Practice Questions

Q63.If z is a non-real complex number, then the minimum value of Im z5 is (Where Im z = Imaginary part of z ) (Im z)5 JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper (1) −2 (2) −4 (3) −5 (4) −1

Q68.The sum of coefficients of integral powers of x in the binomial expansion of (1 −2√x) 50 is (1) 2 1 (250 + 1) (2) 12 (350 + 1) (3) 1 2 (350) (4) 12 (350 −1)

Q69.Locus of the image of the point (2, 3) in the line (2x −3y + 4) + k(x −2y + 3) = 0, k∈R , is a (1) Circle of radius √3 (2) Straight line parallel to x-axis. (3) Straight line parallel to y-axis. (4) Circle of radius √2

Q72.The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the x2 y2 ellipse 9 + 5 = 1, is (1) 27 (2) 274 (3) 18 (4) 272

Q79.The least value of the product xyz (such that x, y and z are positive real numbers) for which the determinant x 1 1 1 y 1 is non-negative is 1 1 z (1) −1 (2) −16√2 (3) −8 (4) −2√2

Q81.Let k and K be the minimum and the maximum values of the function f(x) = (1+x)0.6 in [0, 1], respectively, 1+x0.6 then the ordered pair (k, K) is equal to: (1) (2−0.4, 1) (2) (2−0.6, 1) (3) (2−0.4, 20.6) (4) (1,20.6) JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper 1

Q83.Let f : R →R be a function such that f(2 −x) = f(2 + x) and f(4 −x) = f(4 + x), for all x ∈R and 2 50 ∫ f(x)dx = 5. Then the value of ∫ f(x)dx is 0 10 (1) 100 (2) 125 (3) 80 (4) 200

Q87.Let →a, b and →c be three non - zero vectors such that no two of them are collinear and × →c→a. If θ is the angle between vectors b and →c, then a value of sin θ is = 13 b (→a → → → b) ×→c (1) −2√3 (2) 2√2 3 3 (3) −√2 (4) 2 3 3

Q88.The shortest distance between the z - axis and the line x + y + 2z −3 = 0 = 2x + 3y + 4z −4, is (1) 1 (2) 2 (3) 3 (4) 4

Q89.If the shortest distance between the line x−1α = y+1−1 = 1z , (α ≠−1) , and x + y + z + 1 = 0 = 2x −y + z + 3 is 1 ,then value of α is : √3 (1) −1916 (2) 3219 (3) −1619 (4) 1932

Q90.If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is: (1) 1 (2) 1 69 26 (3) 1 (4) 1 21 15 JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper

Q90.If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is (1) 22( 13 )11 (2) 195 (3) 55( 32 )10 (4) 220( 31 )12 JEE Main 2015 (04 Apr) JEE Main Previous Year Paper

Q6. From a sphere of mass M and radius R, a smaller sphere of radius R2 is carved out such that the cavity made in the original sphere is between its centre and the periphery (See figure). For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is 3R, the gravitational force between the two sphere is: (1) 41GM2 (2) 41GM2 3600R2 450R2 (3) 59GM2 (4) GM2 450R2 225R2 JEE Main 2014 (11 Apr Online) JEE Main Previous Year Paper

Q8. There is a circular tube in a vertical plane. Two liquids which do not mix and of densities d1 and d2 are filled in the tube. Each liquid subtends 90o angle at centre. Radius joining their interface makes an angle α with vertical. Ratio d1 is : d2 (1) 1+sinα (2) 1+cosα 1−sinα 1−cosα (3) 1+tanα (4) 1+sinα 1−tanα 1−cosα

Q9. A cylindrical vessel of cross-section A contains water to a height h. There is a hole in the bottom of radius ' a '. The time in which it will be emptied is: (1) 2 A (2) √2 A πa2 √hg πa2 √hg (3) 2√2 A (4) A g πa2 √h g √2πa2 √h

Q9. A large number of liquid drops each of radius r coalesce to form a single drop of the radius R. The energy released in the process is converted into kinetic energy of the big drop so formed. The speed of the big drop is (given surface tension of the liquid T , density ρ ) ( 1r −1R ) (1) √2Tρ ( 1r −1R ) (2) √6Tρ ( 1r −1R ) (3) √4Tρ ( 1r −1R ) (4) √Tρ

