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 each a point $z_1$. From $z_1$ particle moves $\sqrt{2}$ units in the direction of vector $\hat{i}+\hat{j}$ and then it moves through an angle in anticlockwise 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)

$-6+7i$

Ans.$\left(d\right)$.
$z_0=1+2i=(1,2)=(x_0,y_0)$
$z_1=(x_0+5,y_0+3)=(6,5)=6+5i$
$\sqrt{2}$ in the direction of $i=j$
$\therefore x_1=\sqrt{2}\cos {45}^o$,$y_1=\sqrt{2}\sin {45}^o$
or $(x_1,y_1)=(1,1)$
$\therefore z_2=(7+6i)$. Now rotation about origin through an angle of $\frac{\pi }{2}$ means multiply $z_2$ by $e^{\left(\pi /2\right)i}$
$=\cos {90}^o+i\sin {90}^o=i$
$\therefore$ $z_3=i(7+6i)=-6+7i\Rightarrow \left(d\right)$