C

[figure] The correct sequence of reagents for the above conversion of X to Y is :

1. (i) NaOH(aq), (ii) Jones reagent, (iii) H3O+
2. (i) NaOEt, (ii) B2H6/H2O2, (iii) Jones reagent
3. (i) Jones reagent, (ii) NaOEt, (iii) Hot KMnO4/KOH
4. (i) B2H6/H2O2, (ii) NaOEt, (iii) Jones reagent

Consider the following sequence of reactions. [figure] Assuming that the reaction proceeds to completion, then 137 mg of 4-nitrotoluene will produce mg of B. (Given molar mass in g mol-1 H : 1, C : 12, N : 14, O : 16, Br : 80)

1. 208
2. 228
3. 146
4. 301

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are false
2. Statement I is true but Statement II is false
3. Both Statement I and Statement II are true
4. Statement I is false but Statement II is true

Compound 'P' undergoes the following sequence of reactions : [figure] 'P' is :

1. [figure]
2. [figure]
3. [figure]
4. [figure]

Consider the following compounds : [figure] Arrange these compounds in the increasing order of reactivity with nitrating mixture.

1. c < b < a
2. b < c < a
3. c < a < b
4. b < a < c

The statements that are incorrect about the nickel(II) complex of dimethylglyoxime are : Choose the correct answer from the options given below :

1. C and E Only
2. A, D and B Only
3. B, C and E Only
4. C and D Only

But-2-yne and hydrogen (one mole each) are separately treated with (i) Pd/C and (ii) Na/ liq. NH3 to give the products X and Y respectively. [figure] Identify the incorrect statements. Choose the correct answer from the options given below :

1. B and D Only
2. A and C Only
3. A and B Only
4. B and C Only

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are false
2. Both Statement I and Statement II are true
3. Statement I is false but Statement II is true
4. Statement I is true but Statement II is false

Which of the following statements regarding the energy of the stationary state is true in the following one-electron systems ?

1. +2.18 10-18 J for second orbit of He+ ion
2. +8.72 10-18 J for first orbit of He+ ion
3. -1.09 10-18 J for second orbit of H atom.
4. -2.18 10-18 J for third orbit of Li2+ ion

The crystal field stabilisation energy of [Co(oxalate)3]3- complex is 'n' times that of the [Cr(oxalate)3]3- complex. Here 'n' is . (Assume o P)

For the following gas phase equilibrium reaction at constant temperature, NH3(g) 1/2 N2(g) + 3/2 H2(g) if the total pressure is 3 atm and the pressure equilibrium constant (Kp) is 9 atm, then the degree of dissociation is given as (x 10-2 )-1/2. The value of x is . (nearest integer)

For the thermal decomposition of reactant AB(g), the following plot is constructed. [figure] The half-life of the reaction is 'x' min. x = min. (Nearest integer)

x mg of pure HCl was used to make an aqueous solution. 25.0 mL of 0.1 M Ba(OH)2 solution is used when the HCl solution was titrated against it. The numerical value of x is 10-1. (Nearest integer) Given : Molar mass of HCl and Ba(OH)2 are 36.5 and 171.0 g mol-1 respectively.

Consider all the structural isomers with molecular formula C5H11Br are separately treated with KOH(aq) to give respective substitution products, without any rearrangement. The number of products which can exhibit optical isomerism from these is .

Let the domain of the function f(x) = 3 5 7 (9x - x2 - 13 ) be the interval (m, n). Let the hyperbola x2a2 - y2b2 = 1 have eccentricity n3 and the length of the latus rectum 8m3. Then b2 - a2 is equal to :

1. 7
2. 9
3. 11
4. 5

A rectangle is formed by the lines x = 0, y = 0, x = 3 and y = 4. Let the line L be perpendicular to 3x + y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point ( 12, -5 ) from the line L is equal to :

1. 10
2. 2 5
3. 2 10
4. 3 10

Let the direction cosines of two lines satisfy the equations : 4l + m - n = 0 and 2mn + 10nl + 3lm = 0. Then the cosine of the acute angle between these lines is :

1. 107 38
2. 10 38
3. 103 38
4. 203 38

If α and β (α < β) are the roots of the equation (-2 + 3 ) ( | x - 3 | ) + (x - 6 x ) + (9 - 2 3 ) = 0, x 0, then + is equal to :

