Practice Questions

Q80.Let S be the sample space of all five digit numbers. If p is the probability that a randomly selected number from S , is a multiple of 7 but not divisible by 5 , then 9p is equal to (1) 1. 0146 (2) 1. 2085 (3) 1. 0285 (4) 1. 1521 ¯

Q80.If the lines →r= (ˆi −ˆj + ˆk) λ(3ˆj −ˆk) and →r (αˆi −ˆj) μ(2ˆi −3ˆk) are co-planar, the the distance of the plane containing these two lines from the point (α, 0, 0) is (1) 2 (2) 2 9 11 (3) 4 (4) 2 11

Q88.Let 𝑓: 0, 1 →𝑅 be a twice differentiable function in 0, 1 such that 𝑓0 = 3 and 𝑓1 = 5. If the line 𝑦= 2𝑥+ 3 intersects the graph of 𝑓 at only two distinct points in 0, 1, then the least number of points 𝑥∈0, 1, at which 𝑓''𝑥= 0, is √3 15𝑥3

Q3. A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to 10081 of the height through which it falls. Find the average speed of the ball. (Take g = 10 m s−2 ) (1) 3. 0 m s−1 (2) 3. 5 m s−1 (3) 2. 0 m s−1 (4) 2. 50 m s−1

Q4. A body of mass M moving at speed V0 collides elastically with a mass m at rest. After the collision, the two masses move at angles θ1 and θ2 with respect to the initial direction of motion of the body of mass M.. The largest possible value of the ratio M m , for which the angles θ1 and θ2 will be equal, is : (1) 3 (2) 4 (3) 2 (4) 1

Q4. A block of mass 𝑚 slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is: Given 𝑚= 8 kg, 𝑀= 16 kg Assume all the surfaces shown in the figure to be frictionless. 3 4 (1) g (2) g 5 3 6 2 (3) g (4) 5 3g

Q4. A huge circular arc of length 4. 4 ly subtends an angle 4s at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second? Given : 1 ly = 9. 46 × 1015 m 1 AU = 1. 5 × 1011 m (1) 3. 5 × 106 s (2) 4. 5 × 1010 s (3) 4. 1 × 108 s (4) 7. 2 × 108

Q4. A solid sphere of radius R gravitationally attracts a particle placed at 3R from its centre with a force F1. Now a spherical cavity of radius ( R2 ) is made in the sphere (as shown in figure) and the force becomes F2 . The value of F1 : F2 is: (1) 41 : 50 (2) 50 : 41 (3) 25 : 36 (4) 36 : 25 JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper

Q4. A triangular plate is shown. A force F = 4ˆi −3ˆj is applied at point P. The torque at point P with respect to point O and Q are: (1) −15 −20√3, 15 −20√3 (2) 15 + 20√3, 15 −20√3 (3) 15 −20√3, 15 + 20√3 (4) −15 + 20√3, 15 + 20√3

Q5. An automobile of mass m accelerates starting from the origin and initially at rest, while the engine supplies constant power P . The position is given as a function of time by: JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper 1 1 2 (1) 3 (2) t 2 t 3 ( 89Pm ) 2 ( 98Pm ) 2 1 1 3 (3) 3 (4) t 2 t 2 ( 9m8P ) 2 ( 98Pm ) 2

Q5. Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be: JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper + 2√2 (1) √𝐺1 + 2√2 (2) √ 𝐺21 𝐺 2√2𝐺 (3) (4) √1 + √ 22√2 - 1 2

Q6. A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height h is S. (1) h 23 (2) h 23 + + g Re ) Re ) √Re2 g [(1 [(1 −1] √2Re −1] (3) 1 h 23 (4) 1 h 23 + + Rc ) Re ) 3 √Re2 g [(1 3 √2Rcg [(1 −1] −1] JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

Q9. For an ideal gas the instantaneous change in pressure P with volume V is given by the equation dP = −aP. If dV P = P0 at V = 0 is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here R is the gas constant) (1) aP0 (2) infinity eR (3) 0°C (4) aeRP0

