JEE Main 2025 (Online) 28th January Morning Shift

A bead of mass 'm' slides without friction on the wall of a vertical circular hoop of radius 'R' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is 'R'. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes 'R', would be (spring constant is 'k', g is acceleration due to gravity) [figure]

1. 2Rg + kR2m
2. 3Rg + kR2m
3. 2Rg + 4kR2m
4. 2 gR + kR2m

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

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

In the experiment for measurement of viscosity ' ' of given liquid with a ball having radius R, consider following statements. Choose the correct answer from the options given below:

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

The center of mass of a thin rectangular plate (fig - x) with sides of length a and b, whose mass per unit area ( ) varies as = 0 xab (where 0 is a constant), would be . [figure]

1. ( 23a, 23b )
2. ( 23a, b2 )
3. ( 13a, b2 )
4. ( a2, b2 )

Consider following statements: Choose the correct answer from the options given below:

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

Find the equivalent resistance between two ends of the following circuit [figure]

1. r
2. r6
3. r3
4. r9

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R In the light of the above statements, choose the most appropriate answer from the options given below

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

A proton of mass 'mp' has same energy 'E' as that of a photon of wavelength ' '. If the proton is moving at non-relativistic speed, then ratio of its de Broglie wavelength to the wavelength of photon is

1. 1c Emp
2. 1c 2Emp
3. 12c Emp
4. 1c E2mp

A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is

1. 8/9
2. 9/8
3. 32/27
4. 27/32

Consider a long thin conducting wire carrying a uniform current I. A particle having mass "m" and charge "q" is released at a distance "a" from the wire with a speed v0 along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance x from the wire. The value of x is [ 0 is vacuum permeability]

1. a [1 - mv02q 0 I ]
2. ae- 4 m v0q 0 I
3. a [1 - mv0q 0 I ]
4. a2

Due to presence of an em-wave whose electric component is given by E = 100 ( t - kx) NC-1 a cylinder of length 200 cm holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as

1. 50 ( t - kx) NC-1
2. 400 ( t - kx) NC-1
3. 200 ( t - kx) NC-1
4. 25 ( t - kx) NC-1

Three infinitely long wires with linear charge density are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?

1. (x2 + y2)(y2 + z2)(z2 + x2) = constant
2. xyz = constant
3. xy + yz + zx = constant
4. (x + y)(y + z)(z + x) = constant

For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?

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

Two capacitors C1 and C2 are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are U1 and U2, respectively. Which of the given statements is true? [figure]

1. C2 > C1, U2 > U1
2. C1 > C2, U1 > U2
3. C2 > C1, U2 < U1
4. C1 > C2, U1 < U2

A particle of mass 'm' and charge 'q' is fastened to one end 'A' of a massless string having equilibrium length l, whose other end is fixed at point 'O'. The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the x-axis is [figure]

1. qEl2m
2. qEl4m
3. qElm
4. 2qElm

A thin prism P1 with angle 4 made of glass having refractive index 1.54, is combined with another thin prism P2 made of glass having refractive index 1.72 to get dispersion without deviation. The angle of the prism P2 in degrees is

1. 1.5
2. 16/3
3. 3
4. 4

Which of the following circuits has the same output as that of the given circuit? [figure]

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

A hemispherical vessel is completely filled with a liquid of refractive index . A small coin is kept at the lowest point (O) of the vessel as shown in figure. The minimum value of the refractive index of the liquid so that a person can see the coin from point E (at the level of the vessel) is . [figure]

1. 32
2. 32
3. 3
4. 2

Choose the correct nuclear process from the below options [p : proton, n : neutron, e- : electron, e+ : positron, : neutrino, : antineutrino]

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

A Carnot engine (E) is working between two temperatures 473 K and 273 K. In a new system two engines - engine E1 works between 473 K to 373 K and engine E2 works between 373 K to 273 K. If 12, 1 and 2 are the efficiencies of the engines E, E1 and E2, respectively, then

1. 12 = 1 2
2. 12 = 1 + 2
3. 12 1 + 2
4. 12 < 1 + 2

Two iron solid discs of negligible thickness have radii R1 and R2 and moment of inertia I1 and I2, respectively. For R2 = 2R1, the ratio of I1 and I2 would be 1/x, where x = .

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of [MaLbTc]. If b = -3, the value of c is .

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n = . Consider all the bodies have equal masses.

A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm, respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be x100 where x is .

A double slit interference experiment performed with a light of wavelength 600 nm forms an interference fringe pattern on a screen with 10th bright fringe having its centre at a distance of 10 mm from the central maximum. Distance of the centre of the same 10th bright fringe from the central maximum when the source of light is replaced by another source of wavelength 660 nm would be mm.

The incorrect decreasing order of atomic radii is

1. Si > P > Cl > F
2. Mg > Al > C > O
3. Al > B > N > F
4. Be > Mg > Al > Si

What is the freezing point depression constant of a solvent, 50 g of which contain 1 g non-volatile solute (molar mass 256 g mol-1) and the decrease in freezing point is 0.40 K?

