The displacement-time graphs of two moving particles make angles of $30^{\circ}$ and $45^{\circ}$ with the time-axis as shown in the figure. The ratio of their respective velocities is:

The $x-t$ graph of a particle moving along a straight line is shown in the figure. The speed-time graph of the particle is correctly shown by:

The position-time $(x-t)$ graphs for two children $A$ and $B$ returning from their school $O$ to their homes $P$ and $Q$ respectively are shown in the figure. Choose the correct entries in the brackets below:
$(a)$ $(A/B)$ lives closer to the school than $(B/A)$
$(b)$ $(A/B)$ starts from the school earlier than $(B/A)$
$(c)$ $(A/B)$ walks faster than $(B/A)$
$(d)$ $A$ and $B$ reach home at the (same/different) time
$(e)$ $(A/B)$ overtakes $(B/A)$ on the road (once/twice)

The velocity-time graph for a particle moving along the $x$-axis is shown in the figure. The corresponding displacement-time graph is correctly shown by

The displacement $x$ of a particle varies with time $t$ as $x = a e^{-\alpha t} + b e^{\beta t}$,where $a, b, \alpha, \text{and } \beta$ are positive constants. The velocity of the particle will:

The velocity-displacement $(v-s)$ graph shows the motion of a particle moving in a straight line. The velocity-displacement graph is a circle of radius $2 \ m$ and the center is at $(2, 0) \ m$. The value of acceleration for this particle at a point $(2-\sqrt{2}, \sqrt{2}) \ m$ will be $ms^{-2}$.