$A$ steamboat goes across a lake and comes back $(a)$ on a quiet day when the water is still and $(b)$ on a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was the same,in which case will it complete the journey in lesser time?

$A$ ship $X$ moving due north with speed $v$ observes that another ship $Y$ is moving due west with the same speed $v$. The actual velocity of $Y$ is $........$.

Two cities $X$ and $Y$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. $A$ girl is driving a scooty with a speed of $60 \ km/h$ in the direction $X$ to $Y$ and notices that a bus goes past her every $30$ minutes in the direction of her motion,and every $10$ minutes in the opposite direction. Choose the correct option for the period $T$ of the bus service and the speed (assumed constant) of the buses.

$Assertion$ : The magnitude of velocity of two boats relative to the river is the same. Both boats start simultaneously from the same point on one bank and may reach the opposite bank simultaneously while moving along different paths.
$Reason$ : For boats to cross the river in the same time,the component of their velocity relative to the river in the direction normal to the flow should be the same.

When a train is at rest,rain drops fall vertically. When it moves along east with a velocity equal to that of rain,the rain will appear to fall,for a passenger in the train

$A$ 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.