JEE Main 2026 (Online) 21st January Evening Shift

A spherical body of radius r and density falls freely through a viscous liquid having density and viscosity and attains a terminal velocity v0. Estimated maximum error in the quantity is : (Ignore errors associated with , and g, gravitational acceleration)

1. 2 [ rr - v0v0 ]
2. 2 [ rr + v0v0 ]
3. 2 rr + v0v0
4. 2 rr - v0v0

Keeping the significant figures in view, the sum of the physical quantities 52.01 m, 153.2 m and 0.123 m is :

1. 205.3 m
2. 205 m
3. 205.333 m
4. 205.33 m

A large drum having radius R is spinning around its axis with angular velocity , as shown in figure. The minimum value of so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M as , is : [figure]

1. 2g R
2. g2 R
3. gR
4. g R

Two known resistances of R and 2R and one unknown resistance X are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is X , then the value of X is . [figure]

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

The r.m.s. speed of oxygen molecules at 47 is equal to that of the hydrogen molecules kept at . (Mass of oxygen molecule/mass of hydrogen molecule = 32/2)

1. -100
2. -253
3. -20
4. -235

The total length of potentiometer wire AB is 50 cm in the arrangement as shown in the figure. If P is the point where the galvanometer shows zero reading then the length AP is cm. [figure]

1. 25
2. 30
3. 15
4. 20

Surface tension of two liquids (having same densities), T1 and T2, are measured using capillary rise method utilizing two tubes with inner radii of r1 and r2 where r1 > r2. The measured liquid heights in these tubes are h1 and h2 respectively. [Ignore the weight of the liquid above the lowest point of meniscus]. The heights h1 and h2 and surface tensions T1 and T2 satisfy the relation :

1. h1 > h2 and T1 = T2
2. h1 < h2 and T1 = T2
3. h1 > h2 and T1 < T2
4. h1 = h2 and T1 = T2

A particle having electric charge 3 10-19 C and mass 6 10-27 kg is accelerated by applying an electric potential of 1.21 V. Wavelength of the matter wave associated with the particle is 10-12 m. The value of is . (Take Planck's constant = 6.6 10-34 J s)

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

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is Hz. [take = 227]

1. 88
2. 14
3. 28
4. 176

Two cars A and B each of mass 103 kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is J s.

1. 3.6 105
2. 105
3. 3 105
4. 2 105

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, = 10 mho/m. The ratio of maximum conduction current density to maximum displacement current density is . [Take 14 0 = 9 109 N m2/C2]

The terminal velocity of a metallic ball of radius 6 mm in a viscous fluid is 20 cm/s. The terminal velocity of another ball of same material and having radius 3 mm in the same fluid will be cm/s.

An infinitely long straight wire carrying current I is bent in a planar shape as shown in the diagram. The radius of the circular part is r. The magnetic field at the centre O of the circular loop is : [figure]

1. - 02 Ir( + 1) i
2. - 02 Ir( - 1) i
3. 02 Ir( + 1) i
4. 02 Ir( - 1) i

A diatomic gas ( = 1.4) does 100 J of work when it is expanded isobarically. Then the heat given to the gas J.

The energy of an electron in an orbit of the Bohr's atom is -0.04E0 eV where E0 is the ground state energy. If L is the angular momentum of the electron in this orbit and h is the Planck's constant, then 2 Lh is :

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

In a Young's double slit experiment set up, the two slits are kept 0.4 mm apart and screen is placed at 1 m from slits. If a thin transparent sheet of thickness 20 is introduced in front of one of the slits then center bright fringe shifts by 20 mm on the screen. The refractive index of transparent sheet is given by 10, where is .

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface. [figure] If refractive index of the material of prism is 2, the angle of prism is.

1. 75
2. 90
3. 60
4. 45

A body of mass 2 kg is moving along x-direction such that its displacement as function of time is given by x(t) = t2 + t + m, where = 1 m/s2, = 1 m/s and = 1 m. The work done on the body during the time interval t = 2 s to t = 3 s, is J.

1. 42
2. 24
3. 49
4. 12

A battery with EMF E and internal resistance r is connected across a resistance R. The power consumption in R will be maximum when:

1. R = 2r
2. R = 2r
3. R = r2
4. R = r

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is: (Take k = 14 0 and assume kQ2 > Gm2 where G is the Gravitational constant)

1. kQ24mR (1 + Gm2kQ2 )
2. kQ22mR (1 - Gm22kQ2 )
3. kQ22mR (1 - Gm2kQ2 )
4. kQ24mR (1 - Gm2kQ2 )

A capacitor C is first charged fully with potential difference of V0 and disconnected from the battery. The charged capacitor is connected across an inductor having inductance L. In t s, 25% of the initial energy in the capacitor is transferred to the inductor. The value of t is s.

1. LC3
2. LC2
3. LC2
4. LC6

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are and respectively.

1. 20 s and 100 m
2. 40 s and 100 m
3. 40 s and 200 m
4. 40 s and 0 m

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitude of the acceleration experienced by the blocks is (assume no slipping of string on pulley). [figure]

1. (M - m)g [ ( 83 )M + m ]
2. (M - m)g2M + m
3. (M - m)gM + m
4. (M - m)g [ ( 136 )M + m ]

The charge stored by the capacitor C in the given circuit in the steady state is . [figure]

1. 7.5
2. 5
3. 10
4. 12.5

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

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

On heating a mixture of common salt and K2Cr2O7 in equal amount along with concentrated H2SO4 in a test tube, a gas is evolved. Formula of the gas evolved and oxidation state of the central metal atom in the gas respectively are:

1. Cr2O2Cl2 and +3
2. Cr2O2Cl2 and +6
3. CrO2Cl2 and +6
4. CrO2Cl2 and +5

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

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

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

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

Given below are some of the statements about Mn and Mn2O7. Identify the correct statements. Choose the correct answer from the options given below :

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

[figure] Consider the above sequence of reactions. The number of bromine atom(s) in the final product (P) will be :

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

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

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

The correct increasing order of C-H(A), C-O(B), C=O(C) and C N(D) bonds in terms of covalent bond length is :

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

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

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

For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature?

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

Consider the following data : f H (methane, g) = -X kJ mol-1 Enthalpy of sublimation of graphite = Y kJ mol-1 Dissociation enthalpy of H2 = Z kJ mol-1 The bond enthalpy of C–H bond is given by :

1. X+Y+2Z4
2. X+Y+4Z2
3. -X+Y+Z4
4. X+Y+Z

The correct order of the rate of the reaction for the following reaction with respect to nucleophiles is : CH3Br + Nu- CH3Nu + Br-

1. ClO4- > CH3COO- > OH- > PhO-
2. OH- > PhO- > CH3COO- > ClO4-
3. CH3COO- > PhO- > OH- > ClO4-
4. PhO- > OH- > CH3COO- > ClO4-

By usual analysis, 1.00 g of compound (X) gave 1.79 g of magnesium pyrophosphate. The percentage of phosphorus in compound (X) is : (nearest integer) (Given, molar mass in g mol-1: O = 16, Mg = 24, P = 31)

1. 40
2. 30
3. 20
4. 50

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

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

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

Decomposition of A is a first order reaction at T(K) and is given by A(g) B(g) + C(g). In a closed 1 L vessel, 1 bar A(g) is allowed to decompose at T(K). After 100 minutes, the total pressure was 1.5 bar. What is the rate constant (in min-1) of the reaction? ( 2 = 0.3)

1. 6.9 10-4
2. 6.9 10-3
3. 6.9 10-1
4. 6.9 10-2

Given below are four compounds : (a) n-propyl chloride (b) isopropyl chloride (c) sec-butyl chloride (d) neopentyl chloride Percentage of carbon in the one which exhibits optical isomerism is :

1. 46
2. 40
3. 52
4. 56

The correct order of reactivity of the following benzyl halides towards reaction with KCN is : [figure]

1. a > b > c > d
2. b > a > d > c
3. b > a > c > d
4. a > b > d > c

Consider the following spectral lines for atomic hydrogen : The correct arrangement of the above lines in ascending order of energy is :

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

Aqueous HCl reacts with MnO2(s) to form MnCl2(aq), Cl2(g), and H2O(l). What is the weight (in g) of Cl2 liberated when 8.7 g of MnO2(s) is reacted with excess aqueous HCl solution? (Given Molar mass in g mol-1 Mn = 55, Cl = 35.5, O = 16, H = 1)

1. 14.2
2. 21.3
3. 7.1
4. 71

Identify the metal ions among Co2+, Ni2+, Fe2+, V3+ and Ti2+ having a spin-only magnetic moment value more than 3.0 BM. The sum of unpaired electrons present in the high spin octahedral complexes formed by those metal ions is .

The osmotic pressure of a living cell is 12 atm at 300 K. The strength of sodium chloride solution that is isotonic with the living cell at this temperature is g L-1. (Nearest integer) Given : R = 0.08 L atm K-1 mol-1 Assume complete dissociation of NaCl (Given: Molar mass of Na and Cl are 23 and 35.5 g mol-1 respectively.)

The first and second ionization constants of H2X are 2.5 10-8 and 1.0 10-13 respectively. The concentration of X2- in 0.1 M H2X solution is 10-15 M. (Nearest Integer)

A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y' (molar mass = 300 g mol-1) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is 10-2. (nearest integer) [Given: Kb of the solvent = 5.0 K kg mol-1] Assume the solution to be dilute and no association or dissociation of X takes place in solution.

MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K. MX(s) M+(aq) + X-(aq); Ksp = 10-10 If the standard reduction potential for M+(aq) + e- M(s) is ( E M+/M ) = 0.79 V, then the value of the standard reduction potential for the metal/metal insoluble salt electrode E X-/MX(s)/M is mV. (nearest integer) [Given: 2.303RTF = 0.059 V]

If the line x + 4y = 7, where R, touches the ellipse 3x2 + 4y2 = 1 at the point P in the first quadrant, then one of the focal distances of P is :

1. 1 3 - 12 11
2. 1 3 - 12 5
3. 1 3 + 12 5
4. 1 3 + 12 7

Let and be the roots of the equation x2 + 2ax + (3a + 10) = 0 such that < 1 < . Then the set of all possible values of a is :

1. (- , -115 ) (5, )
2. (- , -115 )
3. (- , -3)
4. (- , -2) (5, )

Let y2 = 12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that OPA = 90 . Then the locus of the centroid of such triangles OPA is:

1. y2 - 4x + 8 = 0
2. y2 - 2x + 8 = 0
3. y2 - 9x + 6 = 0
4. y2 - 6x + 4 = 0

If the system of equations 3x + y + 4z = 3 2x + y - z = -3 x + 2y + z = 4 has no solution, then the value of is equal to:

1. 13
2. 4
3. 19
4. 23

Let z be the complex number satisfying |z - 5| 3 and having maximum positive principal argument. Then 34 | 5z - 125iz + 16 |2 is equal to:

1. 20
2. 26
3. 12
4. 16

If the area of the region \(x, y) : 1 - 2x y 4 - x2, x 0, y 0\ is , , N, ( , ) = 1, then the value of ( + ) is:

1. 73
2. 85
3. 67
4. 91

The positive integer n, for which the solutions of the equation x(x + 2) + (x + 2)(x + 4) + + (x + 2n - 2)(x + 2n) = 8n3 are two consecutive even integers, is :

1. 3
2. 6
3. 9
4. 12

Let one end of a focal chord of the parabola y2 = 16x be (16, 16). If P( , ) divides this focal chord internally in the ratio 5 : 2, then the minimum value of + is equal to:

