JEE Main 2025 (Online) 29th January Morning Shift

If and K are de Broglie wavelength and kinetic energy, respectively, of a particle with constant mass, the correct graphical representation for the particle will be :

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

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

[figure] For the circuit shown above, equivalent GATE is :

1. NOT gate
2. NAND gate
3. AND gate
4. OR gate

The pair of physical quantities not having the same dimensions is :

1. Angular momentum and Planck's constant
2. Torque and energy
3. Surface tension and impulse
4. Pressure and Young's modulus

The expression given below shows the variation of velocity (v) with time (t), v = At2 + BtC + t. The dimension of ABC is :

1. [ M0L1T-2]
2. [ M0L2T-3]
3. [ M0L2T-2]
4. [ M0L1T-3]

The fractional compression ( VV ) of water at the depth of 2.5 km below the sea level is %. Given, the Bulk modulus of water = 2 109 N m-2, density of water = 103 kg m-3, acceleration due to gravity g = 10 m s-2.

1. 1.0
2. 1.25
3. 1.75
4. 1.5

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

Two projectiles are fired with same initial speed from same point on ground at angles of (45 - ) and (45 + ), respectively, with the horizontal direction. The ratio of their maximum heights attained is :

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

An electric dipole of mass m, charge q, and length l is placed in a uniform electric field E = E0 i. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

1. 12 ml2qE0
2. 2 ml2qE0
3. 2 mlqE0
4. 12 2mlqE0

Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire’s cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be :

1. [a/2, 3a]
2. [a/4, 3a/2]
3. [a/2, 2a]
4. [a/4, 2a]

A coil of area A and N turns is rotating with angular velocity in a uniform magnetic field B about an axis perpendicular to B. Magnetic flux and induced emf across it, at an instant when B is parallel to the plane of coil, are :

1. = AB, = NAB
2. = AB, = 0
3. = 0, = 0
4. = 0, = NAB

Consider I1 and I2 are the currents flowing simultaneously in two nearby coils 1 & 2, respectively. If L1 = self inductance of coil 1, M12 = mutual inductance of coil 1 with respect to coil 2, then the value of induced emf in coil 1 will be :

1. e1 = -L1 dI2dt - M12 dI1dt
2. e1 = -L1 dI1dt + M12 dI2dt
3. e1 = -L1 dI1dt - M12 dI1dt
4. e1 = -L1 dI1dt - M12 dI2dt

Let u and v be the distances of the object and the image from a lens of focal length f. The correct graphical representation of u and v for a convex lens when |u| > f, is

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

The work done in an adiabatic change in an ideal gas depends upon only :

1. change in its pressure
2. change in its temperature
3. change in its specific heat
4. change in its volume

A body of mass ‘m’ connected to a massless and unstretchable string goes in vertical circle of radius ‘R’ under gravity g. The other end of the string is fixed at the center of circle. If velocity at top of circular path is n gR, where, n 1, then ratio of kinetic energy of the body at bottom to that at top of the circle is :

1. n2+4n2
2. n+4n
3. n2n2+4
4. nn+4

As shown below, bob A of a pendulum having massless string of length 'R' is released from 60 to the vertical. It hits another bob B of half the mass that is at rest on a frictionless table in the center. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take g as acceleration due to gravity.) [figure]

1. 43 Rg
2. 13 Rg
3. Rg
4. 23 Rg

A container of fixed volume contains a gas at 27 . To double the pressure of the gas, the temperature of gas should be raised to .

The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the bank is cm. (Take g = 10 m/s2)

The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of F = i - j + k acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is .

Two light beams fall on a transparent material block at point 1 and 2 with angle 1 and 2, respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2, d = 4 3 cm and 1 = 2 = -1 ( n22n1 ), where refractive index of the block n2 > refractive index of the outside medium n1, then the thickness of the block is cm. [figure]

In a hydraulic lift, the surface area of the input piston is 6 cm2 and that of the output piston is 1500 cm2. If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is kJ.

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

At the interface between two materials having refractive indices n1 and n2, the critical angle for reflection of an em wave is 1C. The n2 material is replaced by another material having refractive index n3 such that the critical angle at the interface between n1 and n3 materials is 2C. If n3 > n2 > n1; n2n3 = 25 and 2C - 1C = 12, then 1C is :

1. -1 ( 16n1 )
2. -1 ( 13n1 )
3. -1 ( 56n1 )
4. -1 ( 23n1 )

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

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

1.24 g of AX2 (molar mass 124 g mol-1) is dissolved in 1 kg of water to form a solution with boiling point of 100.0156°C, while 25.4 g of AY2 (molar mass 250 g mol-1) in 2 kg of water constitutes a solution with a boiling point of 100.0260°C. Kb(H2O) = 0.52 kg mol-1 Which of the following is correct?