1. 8
2. 10
3. 9
4. 11

Let S = \z : 3 |2z - 3(1 + i)| 7 \ be a set of complex numbers. Then Minz S | (z + 12(5 + 3i) ) | is equal to :

1. 12
2. 52
3. 2
4. 32

Let f(x) = (2 - x2 ) ex ( 1 + x )(1 - x)3/2 dx. If f(0) = 0, then f ( 12 ) is equal to:

1. 2e - 1
2. 2e + 1
3. 3e - 1
4. 3e + 1

The vertices B and C of a triangle ABC lie on the line x1 = 1 - y-2 = z - 23. The coordinates of A and B are (1, 6, 3) and (4, 9, α) respectively and C is at a distance of 10 units from B. The area (in sq. units) of ABC is :

1. 20 13
2. 5 13
3. 15 13
4. 10 13

Let the mean and variance of 8 numbers −10, −7, −1, x, y, 9, 2, 16 be 72 and 2934, respectively. Then the mean of 4 numbers x, y, x + y + 1, |x − y| is :

1. 11
2. 9
3. 10
4. 12

Number of solutions of 3 2 + 8 + 3 3 = 0, [-3 , 2 ] is :

1. 5
2. 4
3. 3
4. 0

Let α and β respectively be the maximum and the minimum values of the function f( ) = 4 ( 4 ( 7 2 - ) + 4(11 + ) ) - 2 ( 6 ( 3 2 - ) + 6(9 - ) ), R. Then α + 2β is equal to :

1. 6
2. 5
3. 4
4. 3

The value of the integral 24 5 24 dx1 + [3] 2x is :

1. 3
2. 18
3. 6
4. 12

Let a = - i + j + 2 k, b = i - j - 3 k, c = a b and d = c a. Then ( a - b ) d is equal to :

1. 2
2. -2
3. 4
4. -4

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

1. 36 days
2. 24 days
3. 30 days
4. 42 days

Among the statements : I: If vmatrix 1 & & \ & 1 & \ & & 1 vmatrix = vmatrix 0 & & \ & 0 & \ & & 0 vmatrix, then 2 + 2 + 2 = 32, and II: If vmatrix x2 + x & x + 1 & x - 2 \ 2x2 + 3x - 1 & 3x & 3x - 3 \ x2 + 2x + 3 & 2x - 1 & 2x - 1 vmatrix = px + q, then p2 = 196q2.

1. both are true
2. both are false
3. only I is true
4. only II is true

Let the line y - x = 1 intersect the ellipse x22 + y21 = 1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

1. - -1 ( 14 )
2. 2 + -1 ( 14 )
3. 2 + 2 -1 ( 14 )
4. 2 - -1 ( 14 )

The sum of all possible values of n N, so that the coefficients of x, x2 and x3 in the expansion of (1 + x2)2(1 + x)n, are in arithmetic progression is :

1. 12
2. 9
3. 3
4. 7

The value of 100C5051 + 100C5152 + + 100C100101 is:

1. 2101101
2. 2100101
3. 2100100
4. 2101100

Let y = y(x) be the solution of the differential equation x4 dy + (4x3y + 2 x) dx = 0, x > 0, y ( 2 ) = 0. Then 4 y ( 3 ) is equal to :

1. 92
2. 72
3. 64
4. 81

Let A = \-2, -1, 0, 1, 2, 3, 4\. Let R be a relation on A defined by xRy if and only if 2x + y 2. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l + m + n is equal to :

1. 34
2. 32
3. 33
4. 35

Let f(x) = cases ax2 + 2ax + 34x2 + 4x - 3 & , x - 32, 12 \ b & , x = - 32, 12 cases be continuous at x = - 32. If f f(x) = 75, then x is equal to:

1. 1.4
2. 2
3. 1
4. 0

Let |A| = 6, where A is a 3 3 matrix. If |adj (3 adj (A2 adj(2A) ) ) | = 2m 3n, m, n N, then m + n is equal to .

Let the area of the region bounded by the curve y = \ x, x\, lines x = 0, x = 3 2, and the x-axis be A. Then, A + A2 is equal to .

Let f be a twice differentiable non-negative function such that (f(x))2 = 25 + 0x ((f(t))2 + (f'(t))2 )dt. Then the mean of f( e(1)), f( e(2)), , f( e(625)) is equal to .

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that a - b 10 is mn, (m, n) = 1, then m + n is equal to .

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is .