Two particles having position vectors $r_1 = (3 \hat{i} + 5 \hat{j}) \text{ m}$ and $r_2 = (-5 \hat{i} - 3 \hat{j}) \text{ m}$ are moving with velocities $v_1 = (4 \hat{i} + 3 \hat{j}) \text{ m/s}$ and $v_2 = (a \hat{i} + 7 \hat{j}) \text{ m/s}$. If they collide after $2 \text{ s}$,then the value of $a$ is:

$A$ train of $150 \ m$ length is going towards the north direction at a speed of $10 \ m/s$. $A$ parrot flies at a speed of $5 \ m/s$ towards the south direction parallel to the railway track. The time taken by the parrot to cross the train is........$s$.

$A$ $120 \, m$ long train is moving towards west with a speed of $10 \, m/s$. $A$ bird flying towards east with a speed of $5 \, m/s$ crosses the train. The time taken by the bird to cross the train will be ........ $sec$.

Two trains,each $50 \, m$ long,are travelling in opposite directions with velocities $10 \, m/s$ and $15 \, m/s$. The time taken for them to cross each other is:

$A$ man goes $10\,km$ downstream in $2\,hrs$ and $30\,km$ upstream in $10\,hrs$. How much time will he take to swim $40\,km$ in still water?

$A$ river of width $200 \ m$ is flowing from west to east with a speed of $18 \ km/h$. $A$ boat, moving with a speed of $36 \ km/h$ in still water, is made to travel one round trip (bank to bank of the river). The minimum time taken by the boat for this journey and the displacement along the river bank are . . . . . . and . . . . . . respectively.