Question

An insect starts from the origin in the argand plane and goes $4km$ ( $N45^{\circ }E$ ) then it moves $3km\left(N45^{\circ }W\right)$ and then takes an angular movement of $\frac{\pi }{3}$ about origin in the anticlockwise direction. The final position of the insect is

• A. $\left(4-3i\right)e^{\frac{i5\pi }{6}}$
• B. $\left(4+3i\right)e^{\frac{i5\pi }{6}}$
• C. $\left(4-3i\right)e^{\frac{i3\pi }{4}}$
• D. $\left(4+3i\right)e^{\frac{i7\pi }{12}}$

Answer 1

Answer: (D)

Travelling $4km\left(N45^{\circ }E\right)$ it reaches at $z_1=4e^{\frac{i\pi }{4}}$
Travelling $3km\left(N45^{\circ }W\right)$ it reaches at $z_2$
So, $\frac{0-z_1}{z_2-z_1}=\frac{4}{3}i\Rightarrow z_2=\left(4+3i\right)e^{\frac{i\pi }{4}}$

After an angular movement of $\frac{\pi }{3}$ in the anticlockwise direction it reaches at $z_3$
$\therefore z_3=z_2{e}^{\frac{i\pi }{3}}=\left(4+3i\right)e^{\frac{i7\pi }{12}}$