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.

$A$ body of mass $5\,kg$ is moving with a momentum of $10\,kg\,m/s$. Now a force of $2\,N$ acts on the body in the direction of its motion for $5\,s$. The increase in the kinetic energy of the body is $...........J$.

$A$ 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 $(g = 10 \text{ ms}^{-2})$.

$A$ block of mass $0.18 \ kg$ is attached to a spring of force constant $2 \ N/m$. The coefficient of friction between the block and the floor is $0.1$. Initially,the block is at rest and the spring is unstretched. The block is pushed as shown in the figure. The block slides a distance of $0.06 \ m$ and comes to rest. If the initial velocity of the block is $V = N/10 \ m/s$,then what is the value of $N$?

$A$ bullet of mass $m_1$ is moving with speed $v_0$ and hits a sand bag of mass $m_2$. If the speed of the bullet after passing through the sand bag is $\frac{v_0}{3}$,then the height $h$ up to which the bag rises is (assume,$g=$ acceleration due to gravity).

$A$ body of mass $100 \ gm$ moves with a constant speed along a circular path of radius $r$. What is the work done during one complete revolution?