Practice Questions

Q3. A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius Re. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become times larger. Due to this the farthest distance from the centre of the earth that the √32 satellite reaches is R. Value of R is : (1) 4Re (2) 2. 5Re (3) 3Re (4) 2Re

Q3. A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate dM(t) v(t) is its instantaneous velocity. The instantaneous acceleration of the satellite is: dt = bv2(t), where (1) −bv3(t) (2) M(t)−bv3 (3) −2bv3M(t) (4) − 2M(t)bv3

Q4. A rod of length l has non-uniform linear mass density given by ρ(x) = a + b( xl )2, where a and b are constants and 0 ≤x ≤l The value of x for the centre of mass of the rod is at: (1) 2 3 ( 2a+ba+b )L (2) 43 ( 3a+b2a+b )L (3) 3 4 ( 2a+3ba+b )L (4) 32 ( 3a+b2a+b )L

Q4. Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d . The ratio I0 of moment of inertia I0 of the system about an IA axis passing the centroid and about center of any of the spheres IA and perpendicular to the plane of the triangle is: (1) 13 (2) 15 23 13 (3) 23 (4) 13 13 15

Q4. As shown in figure. When a spherical cavity (centred at O ) of radius 1 is cut out of a uniform sphere of radius R (centred at C ), the centre of mass of remaining (shaded part of sphere is at G, i.e., on the surface of the JEE Main 2020 (08 Jan Shift 2) JEE Main Previous Year Paper cavity. R can be determined by the equation: (1) (R2 + R + 1)(2 −R) = 1 (2) (R2 −R −1)(2 −R) = 1 (3) (R2 −R + 1)(2 −R) = 1 (4) (R2 + R −1)(2 −R) = 1

Q5. A particle of mass m is projected with a speed u from the ground at an angle θ = π3 w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity uˆi. The horizontal distance covered by the combined mass before reaching the ground is: (1) 3√3 u2 (2) 3√2 u2 8 g 4 g u2 (3) 5 g (4) 2√2 u2g 8

Q5. Two planets have masses M and 16 M and their radii are a and 2a , respectively. The separation between the centres of the planets is 10a . A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is : (1) 2√GMa (2) 4√GMa (3) √GM2ma (4) 23 √5 GMa

Q5. A satellite of mass M is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R ( R = radius of the earth), it ejects a rocket of mass M10 so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( G is the gravitational constant; Me is the mass of the earth): JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper (1) M 20 (u2 + 113200 GMeR ) (2) 5M(u2 −119200 GMeR ) (3) 3M 2 (4) M 2 8 20 3R (u + √5GMe6R ) (u −√2GMe )

Q5. A uniform rod of length ' ℓ′ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed ω the rod makes an angle θwith it (see figure). To find θ equate the rate of change of angular momentum (direction going into the paper) mℓ2 ω2 sin θ about the centre of mass (CM) to the torque 12 provided by the horizontal and vertical forces FH and Fv about the CM. The value of θ is then such that: (1) cos θ = 2g (2) cos θ = g 3𝓁ω2 2ℓω2 (3) cosθ = g (4) cos θ = 3g ℓω2 2ℓω2 JEE Main 2020 (03 Sep Shift 2) JEE Main Previous Year Paper

Q5. A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass m 2 →v collides with A with a velocity which is half ( 2 ) the instantaneous velocity →vof A. The collision is completely inelastic. Then, the combined body: JEE Main 2020 (09 Jan Shift 1) JEE Main Previous Year Paper (1) continues to move in a circular orbit (2) Escapes from the Planet's Gravitational field (3) Falls vertically downwards towards the planet (4) starts moving in an elliptical orbit around the planet

Q6. The mass density of a planet of radius R varies with the distance r from its centre as ρ(r) = ρ0(1 −r2R2 ) Then the gravitational field is maximum at: (1) (2) r = R R r = √34 (3) r = 1 R (4) R √3 r = √59

Q7. Consider a solid sphere of radius R and mass density ρ(r) = ρ0(1 −r2R2 ), 0 < r ≤R. The minimum density of a liquid in which it will float is: (1) ρ0 (2) ρ0 3 5 (3) 2ρ0 (4) 2ρ0 5 3

Q7. Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently, the mean collision time between the gas molecule changes from τ1 to τ2 . If CPCv = γ for this gas then a good estimate for τ1τ2 is given by (1) 2 (2) 12 1 (3) ( 21 )γ (4) ( γ+1 2 ) 2

