JEE Main 2026 (Online) 24th January Morning Shift

A cylindrical block of mass M and area of cross section A is floating in a liquid of density and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is

1. 2 AMg
2. 2M Ag
3. 2 M Ag
4. AMg

The electrostatic potential in a charged spherical region of radius r varies as V = ar3 + b, where a and b are constants. The total charge in the sphere of unit radius is a 0. The value of is . (permittivity of vacuum is 0)

1. -12
2. -8
3. -9
4. -6

The exit surface of a prism with refractive index n is coated with a material having refractive index n2. When this prism is set for minimum angle of deviation, it exactly meets the condition of critical angle. The prism angle is .

1. 60
2. 30
3. 15
4. 45

A brass wire of length 2 m and radius 1 mm at 27 C is held taut between two rigid supports. Initially it was cooled to a temperature of -43 C creating a tension T in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to 1.4T, is C.

1. -71
2. -80
3. -65
4. -86

A spring of force constant 15 N/m is cut into two pieces. If the ratio of their lengths is 1 : 3, then the force constant of smaller piece is N/m.

1. 20
2. 45
3. 60
4. 15

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

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

Two resistors of 100 each are connected in series with a 9 V battery. A voltmeter of 400 resistance is connected to measure the voltage drop across one of the resistors. The voltmeter reading is V.

1. 3
2. 2
3. 4
4. 4.5

An unpolarised light is incident at an interface of two dielectric media having refractive indices of 2 (incident medium) and 2 3 (refracted medium) respectively. To satisfy the condition that reflected and refracted rays are perpendicular to each other, the angle of incidence is

1. 45
2. 60
3. 10
4. 30

Two electrons are moving in orbits of two hydrogen-like atoms with speeds 3 105 m/s and 2.5 105 m/s respectively. If the radii of these orbits are nearly same then the possible order of energy states are respectively.

1. 8 and 10
2. 10 and 12
3. 9 and 8
4. 6 and 5

In a microscope of tube length 10 cm two convex lenses are arranged with focal lengths of 2 cm and 5 cm. Total magnification obtained with this system for normal adjustment is (5)k. The value of k is .

1. 4
2. 5
3. 2
4. 3.5

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

For the series LCR circuit connected with 220 V, 50 Hz a.c. source as shown in the figure, the power factor is 10. The value of is . [figure]

1. 4
2. 8
3. 6
4. 10

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is kg m2. (g = 9.8 m/s2)

1. 8.3 10-3
2. 4.75 10-3
3. 1.86 10-2
4. 9.5 10-3

Three masses 200 kg, 300 kg and 400 kg are placed at the vertices of an equilateral triangle with sides 20 m. They are rearranged on the vertices of a bigger triangle of side 25 m and with the same centre. The work done in this process is J. (Gravitational constant G = 6.7 10-11 N m2/kg2)

1. 4.77 10-7
2. 1.74 10-7
3. 9.86 10-6
4. 2.85 10-7

Two resistors 2 and 3 are connected in the gaps of bridge as shown in figure. The null point is obtained with the contact of jockey at some point on wire XY. When an unknown resistor is connected in parallel with 3 resistor, the null point is shifted by 22.5 cm toward Y. The resistance of unknown resistor is . [figure]

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

There are three concentric conducting spherical shells A, B and C of radii a, b and c respectively (c > b > a) and they are charged with charges q1, q2 and q3 respectively. The potentials of the spheres A, B and C respectively, are :

1. 14 0 ( q1a + q2b + q3c ), 14 0 ( q1 + q2 + q3b ), 14 0 ( q1 + q2 + q3c )
2. 14 0 ( q1 + q2 + q3a ), 14 0 ( q1 + q2 + q3b ), 14 0 ( q1 + q2 + q3c )
3. 14 0 ( q1 + q2 + q3a ), 14 0 ( q1 + q2b + q3c ), 14 0 ( q1a + q2b + q3c )
4. 14 0 ( q1a + q2b + q3c ), 14 0 ( q1 + q2b + q3c ), 14 0 ( q1 + q2 + q3c )

Three charges +2q, +3q and -4q are situated at (0, -3a), (2a, 0) and (-2a, 0) respectively in the xy plane. The resultant dipole moment about origin is .

1. 2qa(7 i - 3 j)
2. 2qa(3 j - 7 i)
3. 2qa(3 i - 7 j)
4. 2qa(3 j - i)

A boy throws a ball into air at 45 from the horizontal to land it on a roof of a building of height H. If the ball attains maximum height in 2 s and lands on the building in 3 s after launch, then value of H is m. (g = 10 m/s2)

1. 20
2. 25
3. 10
4. 15

Densities of water at 4 and 20 are 1000 kg/m3 and 998 kg/m3 respectively. The increase in internal energy of 4 kg of water when it is heated from 4 to 20 is J. (specific heat capacity of water = 4.2 kJ/kg K and 1 atmospheric pressure = 105 Pa)

1. 268799.2
2. 315826.2
3. 234699.2
4. 258700.8

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

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

In the given figure the blocks A, B and C weigh 4 kg, 6 kg and 8 kg respectively. The coefficient of sliding friction between any two surfaces is 0.5. The force F required to slide the block C with constant speed is N. (Use g = 10 m/s2) [figure]

A short bar magnet placed with its axis at 30 with an external field of 800 Gauss, experiences a torque of 0.016 N m. The work done in moving it from most stable to most unstable position is 10-3 J. The value of is .

A voltage regulating circuit consisting of Zener diode, having breakdown voltage of 10 V and maximum power dissipation of 0.4 W, is operated at 15 V. The approximate value of protective resistance in this circuit is .

Sixty four raindrops of radius 1 mm each falling down with a terminal velocity of 10 cm/s coalesce to form a bigger drop. The terminal velocity of bigger drop is cm/s.

A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature 50 . The gas is now heated to double its temperature. The modified pressure is Pa.

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

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

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

Given below are statements about some molecules/ions. Identify the CORRECT statements. Choose the correct answer from the options given below:

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

Arrange the following alkenes in decreasing order of stability. [figure] Choose the correct answer from the options given below:

1. III > I > II > IV
2. III > II > I > IV
3. I > III > II > IV
4. I > III > IV > II

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

A solution is prepared by dissolving 0.3 g of a non-volatile non-electrolyte solute 'A' of molar mass 60 g mol-1 and 0.9 g of a non-volatile non-electrolyte solute 'B' of molar mass 180 g mol-1 in 100 mL H2O at 27 . Osmotic pressure of the solution will be [Given: R = 0.082 L atm K-1 mol-1]

1. 2.46 atm
2. 0.82 atm
3. 1.47 atm
4. 1.23 atm

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

Among the following, the CORRECT combinations are Choose the correct answer from the options given below:

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

A D is an endothermic reaction occurring in three steps (elementary). (i) A B, Hi = +ve (ii) B C, Hii = -ve (iii) C D, Hiii = -ve Which of the following graphs between potential energy (y-axis) vs reaction coordinate (x-axis) correctly represents the reaction profile of A D?

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

Consider a mixture 'X' which is made by dissolving 0.4 mol of [Co(NH3)5SO4]Br and 0.4 mol of [Co(NH3)5Br]SO4 in water to make 4 L of solution. When 2 L of mixture 'X' is allowed to react with excess of AgNO3, it forms precipitate 'Y'. The rest 2 L of mixture 'X' reacts with excess BaCl2 to form precipitate 'Z'. Which of the following statements is CORRECT?

1. 0.1 mol of 'Y' is formed.
2. 0.4 mol of 'Z' is formed.
3. 0.2 mol of 'Z' is formed.
4. 'Y' is BaSO4 and 'Z' is AgBr.

Consider the following two reactions A and B. [figure] Numerical value of [molar mass of x + molar mass of y] is

1. 46
2. 160
3. 88
4. 4

Arrange the following carbanions in the decreasing order of stability. I. p-Br-C6H4- CH2 II. C6H5- CH2 III. p-CH3O-C6H4- CH2 IV. p-CHO-C6H4- CH2 V. p-CH3-C6H4- CH2 Choose the correct answer from the options given below :

1. I > II > IV > V > III
2. IV > II > I > III > V
3. I > IV > II > V > III
4. IV > I > II > V > III

At 27 in presence of a catalyst, activation energy of a reaction is lowered by 10 kJ mol-1. The logarithm of ratio of k (catalysed)k (uncatalysed) is.... (Consider that the frequency factor for both the reactions is same)

