Question

A particle P starts from the point $z_0=1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\frac{\pi }{2}$in anti-clockwise direction on a circle with centre at origin, to reach a point $z_2$. The point $z_2$ is given by

• A. 6 + 7i
• B. -7 + 6i
• C. 7 + 6i
• D. -6 + 7i

Answer 1

The correct option is D -6 + 7i


$z_2^{{}^{\prime }}=(6+\sqrt{2}cos{45}^{\circ },\sqrt{2}sin{45}^{\circ })=(7,6)=7+6i$
By rotation about (0,0)
$\frac{z_2}{z_2^{{}^{\prime }}}=e^{i\frac{\pi }{2}}$
=(7+6i)$(cos\frac{\pi }{2}+isin\frac{\pi }{2})=(7+6i)\left(i\right)=-6+7i$