Q10.The equation of state for a gas is given by PV = nRT + αV , where n is the number of moles and α is a positive constant. The initial temperature and pressure of one mole of the gas contained in a cylinder are T0 and P0 respectively. The work done by the gas when its temperature doubles isobarically will be : (1) P0 T0 ln 2 (2) P0T0RP0+α (3) P0T0R (4) P0 T0R P0−α

Q10.Two soap bubbles coalesce to form a single bubble. If V is the subsequent change in volume of contained air and S change in total surface area, T is the surface tension and P atmospheric pressure, then which of the following relation is correct? (1) 4PV + 3ST = 0 (2) 3PV + 4ST = 0 (3) 2PV + 3ST = 0 (4) 3PV + 2ST = 0

Q11.A black coloured solid sphere of radius R and mass M is inside a cavity with a vacuum inside. The walls of the cavity are maintained at temperature T0 . The initial temperature of the sphere is 3T0 . If the specific heat of the material of the sphere varies as αT 3 per unit mass with the temperature T of the sphere, where α is a constant, then the time taken for the sphere to cool down to temperature 2T0 will be ( σ is Stefan Boltzmann constant) (1) 16πR2σ Mα ln( 32 ) (2) 16πR2σMα ln( 163 ) (3) 4πR2σ Mα ln( 23 ) (4) 4πR2σMα ln( 163 )

Q12.Two bodies of masses 1 kg and 4 kg are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency 25 rad s−1 , and amplitude 1. 6 cm while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take g = 10 m s−2 ). (1) 20 N (2) 60 N (3) 40 N (4) 10 N

Q12.Three rods of Copper, Brass and Steel are welded together to form a Y-shaped structure. Area of cross-section of each rod is 4 cm2 . End of copper rod is maintained at 100°C. Where as ends of brass and steel are kept at JEE Main 2014 (06 Apr) JEE Main Previous Year Paper 0°C. Lengths of the copper, brass and steel rods are 46, 13 and 12 cms respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are 0. 92, 0. 26 and 0. 12 CGS units respectively. Rate of heat flow through copper rod is : (1) 1. 2 Cal /s (2) 2. 4 Cal /s (3) 4. 8 Cal /s (4) 6. 0 Cal /s

Q13.An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section 8.0 × 10−3 m2 . Initially the gas is at 300 K and occupies a volume of 2.4 × 10−3 m3 and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by 0.1 m. The force constant of the spring is 8000 N/m and the atmospheric pressure is 1.0 × 105 N/m2 . The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires JEE Main 2014 (11 Apr Online) JEE Main Previous Year Paper of the heater. The heat capacity of the heater coil is also negligible). (1) 300 K (2) 800 K (3) 500 K (4) 1000 K k r2 k is the spring m − 4 m2 ) where

Q13.The amplitude of a simple pendulum, oscillating in air with a small spherical bob, decreases from 10 cm to 8 cm in 40 seconds. Assuming that Stokes law is valid, and ratio of the coefficient of viscosity of air to that of carbon dioxide is 1.3, the time in which amplitude of this pendulum will reduce from 10 cm to 5 cm in carbondioxide will be close to (ln 5 = 1.601, ln 2 = 0.693). (1) 231 s (2) 142 s (3) 208 s (4) 161 s

Q14.A particle moves with simple harmonic motion in a straight line. In first τ s, after starting from rest it travels a distance a, and in next τ s it travels 2a, in same direction, then : (1) Amplitude of motion is 3a (2) Time period of oscillations is 8τ (3) Amplitude of motion is 4a (4) Time period of oscillations is 6τ

Q17.The gap between the plates of a parallel plate capacitor of area A and distance between plates d, is filled with a dielectric whose relative permittivity varies linearly from ϵ1 at one plate to ϵ2 at the other. The capacitance of the capacitor is JEE Main 2014 (19 Apr Online) JEE Main Previous Year Paper (1) ϵ0(∈2−∈1)A (2) ϵ0(∈2+∈1)A [d ln(∈2/∈1)] 2d (3) ϵ0(∈1+∈2)A (4) ϵ0A d [dln(∈2/∈1)]