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JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Complex Number

The area (in sq. units) of the region S=\ z ∈ ℂ:|z-1| ≤ 2 ;(z+ z...

The area (in sq. units) of the region S={z∈C:∣z−1∣≤2;(z+zˉ)+i(z−zˉ)≤2,Im⁡(z)≥0}S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{Im}(z) \geq 0\} is

Options

  1. A7π3\frac{7 \pi}{3}
  2. B7π4\frac{7 \pi}{4}
  3. C17π8\frac{17 \pi}{8}
  4. D3π2\frac{3 \pi}{2}

Correct answer

D. 3π2\frac{3 \pi}{2}

Step-by-step solution

Put z=x+iy∣z−1∣≤2⇒(x−1)2+y2≤4z=x+i y |z-1| \leq 2 \Rightarrow(x-1)^2+y^2 \leq 4 ...(i) ⇒x−y≤1\Rightarrow x-y \leq 1 ...(ii) Im⁡(z)≥0⇒y≥0\operatorname{Im}(\mathrm{z}) \geq 0 \Rightarrow \mathrm{y} \geq 0 ...(iii)  Required area = Area of semi-circle − area of sector A 12π(2)2−π2=3π2\begin{aligned} & \text{ Required area } \\ & =\text{ Area of semi-circle }- \text{ area of sector A } \\ & \frac{1}{2} \pi(2)^2-\frac{\pi}{2} \\ & =\frac{3 \pi}{2}\end{aligned}
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