1. 1.741
2. 0.1741
3. 17.41
4. 3.482

A student is given one compound among the following compounds that gives positive test with Tollens' reagent. [figure] The compound is :

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

Consider three metal chlorides x, y and z, where x is water soluble at room temperature, y is sparingly soluble in water at room temperature and z is soluble in hot water. x, y and z are respectively

1. AgCl, Hg2Cl2 and PbCl2
2. CuCl2, AgCl and PbCl2
3. AlCl3, PbCl2 and BaCl2
4. MgCl2, AgCl and AlCl3

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

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

The correct stability order of the following diazonium salts is [figure]

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

A hydroxy compound (X) with molar mass 122 g mol-1 is acetylated with acetic anhydride, using a large excess of the reagent ensuring complete acetylation of all hydroxyl groups. The product obtained has a molar mass of 290 g mol-1. The number of hydroxyl groups present in compound (X) is:

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

'W' g of a non-volatile electrolyte solid solute of molar mass 'M' g mol-1 when dissolved in 100 mL water, decreases vapour pressure of water from 640 mm Hg to 600 mm Hg. If aqueous solution of the electrolyte boils at 375 K and Kb for water is 0.52 K kg mol-1, then the mole fraction of the electrolyte solute (x2) in the solution can be expressed as (Given : density of water = 1 g/mL and boiling point of water = 373 K)

1. 162.6 WM
2. 1.38 MW
3. 2.616 MW
4. 1.38 WM

The hydrogen spectrum consists of several spectral lines in Lyman series (L1, L2, L3 ; L1 has lowest energy among Lyman series). Similarly it consists of several spectral lines in Balmer series (B1, B2, B3 ; B1 has lowest energy among Balmer lines). The energy of L1 is x times the energy of B1. The value of x is 10-1. (Nearest integer)

Electricity is passed through an acidic solution of Cu2+ till all the Cu2+ was exhausted, leading to the deposition of 300 mg of Cu metal. However, a current of 600 mA was continued to pass through the same solution for another 28 minutes by keeping the total volume of the solution fixed at 200 mL. The total volume of oxygen evolved at STP during the entire process is mL. (Nearest integer) [Given: Cu2+(aq) + 2e- Cu(s); E red = +0.34 V O2(g) + 4H+ + 4e- 2H2O; E red = +1.23 V Molar mass of Cu = 63.54 g mol-1 Molar mass of O2 = 32 g mol-1 Faraday Constant = 96500 C mol-1 Molar volume at STP = 22.4 L]

Consider two Group IV metal ions X2+ and Y2+. A solution containing 0.01 M X2+ and 0.01 M Y2+ is saturated with H2S. The pH at which the metal sulphide YS will form as a precipitate is . (Nearest integer) (Given: Ksp(XS) = 1 10-22 at 25 , Ksp(YS) = 4 10-16 at 25 , [H2S] = 0.1 M in solution, Ka1 Ka2(H2S) = 1.0 10-21, 2 = 0.30, 3 = 0.48, 5 = 0.70)

In Dumas method for estimation of nitrogen, 0.50 g of an organic compound gave 70 mL of nitrogen collected at 300 K and 715 mm pressure. The percentage of nitrogen in the organic compound is %. (Aqueous tension at 300 K is 15 mm).

X and Y are the number of electrons involved, respectively during the oxidation of I- to I2 and S2- to S by acidified K2Cr2O7. The value of X + Y is .

Let a = 2 i + j - 2 k, b = i + j and c = a b. Let d be a vector such that | d - a| = 11, | c d| = 3 and the angle between c and d is 4. Then a d is equal to

1. 0
2. 1
3. 3
4. 11

Let A(1, 0), B(2, -1) and C ( 73, 43 ) be three points. If the equation of the bisector of the angle ABC is x + y = 5, then the value of 2 + 2 is

1. 5
2. 10
3. 8
4. 13

Let A1 be the bounded area enclosed by the curves y = x2 + 2, x + y = 8 and y-axis that lies in the first quadrant. Let A2 be the bounded area enclosed by the curves y = x2 + 2, y2 = x, x = 2, and y-axis that lies in the first quadrant. Then A1 - A2 is equal to

