Two particles $P_1$ and $P_2$ are moving with velocities $v_1$ and $v_2$ respectively. Which of the following statements about their relative velocity $v_{12}$ is true?

$A$ swimmer swims in still water at a speed of $5 \text{ km/hr}$. He enters a $200 \text{ m}$ wide river,having a river flow speed of $4 \text{ km/hr}$,at point $A$ and proceeds to swim at an angle of $127^{\circ}$ $(\sin 37^{\circ} = 0.6)$ with the river flow direction. Another point $B$ is located directly across $A$ on the other side. The swimmer lands on the other bank at a point $C$,from which he walks the distance $CB$ with a speed of $3 \text{ km/hr}$. The total time in which he reaches from $A$ to $B$ is .......... $\text{minutes}$.

When a car is at rest,its driver sees rain drops falling on it vertically. When driving the car with speed $v$,he sees that rain drops are coming at an angle $60^{\circ}$ from the horizontal. On further increasing the speed of the car to $(1+\beta)v$,this angle changes to $45^{\circ}$. The value of $\beta$ is close to...

$A$ person walks up a stalled escalator in $90 \, s$. When just standing on the same moving escalator,he is carried in $60 \, s$. The time it would take him to walk up the moving escalator will be $....... \, s$.

The figure shows two ships moving in the $x-y$ plane with velocities $V_A$ and $V_B$. The ships move such that $B$ always remains north of $A$. The ratio $\frac{V_A}{V_B}$ is equal to ........

Two trains $A$ and $B$ travel on two parallel rail tracks in opposite directions with speeds $v_1$ and $v_2$,respectively. They take $4 \ s$ to pass each other at these speeds. If the speed of train $A$ is increased by $50 \%$,then they take $3 \ s$ to pass each other. The ratio $v_1 / v_2$ is: