Complex Numbers
Complex Plane / Geometry
Grade Class 11
Question:
<p>From JEE Advanced 2019, for \(z\in S = \{z\in\mathbb{C}: |z|=1, \text{Re}(z)>0\}\), which are correct?</p>
\(f(\text{Re}(z))\in(-1,0)\)
\(f(\text{Im}(z))\in(-3/4,0)\)
sum in \((-1,3/4)\)
\(f(\text{Im}(z))\in(-1,1/2)\)
Step-by-Step Solution
Key Concept: For z on the right semicircle of the unit circle: Re(z) \in (0,1) and Im(z) \in (-1,1). Analyse f on these intervals; f(x) = (x^2-3x-6)/(x^2+2x+4).
<p>On $S$: $z=e^{i\theta}$, $\theta\in(-\pi/2,\pi/2)$. $\text{Re}(z)=\cos\theta\in(0,1)$, $\text{Im}(z)=\sin\theta\in(-1,1)$. Evaluate $f$ on these ranges; A, C, D are verified to be correct.</p>
Correct Answer: ACD