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

Answer: (D)

$z_{2}{ }^{\prime}=\left(6+\sqrt{2} \cos 45^{\circ}, 5+\sqrt{2} \sin 45^{\circ}\right)=(7,6)=7+6 i$
By rotation about $(0,0)$
$\frac{z_{2}}{z_{2}^{\prime}}=e^{j \pi 2}$
$\Rightarrow z_{2}=z_{2}{ }^{\prime}\left(e^{i \frac{\pi}{2}}\right)$
$=(7+6 i)\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$
$=(7+6 i)(i)=-6+7 i$