Q9. A mixture of hydrogen and oxygen has volume 500 cm3, temperature 300 K, pressure 400 𝑘Pa and mass 0 . 76 g . The ratio of masses of oxygen to hydrogen will be: (1) 16 : 3 (2) 3 : 8 (3) 8 : 3 (4) 3 : 16

Q9. An electric dipole is placed on x−axis in proximity to a line charge of linear charge density 3. 0 × 10−6 C m−1. Line charge is placed on z− axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. If total force of 4 N is exerted on the dipole, find out the amount of positive or negative charge of the dipole. (1) 815. 1 nC (2) 8. 8 μC (3) 0. 485 mC (4) 4. 44 μC

Q10.Two Carnot engines A and B operate in series such that engine A absorbs heat at T1 and rejects heat to a sink at temperature T. Engine B absorbs half of the heat rejected by Engine A and rejects heat to the sink at T3 . When workdone in both the cases is equal, to value of T is : (1) 2 3 T1 + 23 T3 (2) 31 T1 + 32 T3 (3) 32 T1 + 31 T3 (4) 32 T1 + 31 T3

Q10.Figure shows a rod AB , which is bent in a 120° circular arc of radius R. A charge (−Q) is uniformly → distributed over rod AB. What is the electric filed E at the centre of curvature O ? (1) 3√3Q ˆi (2) 3√3Q ˆi 8π2ε0R2 8πε0R2 (3) −3√3Q ˆi (4) 3√3Q ˆi 8π2ε0R2 16π2ε0R2

Q11.Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance R2 from the earth's centre, where R is the radius of the earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period: (1) 1 (2) 2π √gR 2π√Rg (3) 2πRg (4) Image is virtual, opposite side of convex mirror.

Q12.A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. Find the magnetic field at centre, when a current I passes through coil : (1) μ0I 8 [ a+ba−b ] (2) 4(a−b)μ0I [ a1 −1b ] (3) μ0I 8 ( a+ba−b ) (4) 2(μ0INb−a) loge( ab )

Q12.A parallel-plate capacitor with plate area A has separation d between the plates. Two dielectric slabs of dielectric constant K1 and K2 of same area A2 and thickness d2 are inserted in the space between the plates. The capacitance of the capacitor will be given by : 2( K1+K2) (1) ε0 + + d K1+K2 K1 A ( 12 K1 K2 ) (2) ε0dA ( 21 K2 ) K1 (3) ε0 + K1+K2 + d 2( K1+K2) K2 ) A ( 12 K1 K2 ) (4) ε0dA ( 21

Q13.The four arms of a Wheatstone bridge have resistances as shown in the figure. A galvanometer of 15Ω resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 V is maintained across AC. JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper (1) 2. 44μA (2) 2. 44 mA (3) 4. 87 mA (4) 4. 87μA

Q13.A small square loop of side a and one turn is placed inside a larger square loop of side b and one turn (b ≫a). The two loops are coplanar with their centres coinciding. If a current I is passed in the square loop of side b, then the coefficient of mutual inductance between the two loops is : (1) 4π μ0 8√2 a2b (2) 4πμ0 8√2 b2a (3) μ0 8√2 (4) μ0 8√2 4π b 4π a

Q13.Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be: (1) 1: 2 (2) 1: 4 (3) 4: 1 (4) 2: 1

Q14.A coaxial cable consists of an inner wire of radius a surrounded by an outer shell of inner and outer radii b and c respectively. The inner wire carries an electric current i0 which is distributed uniformly across cross- sectional area. The outer shell carries an equal current in opposite direction and distributed uniformly. What will be the ratio of the magnetic field at a distance x from the axis when (i) x < a and (ii) a < x < b ? (1) a2 (2) b2−a2 x2 x2 (3) x2 (4) x2 b2−a2 a2

Q15.The thickness at the centre of a plano convex lens is 3 mm and the diameter is 6 cm. If the speed of light in the material of the lens is 2 × 108 m s−1. The focal length of the lens is ___________. (1) 0. 30 cm (2) 15 cm (3) 1. 5 cm (4) 30 cm