$(a)$ How much work is done when a force of $1\, N$ moves a body through a distance of $1\, m$ in its direction?
$(b)$ Is it possible that a force is acting on a body but still the work done is zero? Explain giving one example.

(N/A) Work done is calculated by the formula $W = F \times s$. Given $F = 1\, N$ and $s = 1\, m$,the work done is $1\, N \times 1\, m = 1\, J$.
$(b)$ Yes,it is possible for the work done to be zero even if a force is acting on a body. This occurs when the force is applied at an angle of $90^{\circ}$ to the direction of displacement. Since $W = Fs \cos(\theta)$,if $\theta = 90^{\circ}$,then $\cos(90^{\circ}) = 0$,making the work done zero.
Example: When a satellite moves in a circular orbit around the Earth,the gravitational force acts towards the center of the Earth (perpendicular to the motion),while the displacement is along the tangent to the orbit. Thus,the work done by gravity on the satellite is zero.