Q9. A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R > r) . If the surface charge densities on the two shells are equal, the electric potential at the common centre is : (R+r) (2R+r) 1 1 (1) Q (2) Q 2(R2+r2) (R2+r2) 4πϵ0 4πϵ0 (3) 1 (R+2r)Q (4) 1 (R+r) Q 4πϵ0 2(R2+r2) 4πϵ0 (R2+r2)

Q9. A small spherical droplet of density d is floating exactly half immersed in a liquid of density ρ and surface tension T. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet): r = (1) r = √ 3(d+ρ)g2T (2) √ (d−ρ)gT (3) T (4) 3T r = r = √ (d+ρ)g √ (2d−ρ)g

Q9. The mass density of a spherical galaxy varies as K over a large distance r from its center. In that region, a small r star is in a circular orbit of radius R. Then the period of revolution, T depends on R as: (1) T 2 ∝ R (2) T 2 ∝ R3 (3) T 2 ∝ 1 (4) T ∝R R3

Q10.Consider a sphere of radius R which carries a uniform charge density ρ . If a sphere of radius R is carved out 2 −→ −−EA → → of it, as shown, the ratio of magnitude of electric field EA and EB , respectively, at points A and B due to− → EB the remaining portion is: (1) 21 (2) 18 34 34 (3) 17 (4) 18 54 54 + × 10−29 C m at the origin (0,0, 0) . The electric field due

Q10.Assume that the displacement (s) of air is proportional to the pressure difference (Δp) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of air (ρ) and the frequency (f). If Δp~10 Pa, n~300 m/s, p~1 kg/m3 f~1000Hz, then s will be of the order of (take the multiplicative constant to be 1 ) (1) 1003 mm (2) 10 mm (3) 101 mm (4) 1 mm

Q10.When a particle of mass m is attached to a vertical spring of spring constant k and released, its motion, is described by y(t) = y0 sin2 ωt, where ' y' is measured from the lower end of upstretched spring. Then ω is : (1) 1 (2) 2 √gy0 √gy0 (3) √ 2y0g (4) √2gy0

Q10.A particle of mass m and charge q has an initial velocity →v= v0ˆj . If an electric field E = E0ˆi and magnetic → field B = B0ˆi act on the particle, its speed will double after a time (1) 2mv0 (2) 3mv0 qE0 qE0 (3) √3mv0 (4) √2mv0 qE0 qE0

Q10.An elliptical loop having resistance R, of semi major axis a , and semi minor axis b is placed in a magnetic field as shown in the figure. If the loop is rotated about the x-axis with angular frequency ω, the average power loss in the loop due to Joule heating is : (1) π2a2b2B2ω2 (2) zero 2R (3) π bB ω (4) π2a2b2B2ω2 R R

Q10. A parallel plate capacitor has plates of area A separated by distance d between them. It is filled with a dielectric which has a dielectric constant that varies as K(x) = K0(1 + αx) where x is the distance measured from one of the plates. If (αd) << 1, the total capacitance of the system is best given by the expression: (1) AK0ε0 d (1 + αd2 ) (2) AK0ε0d [1 + ( αd2 ) 2] α2d2 (3) AK0ε0 + d (1 + αd) d (1 2 ) (4) AK0ε0 JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper

Q11.Two identical electric point dipoles have dipole have dipole moments →p1 = pˆi and →p2 = −pˆi and are held on the x-axis at distance ' a ' from each other. When released, they move along the x-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is ' m', their speed when they are infinitely far apart is : (1) P 1 (2) P 1 a √ πε0 ma a √ 2πε0 ma (3) P 2 (4) P 3 a √ πε0 ma a √ 2πε0 ma

Q12.A particle of charge q and mass m is moving with a velocity −vˆi(v ≠0) → Y −Z plane at distance d. If there is magnetic field B = B0ˆk, the minimum value of v for which the particle will not hit the screen is : (1) qdB0 (2) 2qdB0 3m m (3) qdB0 (4) qdB0 m 2m

Q12.A capacitor is made of two square plates each of side ‘ a ’ making a very small angle α between them, as shown in figure. The capacitance will be close to: (1) ∈0a2 d (1 −αa2d ) (2) ∈0a2d (1 −αa4d ) (3) ∈0a2 d (1 + αad ) (4) ∈0a2d (1 −3αa2d )