Fill in the blanks:
$(a)$ If the distance covered by a particle is zero,then its displacement is ....... .
$(b)$ For an object moving with uniform acceleration $a$,the change in velocity during a time interval $\Delta t$ is ......... .
$(c)$ The time rate of change of position of an object is called .......... .

$A$ car moves towards north at a speed of $54 \,km/h$ for $1 \,h$. Then it moves eastward with the same speed for the same duration. The average speed and velocity of the car for the complete journey is ..........

Which of the following statements are correct?
$(a)$ Average speed of a particle in a given time interval is never less than the magnitude of average velocity.
$(b)$ It is possible to have a situation in which $|d\vec{v}/dt| \neq 0$ but $d/dt|\vec{v}| = 0$.
$(c)$ The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval.
$(d)$ The average velocity of a particle moving on a straight line is zero in a time interval. It is possible that the velocity is never zero in the interval. (Infinite accelerations are not allowed).

$A$ horse rider covers half the distance with $5\,m/s$ speed. The remaining part of the distance was travelled with speed $10\,m/s$ for half the time and with speed $15\,m/s$ for the other half of the time. The mean speed of the rider averaged over the whole time of motion is $\frac{x}{7}\,m/s$. The value of $x$ is

$A$ passenger arriving in a new town wishes to go from the station to a hotel located $10 \;km$ away on a straight road from the station. $A$ dishonest cabman takes him along a circuitous path $23 \;km$ long and reaches the hotel in $28 \;min$. What is
$(a)$ the average speed of the taxi,
$(b)$ the magnitude of average velocity? Are the two equal?

During the first $18\,min$ of a $60\,min$ trip,a car has an average speed of $11\,m/s$. What should be the average speed for the remaining $42\,min$ so that the car has an average speed of $21\,m/s$ for the entire trip (in $,m/s$)?