1. AX2 and AY2 (both) are completely unionised.
2. AX2 and AY2 (both) are fully ionised.
3. AX2 is completely unionised while AY2 is fully ionised.
4. AX2 is fully ionised while AY2 is completely unionised.

For a Mg | Mg2+(aq) || Ag+(aq) | Ag the correct Nernst Equation is :

1. Ecell = E cell - RT2F [Ag+]2[Mg2+]
2. Ecell = E cell - RT2F [Mg2+][Ag+]
3. Ecell = E cell - RT2F [Ag+][Mg2+]
4. Ecell = E cell + RT2F [Ag+]2[Mg2+]

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

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

The correct increasing order of stability of the complexes based on 0 value is : I. [Mn(CN)6]3- II. [Co(CN)6]4- III. [Fe(CN)6]4- IV. [Fe(CN)6]3-

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

500 J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm. The final temperature and the change in internal energy respectively are: Given: R = 8.3 J K-1 mol-1

1. 368 K and 500 J
2. 348 K and 300 J
3. 378 K and 300 J
4. 378 K and 500 J

The molar conductivity of a weak electrolyte when plotted against the square root of its concentration, which of the following is expected to be observed?

1. Molar conductivity increases sharply with increase in concentration.
2. Molar conductivity decreases sharply with increase in concentration.
3. A small increase in molar conductivity is observed at infinite dilution.
4. A small decrease in molar conductivity is observed at infinite dilution.

The correct option with order of melting points of the pairs (Mn, Fe), (Tc, Ru) and (Re, Os) is :

1. Fe < Mn, Ru < Tc and Re < Os
2. Fe < Mn, Ru < Tc and Os < Re
3. Mn < Fe, Tc < Ru and Re < Os
4. Mn < Fe, Tc < Ru and Os < Re

The reaction A2 + B2 2AB follows the mechanism: A2 k-1 k1 A + A (fast) A + B2 k2 AB + B (slow) A + B AB (fast) The overall order of the reaction is:

1. 3
2. 2.5
3. 1.5
4. 2

If a0 is denoted as the Bohr radius of hydrogen atom, then what is the de Broglie wavelength (λ) of the electron present in the second orbit of hydrogen atom? [n : any integer]

1. 8 a0n
2. 2a0n
3. na04
4. 4 a0n

The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.

1. E Sn4+/Sn2+ = +0.15 V
2. E Al3+/Al = -1.66 V
3. E Pb4+/Pb2+ = +1.67 V
4. E Tl3+/Tl = +1.26 V

Choose the correct statements. Choose the correct answer from the options given below :

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

In the following substitution reaction : [figure] Product 'P' formed is :

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

At temperature T, compound AB2(g) dissociates as AB2(g) AB(g) + 12B2(g) having degree of dissociation x (small compared to unity). The correct expression for x in terms of Kp and p is:

1. Kp
2. [3] 2Kp2p
3. [3] 2Kpp
4. [4] 2Kpp

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

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

The product (P) formed in the following reaction is : [figure]

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

The steam volatile compounds among the following are : Choose the correct answer from the options given below :

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

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

Total number of nucleophiles from the following is : NH3, PhSH, (H3C)2S, H2C=CH2, OH, H3O , (CH3)2 CO, [figure]

1. 4
2. 6
3. 5
4. 7

An element 'E' has the ionisation enthalpy value of 374 kJ mol-1. 'E' reacts with elements A, B, C and D with electron gain enthalpy values of −328, −349, −325 and −295 kJ mol-1, respectively. The correct order of the products EA, EB, EC and ED in terms of ionic character is :

1. EB > EA > EC > ED
2. EA > EB > EC > ED
3. ED > EC > EA > EB
4. ED > EC > EB > EA

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

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

The molar mass of the water insoluble product formed from the fusion of chromite ore (FeCr2O4) with Na2CO3 in presence of O2 is g mol-1.

Given below are some nitrogen containing compounds : [figure] Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume mg of HCl. (Given molar mass in g mol-1 C : 12, H : 1, O : 16, Cl : 35.5)

If A2B is 30% ionised in an aqueous solution, then the value of van't Hoff factor (i) is 10-1.

[figure] 0.1 mole of compound 'S' will weigh g. (Given molar mass in g mol-1 C : 12, H : 1, O : 16)

The sum of sigma ( ) and pi ( ) bonds in Hex-1,3-dien-5-yne is .