1. 4.43 K kg mol-1
2. 3.72 K kg mol-1
3. 5.12 K kg mol-1
4. 1.86 K kg mol-1

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

A weak acid HA has degree of dissociation x. Which option gives the correct expression of (pH - pKa)?

1. ( 1-xx )
2. 0
3. (1+2x)
4. ( x1-x )

array|c|c| [A]0/ mol L-1 & t1/2/min \ 0.100 & 200 \ 0.025 & 100 \ array For a given reaction R P, t1/2 is related to [A]0 as given in table. Given: 2 = 0.30 Which of the following is true? Choose the correct answer from the options given below:

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

A molecule (P) on treatment with acid undergoes rearrangement and gives (Q). (Q) on ozonolysis followed by reflux under alkaline condition gives (R). The structure of (R) is given below. [figure] The structure of (P) is

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

The products A and B in the following reactions, respectively are A AgNO2 CH3-CH2-CH2-Br AgCN B

1. CH3-CH2-CH2-ONO, CH3-CH2-CH2-CN
2. CH3-CH2-CH2-NO2, CH3-CH2-CH2-CN
3. CH3-CH2-CH2-ONO, CH3-CH2-CH2-NC
4. CH3-CH2-CH2-NO2, CH3-CH2-CH2-NC

In a multielectron atom, which of the following orbitals described by three quantum numbers will have same energy in absence of electric and magnetic fields? Choose the correct answer from the options given below:

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

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

Consider the following elements In, Tl, Al, Pb, Sn and Ge. The most stable oxidation states of elements with highest and lowest first ionisation enthalpies, respectively, are

1. +4 and +1
2. +4 and +3
3. +1 and +4
4. +2 and +3

Both acetaldehyde and acetone (individually) undergo which of the following reactions? Choose the correct answer from the options given below:

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

The compounds that produce CO2 with aqueous NaHCO3 solution are: Choose the correct answer from the options given below:

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

The metal ion whose electronic configuration is not affected by the nature of the ligand and which gives a violet colour in non-luminous flame under hot condition in borax bead test is

1. Cr3+
2. Ti3+
3. Ni2+
4. Mn2+

Consider 'n' is the number of lone pair of electrons present in the equatorial position of the most stable structure of ClF3. The ions from the following with 'n' number of unpaired electrons are Choose the correct answer from the options given below:

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

The correct order of stability of following carbocations is : [figure]

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

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

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

1. Both Statement I and Statement II are correct
2. Statement I is incorrect but Statement II is correct
3. Both Statement I and Statement II are incorrect
4. Statement I is correct but Statement II is incorrect

Which of the following oxidation reactions are carried out by both K2Cr2O7 and KMnO4 in acidic medium? Choose the correct answer from the options given below :

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

The molecules having square pyramidal geometry are

1. BrF5 & PCl5
2. SbF5 & PCl5
3. BrF5 & XeOF4
4. SbF5 & XeOF4

Ice and water are placed in a closed container at a pressure of 1 atm and temperature 273.15 K. If pressure of the system is increased 2 times, keeping temperature constant, then identify correct observation from following

1. Liquid phase disappears completely.
2. The amount of ice decreases.
3. The solid phase (ice) disappears completely.
4. Volume of system increases.

The formation enthalpies, Hf for H(g) and O(g) are 220.0 and 250.0 kJ mol-1, respectively, at 298.15 K, and Hf for H2O(g) is -242.0 kJ mol-1 at the same temperature. The average bond enthalpy of the O-H bond in water at 298.15 K is kJ mol-1 (nearest integer).

Consider the following sequence of reactions: [figure] 11.25 mg of chlorobenzene will produce 10-1 mg of product B. (Consider the reactions result in complete conversion.) [Given molar mass of C, H, O, N and Cl as 12, 1, 16, 14 and 35.5 g mol-1 respectively]

The molarity of a 70\% (mass/mass) aqueous solution of a monobasic acid (X) is 10-1 M (Nearest integer) [Given: Density of aqueous solution of (X) is 1.25 g mL-1 Molar mass of the acid is 70 g mol-1]

Given below is the plot of the molar conductivity vs concentration for KCl in aqueous solution. [figure] If, for the higher concentration of KCl solution, the resistance of the conductivity cell is 100 , then the resistance of the same cell with the dilute solution is 'x' . The value of x is (Nearest integer)

Quantitative analysis of an organic compound (X) shows following % composition. C : 14.5\% Cl : 64.46\% H : 1.8\% Empirical formula mass of the compound (X) is 10-1 (Given molar mass in g mol-1 of C : 12, H : 1, O : 16, Cl : 35.5)

Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, -1) and C = (2, 0, -2). Then among the statements (S1) : ABC is an isosceles right angled triangle, and (S2) : the area of ABC is 9 22,

