MHTCET 25 Apr 2025 Shift 2 Paper

Concept:Use stepwise movement on the complex plane, and treat the final $\frac{\pi}{2}$ anticlockwise rotation about the origin as multiplication by $i$.Explanation:Initially, $Z_0 = 1 + 2i$, so the point is $(1, 2)$.Moving $5$ units horizontally away from the origin means moving along the positive real axis: $1 + 5 = 6$.So the point becomes $6 + 2i$.Then moving $3$ units vertically upwards adds $3i$, giving $Z_1 = 6 + 5i$.The direction $\hat{i} + \hat{j}$ has unit vector $\frac{\hat{i} + \hat{j}}{\sqrt{2}}$.Moving $\sqrt{2}$ units in this direction gives displacement $(1 + i)$.Hence, the point reached is $Z_1 + (1 + i) = (6 + 5i) + (1 + i) = 7 + 6i$.Moving through angle $\frac{\pi}{2}$ anticlockwise on a circle centred at the origin is a $90^\circ$ rotation, which multiplies the complex number by $i$.So, $Z_2 = i(7 + 6i) = -6 + 7i$.Answer:$Z_2 = -6 + 7i$Thus, the correct option is C.