State whether the following statements are true or false:
$(a)$ Power is the vector product of force and velocity.
$(b)$ $1\,kg\,m = 9.8\,J$
$(c)$ When a teacher rubs a blackboard with a duster,they do no work.
$(d)$ Kinetic energy can be negative.

An object of mass $m$ is sliding down a hill of arbitrary shape and after traveling a certain horizontal path stops because of friction. The friction coefficient is different for different segments of the entire path but it is independent of the velocity and direction of motion. The work done that a force must perform to return the object to its initial position along the same path would be:

Is it possible for the kinetic energy of an object not to increase when work is done on it? If so,when?

$A$ bullet moving with a velocity of $100\, m/s$ can just penetrate two planks of equal thickness. The number of such planks penetrated by the same bullet,when the velocity is doubled,will be

Three objects $A$,$B$,and $C$ are kept in a straight line on a frictionless horizontal surface. These have masses $m$,$2m$,and $m$,respectively. The object $A$ moves towards $B$ with a speed of $9 \ m/s$ and makes an elastic collision with it. Thereafter,$B$ makes a completely inelastic collision with $C$. All motions occur on the same straight line. Find the final speed (in $m/s$) of the object $C$.

$A$ simple pendulum of length $L = \frac{10}{3} \text{ m}$ with a bob of mass $M = 3m$ is hanging freely from a rigid support. $A$ bullet of mass $m$ is fired with a velocity $u = 50 \text{ ms}^{-1}$ from the ground at an angle $\theta$ with the horizontal. When the bullet is at its highest point of its trajectory,it collides head-on with the bob of the pendulum and gets embedded in the bob. After collision,if the pendulum moves through a maximum angle of $120^{\circ}$,then the value of $\theta$ is $(g = 10 \text{ ms}^{-2})$.