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Complex Numbers - JEE Mathematics question (JEE Advanced 2013 Paper 2)

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

Let $S=S_1\cap S_2\cap S_3$, where $S_1=\{z\in\mathbb{C}:|z|<4\}$, $S_2=\left\{z\in\mathbb{C}:\operatorname{Im}\left[\frac{z-1+\sqrt3 i}{1-\sqrt3 i}\right]>0\right\}$ and $S_3=\{z\in\mathbb{C}:\operatorname{Re}z>0\}$.
JEE Advanced 2013 Paper 2Question 1
Area of $S=$
Complex Numbers
Options for question 1, choose one

Solution

$\operatorname{Im}\frac{z-(1-\sqrt3 i)}{1-\sqrt3 i}>0$: the line through $1-\sqrt3 i$ with direction $1-\sqrt3 i$ passes through $0$, namely $y=-\sqrt3x$, and $S_2$ is the side containing $(1,0)$. With $x>0$ and $|z|<4$, $S$ is the sector of radius $4$ from angle $-\frac\pi3$ to $\frac\pi2$ (angle $\frac{5\pi}6$): area $=\frac12\cdot16\cdot\frac{5\pi}6=\frac{20\pi}3$.

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