1. both are false
2. only (S2) is true
3. only (S1) is true
4. both are true

If f(x) = 2x2x + 2, x R, then k=181 f ( k82 ) is equal to

1. 82
2. 81 2
3. 41
4. 812

The sum, of the squares of all the roots of the equation x2 + |2x - 3| - 4 = 0, is

1. 6(2 - 2)
2. 3(3 - 2)
3. 3(2 - 2)
4. 6(3 - 2)

The relation R = \(x, y) : x, y Z and x + y is even\ is:

1. reflexive and transitive but not symmetric
2. reflexive and symmetric but not transitive
3. an equivalence relation
4. symmetric and transitive but not reflexive

Let an be a sequence such that a0 = 0, a1 = 12 and 2an+2 = 5an+1 - 3an, n = 0, 1, 2, 3, . Then k=1100 ak is equal to

1. 3a100 + 100
2. 3a100 - 100
3. 3a99 - 100
4. 3a99 + 100

( -1 35 + -1 513 + -1 3365 ) is equal to:

1. 3365
2. 1
3. 3265
4. 0

If - 2 2 96x2 2 x(1 + ex) dx = ( 2 + ), , Z, then ( + )2 equals

1. 196
2. 100
3. 64
4. 144

Let Tr be the rth term of an A.P. If for some m, Tm = 125, T25 = 120, and 20 r=125Tr = 13, then 5m r=m2mTr is equal to

1. 98
2. 126
3. 112
4. 142

Let nCr-1 = 28, nCr = 56 and nCr+1 = 70. Let A(4 t, 4 t), B(2 t, -2 t) and C (3r - n, r2 - n - 1 ) be the vertices of a triangle ABC, where t is a parameter. If (3x - 1)2 + (3y)2 = , is the locus of the centroid of triangle ABC, then equals

1. 18
2. 8
3. 20
4. 6

Two numbers k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = -1) is non-zero, equals

1. 34
2. 12
3. 14
4. 23

Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denotes the number of defective oranges, then the variance of x is

1. 26/75
2. 14/25
3. 18/25
4. 28/75

Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length 254 and it passes through the point (1, 0), then the area of ABCD is

1. 758
2. 1258
3. 252
4. 754

If the image of the point (4, 4, 3) in the line x - 12 = y - 21 = z - 13 is ( , , ), then + + is equal to

1. 12
2. 9
3. 7
4. 8

Let for some function y = f(x), 0x t f(t) dt = x2 f(x), x > 0 and f(2) = 3. Then f(6) is equal to

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

The sum of all local minimum values of the function f(x) = cases 1 - 2x, & x < -1 \ 13(7 + 2|x|), & -1 x 2 \ 1118(x - 4)(x - 5), & x > 2 cases is

1. 16772
2. 15772
3. 17172
4. 13172

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

1. 5720
2. 5719
3. 4608
4. 4607

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 - x + y + = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to

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

Let f : R R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a-1 )x + b, a 1. If f(x + y) = f(x) + f(y) + 1 - 27xy, then the value of 28 i=15|f(i)| is

1. 735
2. 675
3. 715
4. 545

Let O be the origin, the point A be z1 = 3 + 2 2i, the point B(z2) be such that 3|z2| = |z1| and (z2) = (z1) + 6. Then

1. area of triangle ABO is 114
2. area of triangle ABO is 11 3
3. ABO is a scalene triangle
4. ABO is an obtuse angled isosceles triangle

The area (in sq. units) of the region \(x, y) : 0 y 2|x| + 1, 0 y x2 + 1, |x| 3\ is

1. 323
2. 643
3. 173
4. 803

Let M denote the set of all real matrices of order 3 3 and let S = \-3, -2, -1, 1, 2\. Let S1 = \A = [aij] M : A = AT and aij S, i, j\, S2 = \A = [aij] M : A = -AT and aij S, i, j\, S3 = \A = [aij] M : a11 + a22 + a33 = 0 and aij S, i, j\. If n(S1 S2 S3) = 125 , then equals .

Let f(x) = cases 3x, & x < 0 \ \1 + x + [x], x + 2[x]\, & 0 x 2 \ 5, & x > 2 cases where [.] denotes greatest integer function. If and are the number of points, where f is not continuous and is not differentiable, respectively, then + equals .

If = 1 + r=16(-3)r-1 12C2r-1, then the distance of the point (12, 3) from the line x - 3y + 1 = 0 is .

Let E1 : x29 + y24 = 1 be an ellipse. Ellipses Ei's are constructed such that their centres and eccentricities are same as that of E1, and the length of minor axis of Ei is the length of major axis of Ei+1 (i 1). If Ai is the area of the ellipse Ei, then 5 ( i=1 Ai ), is equal to .

Let a = i + j + k, b = 2 i + 2 j + k and d = a b. If c is a vector such that a c = | c|, | c - 2 a|2 = 8 and the angle between d and c is 4, then |10 - 3 b c| + | d c|2 is equal to .