Practice Questions

Q88.The plane 2x −y + z = 4 intersects the line segment joining the points A(a, −2, 4) and B(2, b, −3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is √5 . If ab < 0 and P is the point (a −b, b, 2 b −a) then CP2 is equal to: (1) 17 (2) 16 3 3 (3) 73 (4) 97 3 3

Q88.Let the plane P : 4x −y + z = 10 be rotated by an angle π2 about its line of intersection with the plane x + y −z = 4 . If α is the distance of the point (2, 3, −4) from the new position of the plane P , then 35α is equal to (1) 85 (2) 105 (3) 126 (4) 90

Q89.Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is (1) 5 (2) 2 24 15 (3) 1 (4) 5 6 36

Q89.The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the point A , B, C is (2, a, 4), a ∈N . If the volume of the tetrahedron OABC is 144 unit3 , then which of the following points is NOT on P ? (1) (0, 4, 4) (2) (3, 0, 4) (3) (0, 6, 3) (4) (2, 2, 4)

Q90.Let S = {w1, w2, … . } be the sample space associated to a random experiment. Let P(wn) = P(wn−1)2 , . Let A = {2k + 3l; k, l ∈N} and B = {wn; n ∈A} . Then P(B) is equal to (1) 3 (2) 3 32 64 (3) 1 (4) 1 16 32 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q3. A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration ac is varying with time t as ac = k2rt2 , where k is a constant. The power delivered to the particle by the force acting on it is - (1) Zero (2) 2mk2r2t (3) mk2r2t (4) 2mk2rt

Q3. A body of mass 10 kg is projected at an angle of 45° with the horizontal. The trajectory of the body is observed to pass through a point (20, 10). If T is the time of flight, then its momentum vector, at time t = T , is _____ . √2 [Take g = 10 m s−2 ] N s (1) 100ˆi + (100√2 −200)ˆj N s (2) 100√2ˆi + (100 −200√2)ˆj N s (3) 100ˆi + (100 −200√2)ˆj N s (4) 100√2ˆi + (100√2 −200)ˆj

Q5. A √34 m long ladder weighing 10 kg leans on a frictionless wall. Its feet rest on the floor 3 m away from the wall as shown in the figure. If Ff and Fw are the reaction forces of the floor and the wall, then ratio of Fw will Ff be : (Use g = 10 m s−2 .) (1) 6 (2) 3 √110 √113 (3) 3 (4) 2 √109 √109

Q7. An ice cube of dimensions 60 cm × 50 cm × 20 cm is placed in an insulation box of wall thickness 1 cm. The box keeping the ice cube at 0°C of temperature is brought to a room of temperature 40°C. The rate of melting of ice is approximately: (Latent heat of fusion of ice is 3 . 4 × 105 J kg-1 and thermal conducting of insulation wall is 0 . 05 W m-1 °C-1) JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper (1) 61 × 10-1 kg s-1 (2) 61 × 10-5 kg s-1 (3) 208 kg s-1 (4) 30 × 10-5 kg s-1

Q8. A drop of liquid of density ρ is floating half immersed in a liquid of density σ and surface tension 7. 5 × 10−4 N cm−1 . The radius of drop in cm will be : (Take : g = 10 ms−2 ) (1) 15 (2) 15 √2ρ−σ √ρ−σ (3) 3 (4) 3 2√ρ−σ 20√2ρ−σ

Q10.Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is √2 times the speed of sound, then the value of n will be (1) 1 (2) 2 (3) 3 (4) 4

Q10.Two identical positive charges Q each are fixed at a distance of 2a apart from each other. Another point charge q0 with mass m is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge q0 executes SHM. The time period of oscillation of charge q0 will be (1) √4π3ε0ma3q0Q (2) √ 4π3ε0ma3q0Q (3) √2π2ε0ma3q0Q (4) √8π3ε0ma3q0Q

Q11.A spherically symmetric charge distribution is considered with charge density varying as ρ0( 34 −rR ) for r ≤R ρ(r) = { Zero for r > R Where, r(r < R) is the distance from the centre O (as shown in figure). The electric field at point P will be : (1) 4ε0 ρ0r ( 43 −rR ) (2) ρ0r3ε0 ( 43 −rR ) (3) ρ0r 4ε0 (1 −rR ) (4) ρ0r5ε0 (1 −rR )

Q11.A source of potential difference 𝑉 is connected to the combination of two identical capacitors as shown in the figure. When key 𝐾 is closed, the total energy stored across the combination is 𝐸1. Now key 𝐾 is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored 𝐸1 across the combination is now 𝐸2. The ratio will be 𝐸2 1 2 (1) (2) 10 5 5 5 (3) (4) 13 26

Q11.Two point charges Q each are placed at a distance d apart. A third point charge q is placed at a distance x from mid-point on the perpendicular bisector. The value of x at which charge q will experience the maximum Coulomb's force is: (1) d (2) d 2 (3) d (4) d √2 2√2

