The displacement x of a particle varies with time t as x = ae^-αt + be^βt where a, b, $\mathrm{\alpha }$ and β are positive constants. The velocity of the particle will

• decrease with time

• be independent of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

• drop to zero when $\mathrm{\alpha } = \mathrm{\beta }$

• increase with time

If A + B = C and that C is perpendicular to A. What is the angle between A and B, if $\left|\mathrm{A}\right| = \left|\mathrm{B}\right|$

• $\frac{\mathrm{\pi }}{4}$ rad

• $\frac{\mathrm{\pi }}{2}$ rad

• $\frac{3 \mathrm{\pi }}{4}$ rad

• $\mathrm{\pi }$ rad

A body is whirled in a horizontal circle of radius 25 cm. It has an angular velocity of 13 rad/s. What is its linear velocity at any point on circular path?

• 2 m/s

• 3 m/s

• 3.25 m/s

• 4.25 m/s

Two wires are stretched through same distance. The force constant ofsecond wire is half as that of the first wire. The ratio of work done to stretch first wire and second wire will be

• 2 : 1

• 1 : 2

• 3 : 1

• 1 : 3

A particle slides down on a smooth incline of inclination 30°, fixed in an elevator going up with an acceleration 2 m/s². The box of incline has a length 4m. The time taken by the particle to reach the bottom will be

• ( 8/9 ) √3 s

• ( 9/8 ) √3 s

• ( 4/3 ) √(√3/2 ) s

• ( 3/4 ) √(√3/2 ) s

A stone projected with a velocity u at an angle θ with the horizontal reaches maximum height H₁. When it is projected with velocity u at an angle $\left(\frac{\mathrm{\pi }}{2} - \mathrm{\theta }\right)$ with the horizontal, it reaches maximum height H₂. The relation between the horizontal range R of the projectile, H₁ and H₂ is 

• R = $4 \sqrt{{\mathrm{H}}_1 {\mathrm{H}}_2}$

• R = 4 ( H₁ - H₂ )

• R = 4( H₁ + H₂ )

• R = $\frac{{\mathrm{H}}_1^2}{{\mathrm{H}}_2^2}$

A light string passes over a frictionless pulley. To one of its ends a mass of 8 kg is attached. To its other end two masses of 7 kg each are attached. The acceleration of the system will be

• 10.2 g

• 5.10 g

• 20.36 g

• 0.27 g

A sphere of mass m moving with velocity v hits inelastically with another stationary sphere of same mass. The ratio of their final velocities will be (in terms of e)

• $\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2} = \frac{1 + \mathrm{e}}{1 - \mathrm{e}}$

• $\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2} = \frac{1 - \mathrm{e}}{1 + \mathrm{e}}$

• $\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2} = \frac{1 + \mathrm{e}}{2}$

• $\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2} = \frac{1 - \mathrm{e}}{2}$

Particles of masses m, 2m, 3m, ... , nm are placed on the same line at distances L, 2L, 3L, ... , nL from O. The distance of centre of mass from O is

A ball of radius R rolls without slipping. Find the fraction of total energy associated with its rotational energy, if the radius of the gyration of the ball about an axis passing through its centre of mass is K.

• $\frac{{\mathrm{K}}^2}{{\mathrm{K}}^2 + {\mathrm{R}}^2}$

• $\frac{{\mathrm{R}}^2}{{\mathrm{K}}^2 + {\mathrm{R}}^2}$

• $\frac{{\mathrm{K}}^2 + {\mathrm{R}}^2}{{\mathrm{R}}^2}$

• $\frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$