1. 5
2. 7
3. 22
4. 16

Let A = \x : |x2 - 10| 6\ and B = \x : |x - 2| > 1\. Then

1. A B = (- , 1] (2, )
2. B - A = (- , -4) (-2, 1) (4, )
3. A - B = [2, 3)
4. A B = [-4, -2] [3, 4]

Let the line L pass through the point (-3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the point (-2, r, 1) is 143, then the sum of all possible values of r is :

1. 16
2. 12
3. 6
4. 10

Let f : R R be a twice differentiable function such that f''(x) > 0 for all x R and f'(a - 1) = 0, where a is a real number. Let g(x) = f( 2 x - 2 x + a), 0 < x < 2. Consider the following two statements: (I) g is increasing in (0, 4 ) (II) g is decreasing in ( 4, 2 ) Then,

1. Both (I) and (II) are True
2. Neither (I) nor (II) is True
3. Only (I) is True
4. Only (II) is True

Let a1, a22, a322, , a1029 be a G.P. of common ratio 1 2. If a1 + a2 + + a10 = 62, then a1 is equal to:

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

Let y = y(x) be the solution of the differential equation x dydx - 2y = 2 + 3 x, x (- 2, 2 ), y(0) = - 74. Then y ( 6 ) is equal to :

1. - 52
2. -3 2 - 7
3. - 54
4. -3 3 - 7

Let f(x) = x3 + x2 f'(1) + 2x f''(2) + f'''(3), x R. Then the value of f'(5) is :

1. 6575
2. 1175
3. 25
4. 625

Let A = \2, 3, 5, 7, 9\. Let R be the relation on A defined by xRy if and only if 2x 3y. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:

1. 21
2. 25
3. 23
4. 27

For the matrices A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B = bmatrix -29 & 49 \ -13 & 18 bmatrix, if (A15 + B) bmatrix x \ y bmatrix = bmatrix 0 \ 0 bmatrix, then among the following which one is true?

1. x = 16, y = 3
2. x = 5, y = 7
3. x = 11, y = 2
4. x = 18, y = 11

A random variable X takes values 0, 1, 2, 3 with probabilities 2a+130, 8a-130, 4a+130, b respectively, where a, b R. Let and respectively be the mean and standard deviation of X such that 2 + 2 = 2. Then ab is equal to:

1. 12
2. 60
3. 30
4. 3

For a triangle ABC, let p = BC, q = CA and r = BA. If | p| = 2 3, | q| = 2 and = 1 3, where is the angle between p and q, then | p ( q - 3 r)|2 + 3| r|2 is equal to :

1. 200
2. 220
3. 410
4. 340

The largest n N, for which 7n divides 101!, is :

1. 18
2. 15
3. 19
4. 16

Let the line L1 be parallel to the vector -3 i + 2 j + 4 k and pass through the point (2, 6, 7), and the line L2 be parallel to the vector 2 i + j + 3 k and pass through the point (4, 3, 5). If the line L3 is parallel to the vector -3 i + 5 j + 16 k and intersects the lines L1 and L2 at the points C and D, respectively, then | CD |2 is equal to:

1. 290
2. 171
3. 89
4. 312

Let [ ] denote the greatest integer function and f(x) = n 1n3 k=1n [ k23x ]. Then 12 j=1 f(j) is equal to .

If P is a point on the circle x2 + y2 = 4, Q is a point on the straight line 5x + y + 2 = 0 and x - y + 1 = 0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is .

If ( 115C0 + 115C1 ) ( 115C1 + 115C2 ) ( 115C12 + 115C13 ) = 1314C0 14C1 14C12, then 30 is equal to .

If 01 4 -1 (1 - 2x + 4x2 )dx = a -1(2) - b e(5), where a, b N, then (2a + b) is equal to .

Let the maximum value of ( -1x )2 + ( -1x )2 for x [- 32, 1 2 ] be mn 2, where (m, n) = 1. Then m + n is equal to .