Complex Numbers
Range of Real Part of Complex Expression with Fixed Real Part
nta_pyq_2023_apr
Grade 11
Question:
If the set $\left\{\text{Re}\left(\dfrac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):\ z\in\mathbb{C},\ \text{Re}\,z=3\right\}$ is equal to the interval $(\alpha,\beta]$, then $24(\beta-\alpha)$ is equal to
Step-by-Step Solution
Key Concept: Set $z=3+iy$. After simplification, the real part expression reduces to $f(y)=\frac{1}{8}\cdot\frac{9-y^2}{1+y^2}$. Find the range of $f(y)$.
$f(y)\in(-\frac{1}{8},\frac{9}{8}]$. $24(\frac{9}{8}+\frac{1}{8})=24\times\frac{10}{8}=30$.
Correct Answer: 3