Practice Questions

Q79.If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line x+1 2 = y−31 = z+2−1 and containing the line x−23 = 1−y2 = z+11 is αx + βy + γz = 24 then α + β + γ is equal to: (1) 20 (2) 19 (3) 18 (4) 21

Q80.Let 𝑆= {1, 2, 3, 4, 5, 6} . Then the probability that a randomly chosen onto function 𝑔 from 𝑆 to 𝑆 satisfies 𝑔3 = 2 𝑔1 is : 1 1 (1) (2) 15 5 (3) 1 (4) 1 30 10

Q80.Let A be a set of all 4 -digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is: (1) 1 (2) 122 5 297 (3) 97 (4) 2 297 9

Q80.Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The probability that such formed matrices have all different entries and are non-singular, is: (1) 45 (2) 23 162 81 (3) 22 (4) 43 81 162

Q80.The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is: (1) 65 (2) 65 28 27 (3) 35 (4) 135 27 29

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

Q3. A small ball of mass m is thrown upward with velocity u from the ground. The ball experiences a resistive force mkv2 where v is it speed. The maximum height attained by the ball is : ku2 (1) + 2k 1 tan−1 ku2g (2) k1 ln(1 2g ) ku2 (3) + k 1 tan−1 ku22g (4) 2k1 ln(1 g )

Q3. A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4Cx2 and rotating with angular speed ω (see figure). The value of ω is (neglect friction) (1) 2√2gC (2) √2gC (3) √2gCab (4) √2gC

Q3. Particle A of mass m1 moving with velocity (√3ˆi +ˆj) ms−1 collides with another particle which is at rest initially. Let →v1 and →v2 be the velocities of particles A and B after collision respectively. If + and →v2 is : m1 = 2m2 and after collision →v1 −(ˆi √3ˆj) ms−1 , the angle between →v1 (1) 15° (2) 60° (3) −45° (4) 105°

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 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. 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 )

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

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

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

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)

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. 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

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.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