Let ${ABCDEF}$ be a regular hexagon, with the vertices taken in order. The resultant of the vectors: $\overrightarrow{AB},~\overrightarrow{BC},~\overrightarrow{CD},~\overrightarrow{DE}$ equals, in magnitude, the vector:

Let $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}},$ then:

A particle starting from the origin $(0,0)$ moves in a straight line in the $(x,y)$ plane. Its coordinates at a later time are ($\sqrt{3}$, $3).$ The path of the particle makes an angle of __________ with the $x$-axis:
1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $0$

The position of a moving particle at time $t$ is $\overrightarrow{r}=3\hat{i}+4t^{2}\hat{j}-t^{3}\hat{k}.$ Its displacement during the time interval $t=1$ s to $t=3$ s will be:

Three girls skating on a circular ice ground of radius $200$ m start from a point $P$ on the edge of the ground and reach a point $Q$ diametrically opposite to $P$ following different paths as shown in the figure. The correct relationship among the magnitude of the displacement vector for three girls will be:

1. $A > B > C$
2. $C > A > B$
3. $B > A > C$
4. $A = B = C$

A cat is situated at point $A$ ($0,3,4$) and a rat is situated at point $B$ ($5,3,-8$). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. $5$ unit
2. $12$ unit
3. $13$ unit
4. $17$ unit

A particle is moving on a circular path of radius $R.$ When the particle moves from point $A$ to $B$ (angle $\theta$), the ratio of the distance to that of the magnitude of the displacement will be:

         
1. $\dfrac{\theta}{\sin\frac{\theta}{2}}$
2. $\dfrac{\theta}{2\sin\frac{\theta}{2}}$
3. $\dfrac{\theta}{2\cos\frac{\theta}{2}}$
4. $\dfrac{\theta}{\cos\frac{\theta}{2}}$

A particle is moving such that its position coordinates $(x,y)$ are $​ (2~\text m,  3~\text m)​$ at time $t=0,$   $​ (6~\text m,  7~\text m)​$ at time $t=2~\text s$  and $​ (13~\text m,  14~\text m)​$ at time $t=5~\text s.$  The average velocity vector $(v_{avg})$ from $t=0$ to $t=5~\text s$ is: