Practice Questions

Q89.Let the line 𝐿: = = intersect the plane 2𝑥+ 𝑦+ 3𝑧= 16 at the point 𝑃. Let the point 𝑄 be the 2 -1 1 foot of perpendicular from the point 𝑅1, - 1, - 3 on the line 𝐿. If 𝛼 is the area of triangle 𝑃𝑄𝑅. then 𝛼2 is equal to _____ .

Q89.Let 𝑦= 𝑦𝑥 be a solution of the differential equation 𝜋 𝜋 𝜋 𝜋 𝜋 is equal to (𝑥 cos𝑥)𝑑𝑦+ (𝑥𝑦 sin𝑥+ 𝑦 cos𝑥- 1)𝑑𝑥= 0, 0 < 𝑥< 2 . If 3𝑦 3 = √3, then 6𝑦" 6 + 2𝑦'𝜋6 _______ .

Q89.Let the tangent at any point P on a curve passing through the points 1, 1 and 10, 100, intersect positive x-axis and y-axis at the points A and B respectively. If 𝑃 𝐴: 𝑃 𝐵= 1: 𝑘 and y = yx is the solution of the 𝑑𝑦 𝑘 differential equation 𝑒 𝑑𝑥= 𝑘𝑥+ 2, 𝑦0 = 𝑘, then 4y1 - 5loge3 is equal to _______________

Q89.The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(−3, −6, 1) and B(2, 4, −3) divides the line segment AB internally in the ratio k : 1. If a, b, c (|a|, |b|, |c| are coprime) are the direction ratios of the perpendicular from the point C on the line 1−x 1 = y+42 = z+23 , then |a + b + c| is equal to _____ .

Q90.Let the plane P contain the line 2x + y −z −3 = 0 = 5x −3y + 4z + 9 and be parallel to the line x+2 2 = 3−y−4 = z−75 . Then the distance of the point A(8, −1, −19) from the plane P measured parallel to the line −3 x = y−54 = −122−z is equal to _________. JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let a line 𝐿 pass through the origin and be perpendicular to the lines 𝐿1: →𝑟= ^𝑖- 11 ^𝑗- 7 ^𝑘+ 𝜆 ^𝑖+ 2 ^𝑗+ 3 ^𝑘, 𝜆∈ℝ and 𝐿2: →𝑟= - ^𝑖+ ^𝑘+ 𝜇2 ^𝑖+ 2 ^𝑗+ ^𝑘, 𝜇∈ℝ. If 𝑃 is the point of intersection of 𝐿 and 𝐿1, and ,𝑄𝛼, 𝛽, 𝛾 is the foot of perpendicular from 𝑃 on 𝐿2, then 9𝛼+ 𝛽+ 𝛾 is equal to ________. JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the foot of perpendicular from the point 𝐴4, 3, 1 on the plane 𝑃: 𝑥- 𝑦+ 2𝑧+ 3 = 0 be 𝑁. If 𝐵( 5, 𝛼, 𝛽) , 𝛼, 𝛽∈ℤ is a point on plane 𝑃 such that the area of the triangle 𝐴𝐵𝑁 is 3√2, then 𝛼2 + 𝛽2 + 𝛼𝛽 is equal to _____________ JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let 𝑦= 𝑝𝑥 be the parabola passing through the points – 1, 0, 0, 1 and 1, 0. If the area of the region 𝑥, 𝑦: 𝑥+ 12 + 𝑦- 12 ≤1, 𝑦≤𝑝𝑥 is 𝐴, then 12𝜋- 4𝐴 is equal to ________ . JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

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

Q90.If 𝑦= 𝑦( 𝑥) is the solution of the differential equation 𝑑𝑦 4𝑥 𝑥+ 25 , 𝑥> 1 such that 𝑑𝑥+ 𝑥2 - 1𝑦= 𝑥2 - 1 2 2 𝑦(2) = 9log𝑒2 + √3 and 𝑦√2 = 𝛼log𝑒√𝛼+ 𝛽+ 𝛽- √𝛾, 𝛼, 𝛽, 𝛾∈ℕ, then 𝛼𝛽𝛾 is equal to JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q90.If 𝜆1 < 𝜆2 are two values of 𝜆 such that the angle between the planes 𝑃1: →𝑟(3 ^i - 5 ^𝑗+ ^𝑘) = 7 and , then the square of the length of perpendicular from the point 𝑃2: →𝑟· (𝜆 ^𝑖+ ^𝑗- 3 ^𝑘) = 9 is sin-12√65 38𝜆1, 10𝜆2, 2 to the plane 𝑃1 is JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

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

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

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.Two identical metallic spheres 𝐴 and 𝐵 when placed at certain distance in air repel each other with a force of 𝐹 . Another identical uncharged sphere 𝐶 is first placed in contact with 𝐴 and then in contact with 𝐵 and finally placed at midpoint between spheres 𝐴and 𝐵. The force experienced by sphere 𝐶 will be : 3𝐹 3𝐹 (1) (2) 2 4 (3) 𝐹 (4) 2𝐹

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.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

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

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