Two insulated wires perpendicular to each other in the same plane carry equal currents as shown in figure. Is there a region where the magnetic field is zero? If so, where is this region? If not, explain why the field is never zero?

In the upper left and lower right quadrants, the fields due to the two currents are in opposite directions. (The fields from both currents in the lower left and upper right quadrants are in the same direction.) It is possible for the fields to cancel in the upper left and lower right quadrants.
For the fields to cancel, the magnitudes must be equal. The magnetic field due to a long, straight current depends inversely on the distance from the current. Therefore, The magnetic field will cancel where the distance to both currents is the same. A diagonal line from lower right to upper left will be equidistant from both currents. The magnetic field along this diagonal line is zero.