The figure shows a velocity-time graph of a particle moving along a straight line  The correct displacement-time graph of the particle is shown as

The length of second's hand in watch is $1 \,cm.$ The change in velocity of its tip in $15$ seconds is

The initial velocity of a projectile is $\vec u = (4\hat i + 3\hat j)\,m/s$ it is moving with uniform acceleration $\vec a = (0.4\hat i + 0.3\hat j)\, m/s^2$ The magnitude of its velocity after $10\,s$ is.........$m/s$

For any arbitrary motion in space, which of the following relations are true

$(a)$ $\left. v _{\text {average }}=(1 / 2) \text { (v }\left(t_{1}\right)+ v \left(t_{2}\right)\right)$

$(b)$ $v _{\text {average }}=\left[ r \left(t_{2}\right)- r \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

$(c)$ $v (t)= v (0)+ a t$

$(d)$ $r (t)= r (0)+ v (0) t+(1 / 2)$ a $t^{2}$

$(e)$ $a _{\text {merage }}=\left[ v \left(t_{2}\right)- v \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

(The 'average' stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

Read each statement below carefully and state, with reasons and examples, if it is true or false :

A scalar quantity is one that

$(a)$ is conserved in a process

$(b)$ can never take negative values

$(c)$ must be dimensionless

$(d)$ does not vary from one point to another in space

$(e)$ has the same value for observers with different orientations of axes.