Question

An insect starts from the origin in the argand plane and goes $4 \, km$ ( $N \, 45^\circ \, E$ ) then it moves $3 \, km \, \left(N \, 45 ^\circ W\right)$ and then takes an angular movement of $\frac{\pi }{3}$ about origin in the anticlockwise direction. The final position of the insect is

• A. $\left(4 - 3 i\right)e^{\frac{i 5 \pi }{6}}$
• B. $\left(4 + 3 i\right)e^{\frac{i 5 \pi }{6}}$
• C. $\left(4 - 3 i\right)e^{\frac{i 3 \pi }{4}}$
• D. $\left(4 + 3 i\right)e^{\frac{i 7 \pi }{12}}$

Answer 1

Answer: (D)

Travelling $4 \, km \, \left(N \, 45 ^\circ E\right)$ it reaches at $z_{1}=4e^{\frac{i \pi }{4}}$
Travelling $3 \, km \, \left(N \, 45 ^\circ W\right)$ it reaches at $z_{2}$
So, $\frac{0 - z_{1}}{z_{2} - z_{1}}=\frac{4}{3}i\Rightarrow z_{2}=\left(4 + 3 i\right)e^{\frac{i \pi }{4}}$

After an angular movement of $\frac{\pi }{3}$ in the anticlockwise direction it reaches at $z_{3}$
$\therefore z_{3}=z_{2}e^{\frac{i \pi }{3}}=\left(4 + 3 i\right)e^{\frac{i 7 \pi }{12}}$