Let L1 : x-11 = y-2-1 = z-12 and L2 : x+1-1 = y-22 = z1 be two lines. Let L3 be a line passing through the point ( , , ) and be perpendicular to both L1 and L2. If L3 intersects L1, then |5 - 11 - 8 | equals :

1. 25
2. 20
3. 16
4. 18

Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P is :

1. 164
2. 158
3. 161
4. 173

Let A = [aij] = bmatrix 5 128 & 4 5 \ 5 8 & 4 25 bmatrix. If Aij is the cofactor of aij, Cij = k=12 aikAjk, 1 i, j 2, and C = [Cij], then 8|C| is equal to :

1. 288
2. 262
3. 222
4. 242

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is :

1. 108
2. 90
3. 122
4. 84

Let the ellipse E1 : x2a2 + y2b2 = 1, a > b and E2 : x2A2 + y2B2 = 1, A < B have same eccentricity 1 3. Let the product of their lengths of latus rectums be 32 3 and the distance between the foci of E1 be 4. If E1 and E2 meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

1. 24 65
2. 18 65
3. 6 6
4. 12 65

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

1. 384
2. 392
3. 96
4. 192

Let be a triangle formed by the lines 7x - 6y + 3 = 0, x + 2y - 31 = 0 and 9x - 2y - 19 = 0. Let the point (h, k) be the image of the centroid of in the line 3x + 6y - 53 = 0. Then h2 + k2 + hk is equal to :

1. 47
2. 37
3. 40
4. 36

The number of solutions of the equation ( 9x - 9 x + 2 ) ( 2x - 7 x + 3 ) = 0 is :

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

The least value of n for which the number of integral terms in the Binomial expansion of ( [3]7 + [12]11)n is 183, is :

1. 2184
2. 2172
3. 2196
4. 2148

The value of n ( k=1n k3 + 6k2 + 11k + 5(k+3)! ) is :

1. 5/3
2. 2
3. 4/3
4. 7/3

Let y = y(x) be the solution of the differential equation : x( e( x))2 dy + ( x - 3y x e( x))dx = 0, x (0, 2 ). If y ( 4 ) = - 1 e 2, then y ( 6 ) is equal to :

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

Let |z1 - 8 - 2i| 1 and |z2 - 2 + 6i| 2, z1, z2 C. Then the minimum value of |z1 - z2| is :

1. 3
2. 10
3. 7
4. 13

Let x1, x2, , x10 be ten observations such that i=110(xi - 2) = 30, i=110(xi - )2 = 98, > 2, and their variance is 45. If and 2 are respectively the mean and the variance of 2(x1 - 1) + 4 , 2(x2 - 1) + 4 , , 2(x10 - 1) + 4 , then 2 is equal to :

1. 100
2. 90
3. 120
4. 110

Let a = i + 2 j + k and b = 2 i + 7 j + 3 k. Let L1 : r = (- i + 2 j + k) + a, R and L2 : r = ( j + k) + b, R be two lines. If the line L3 passes through the point of intersection of L1 and L2 and is parallel to a + b, then L3 passes through the point :

1. (-1, -1, 1)
2. (2, 8, 5)
3. (8, 26, 12)
4. (5, 17, 4)

Let the area of the region \(x, y) : 2y x2 + 3, y + |x| 3, y |x - 1|\ be A. Then 6A is equal to :

1. 14
2. 18
3. 16
4. 12

Let a = 2 i - j + 3 k, b = 3 i - 5 j + k and c be a vector such that a c = c b and ( a + c) ( b + c) = 168. Then the maximum value of | c|2 is :

1. 77
2. 154
3. 308
4. 462

Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

1. 14
2. 3 7
3. 2 14
4. 5 7

Define a relation R on the interval [0, 2 ) by x R y if and only if 2 x - 2 y = 1. Then R is :

1. both reflexive and symmetric but not transitive
2. both reflexive and transitive but not symmetric
3. reflexive but neither symmetric nor transitive
4. an equivalence relation

Let M and m respectively be the maximum and the minimum values of f(x) = vmatrix 1 + 2 x & 2 x & 4 4x \ 2 x & 1 + 2 x & 4 4x \ 2 x & 2 x & 1 + 4 4x vmatrix, x R Then M4 - m4 is equal to :

1. 1280
2. 1040
3. 1215
4. 1295

The integral 80 0 4 ( + 9 + 16 2 )d is equal to :

1. 3 4
2. 4 3
3. 6 43
4. 2 3

Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which x 0+ (x ( [ 1x ] + [ 2x ] + + [ px ] ) - x2 ( [ 1x2 ] + [ 22x2 ] + + [ 92x2 ] ) ) 1 is equal to .

Let S = \m Z : Am2 + Am = 3I - A-6 \, where A = bmatrix 2 & -1 \ 1 & 0 bmatrix. Then n(S) is equal to .

Let S = \x : -1x = + -1x + -1(2x + 1) \. Then x S(2x - 1)2 is equal to .

Let f : (0, ) R be a twice differentiable function. If for some a 0, 01 f( x) d = af(x), f(1) = 1 and f(16) = 18, then 16 - f' ( 116 ) is equal to .

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is .