Q12.A long cylindrical volume contains a uniformly distributed charge of density ρ. The radius of cylindrical volume is R. A charge particle (q) revolves around the cylinder in a circular path. The kinetic energy of the particle is : (1) ρqR2 (2) ρqR2 4εo 2εo (3) qρ (4) 4ε0R2 4ε0R2 qρ

Q13.As shown in the figure, a metallic rod of linear density 0 . 45 kg m-1 is lying horizontally on a smooth incline plane which makes an angle of 45° with the horizontal. The minimum current flowing in the rod required to keep it stationary, when 0 . 15 T magnetic field is acting on it in the vertical upward direction, will be (Use 𝑔= 10 m s-2) (1) 30 A (2) 15 A (3) 10 A (4) 3 A JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q13.A small square loop of wire of side 𝑙 is placed inside a large square loop of wire 𝐿𝐿≫𝑙. Both loops are coplanar and their centres coincide at point 𝑂 as shown in figure. The mutual inductance of the system is (1) 2√2𝜇0𝐿2 (2) 𝜇0𝑙2 𝜋𝑙 2√2𝜋𝐿 (3) 𝜇0𝐿2 (4) 2√2𝜇0𝑙2 2√2𝜋𝑙 𝜋𝐿

Q14.A coil of inductance 1 H and resistance 100 Ω is connected to a battery of 6 V . Determine approximately : (a) The time elapsed before the current acquires half of its steady-state value (b) The energy stored in the magnetic field associated with the coil at an instant 15 ms after the circuit is switched on. (Given ln 2 = 0. 693, e −32 = 0. 25) (1) t = 10 ms; U = 2 mJ (2) t = 10 ms; U = 1 mJ (3) t = 7 ms; U = 1 mJ (4) t = 7 ms; U = 2 mJ

Q14.A coil is placed in a time varying magnetic field. If the number of turns in the coil were to be halved and the radius of wire doubled, the electrical power dissipated due to the current induced in the coil would be (Assume the coil to be short circuited.) (1) Doubled (2) The same (3) Quadrupled (4) Halved

Q14.A velocity selector consists of electric field 𝐸= 𝐸 ^k and magnetic field 𝐵= 𝐵 ^j with 𝐵= 12 mT. The value 𝐸 required for an electron of energy 728eV moving along the positive x-axis to pass undeflected is (Given, mass of electron = 9 . 1 × 10-31 kg) (1) 192 kV m-1 (2) 192 mV m-1 (3) 9600 kV m-1 (4) 16 kV m-1

Q15.Two long parallel conductors S1 and S2 are separated by a distance 10 cm and carrying currents of 4 A and 2 A respectively. The conductors are placed along x-axis in X −Y plane. There is a point P located between the conductors (as shown in figure). + m s−1 ; where ˆi & A charge particle of 3π coulomb is passing through the point P with velocity →v= (2ˆi 3ˆj) ˆj represents unit vector along x & yaxis respectively. The force acting on the charge particle is 4π × + N. The value of x is 10−5(−xˆi 2ˆj) (1) 2 (2) 1 (3) 3 (4) −3

Q15.If Electric field intensity of a uniform plane electro magnetic wave is given as E = −301. 6 sin(kz −ωt)ˆax + 452. 4 sin(kz −ωt)ˆay V m−1. Then, magnetic intensity H of this wave in A m−1 will be [Given : Speed of light in vacuum c = 3 × 108 m s−1 , Permeability of vacuum μ0 = 4π × 10−7 N A−2 ] (1) +0. 8 sin(kz −ωt)ˆay + 0. 8 sin(kz −ωt)ˆax (2) +1. 0 × 10−6 sin(kz −ωt)ˆay + 1. 5 × 10−6(kz −ωt)ˆax (3) −0. 8 sin(kz −ωt)ˆay −1. 2 sin(kz −ωt)ˆax (4) −1. 0 × 10−6 sin(kz −ωt)ˆay −1. 5 × 10−6 sin(kz −ωt)ˆa λ, then this phenomenon is called as

Q17.For an object placed at a distance 2. 4 m from a lens, a sharp focused image is observed on a screen placed at a distance 12 cm from the lens. A glass plate of refractive index 1. 5 and thickness 1 cm is introduced between lens and screen such that the glass plate plane faces parallel to the screen. By what distance should the object be shifted so that a sharp focused image is observed again on the screen? (1) 0. 8 m (2) 3. 2 m (3) 1. 2 m (4) 5. 6 m

Q21.In an experiment of determine the Young's modulus of wire of a length exactly 1 m, the extension in the length of the wire is measured as 0. 4 mm with an uncertainty of ±0. 02 mm when a load of 1 kg is applied. The JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper diameter of the wire is measured as 0. 4 mm with an uncertainty of ±0. 01 mm. The error in the measurement of Young's modulus (ΔY) is found to be x × 1010 N m−2 . The value of x is _____ .