1. 23(2 2 + 1)
2. 23(3 2 + 1)
3. 23( 2 + 1)
4. 23(4 2 + 1)

Let a circle of radius 4 pass through the origin O, the points A(- 3a, 0) and B(0, - 2b), where a and b are real parameters and ab 0. Then the locus of the centroid of OAB is a circle of radius

1. 73
2. 113
3. 53
4. 83

The number of the real solutions of the equation: x|x + 3| + |x - 1| - 2 = 0 is

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

The value of 3 cosec 20 - 20 20 40 60 80 is equal to

1. 32
2. 64
3. 12
4. 16

If the domain of the function f(x) = (10x2 - 17x + 7) (18x2 - 11x + 1 ) is (- , a) (b, c) (d, ) - \e\, then 90(a + b + c + d + e) equals:

1. 170
2. 316
3. 177
4. 307

If x = 512 for some x ( , 3 2 ), then 7x ( 13x2 + 13x2 ) + 7x ( 13x2 - 13x2 ) is equal to

1. 5 13
2. 6 26
3. 4 26
4. 1 13

Let , R be such that the function f(x) = cases 2 (x2 - 2 ) + 2 x, & x < 1 \ ( + 3)x + ( - ), & x 1 cases be differentiable at all x R. Then 34( + ) is equal to

1. 48
2. 84
3. 36
4. 24

Let f(t) = ( 1 - ( e t)1 - ( e t) ) dt, t > 1. If f (e /2 ) = -e /2 and f (e /4 ) = e /4, then equals

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

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. 6
2. 15
3. 12
4. 18

From a lot containing 10 defective and 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. 73108
2. 67108
3. 7107
4. 81108

Let S = 125! + 13! 23! + 15! 21! + up to 13 terms. If 13 S = 2kn!, k N, then n + k is equal to

1. 50
2. 52
3. 49
4. 51

Let 729, 81, 9, 1, be a sequence and Pn denote the product of the first n terms of this sequence. If 2 n=140(Pn) 1n = 3 -13 and ( , ) = 1, then + is equal to

1. 73
2. 74
3. 75
4. 76

Let the lines L1: r = i + 2 j + 3 k + (2 i + 3 j + 4 k), R and L2: r = (4 i + j) + (5 i + 2 j + k), R, intersect at the point R. Let P and Q be the points lying on lines L1 and L2, respectively, such that | PR| = 29 and | PQ| = 473. If the point P lies in the first octant, then 27(QR)2 is equal to

1. 360
2. 348
3. 320
4. 340

Let S = \z C : | z-6iz-2i | = 1 and | z-8+2iz+2i | = 35 \. Then z S |z|2 is equal to :

1. 413
2. 398
3. 385
4. 423

Consider an A.P.: a1, a2, , an; a1 > 0. If a2 - a1 = -34, an = 14a1, and i=1n ai = 5252, then i=117 ai is equal to

1. 238
2. 136
3. 476
4. 952

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation in this data is replaced by , then the mean and variance become 10.1 and 1.99, respectively. Then + equals

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

Let each of the two ellipses E1: x2a2 + y2b2 = 1, (a > b) and E2: x2A2 + y2B2 = 1, (A < B) have eccentricity 45. Let the lengths of the latus recta of E1 and E2 be l1 and l2, respectively, such that 2l12 = 9l2. If the distance between the foci of E1 is 8, then the distance between the foci of E2 is

1. 965
2. 85
3. 165
4. 325

If the function f(x) = ex (e x - x - 1 ) + e( x + x) - x x - x is continuous at x = 0, then the value of f(0) is equal to

1. 23
2. 12
3. 2
4. 32

Let (2 , ) be the largest interval in which the function f(t) = |t+1|t2, t < 0, is strictly decreasing. Then the local maximum value of the function g(x) = 2 e(x-2) + x2 + 4x - , x > 2, is

Let a line L passing through the point P(1, 1, 1) be perpendicular to the lines x-44 = y-11 = z-11 and x-171 = y-711 = z0. 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 .

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 ATA is 5, is

Let a differentiable function f satisfy the equation 036 f ( tx36 )dt = 4 f(x). If y = f(x) is a standard parabola passing through the points (2, 1) and (-4, ), then is equal to .

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is