$A$ magnetic compass needle is placed in the plane of paper near point $A$ as shown in the figure. In which plane should a straight current-carrying conductor be placed so that it passes through $A$ and there is no change in the deflection of the compass? Under what condition is the deflection maximum and why?

(N/A) The conductor should be placed in the plane of the paper itself. Since the axis of the compass needle is vertical,the magnetic field produced by a current-carrying conductor lying in the same plane as the compass will also be vertical at the position of the needle. $A$ vertical magnetic field cannot exert a torque on a compass needle that is pivoted to rotate only in the horizontal plane,hence there is no change in deflection.
The deflection is maximum when the conductor passing through $A$ is placed perpendicular to the plane of the paper. In this orientation,the magnetic field lines produced by the current form concentric circles in the plane of the paper,exerting the maximum horizontal torque on the compass needle.