Q44 / 110 · VITEEE 2023 · Mathematics

If \(f(z)=\frac{7-z}{1-z^2}\), where \(z=1+2 i\), then \(|f(z)|\) is equal to :

  • Correct answer
    A
    \(\frac{|z|}{2}\)
  • B
    \(|z|\)
  • C
    \(2|z|\)
  • D
    None

Solution

\(\begin{aligned} & \mathrm{z}=1+2 \mathrm{i} \Rightarrow|\mathrm{z}|=\sqrt{1+4}=\sqrt{5} \\ & \therefore \mathrm{f}(\mathrm{z})=\frac{7-\mathrm{z}}{1-\mathrm{z}^2}=\frac{7-1-2 \mathrm{i}}{1-(1+2 \mathrm{i})^2} \\ & =\frac{6-2 \mathrm{i}}{1-(1-4+4 \mathrm{i})}=\frac{6-2 \mathrm{i}}{4-4 \mathrm{i}}=\frac{3-\mathrm{i}}{2-2 \mathrm{i}} \\ & \Rightarrow|\mathrm{f}(\mathrm{z})|=\left|\frac{3-\mathrm{i}}{2-2 \mathrm{i}}\right|=\frac{|3-\mathrm{i}|}{|2-2 \mathrm{i}|} \\ & =\frac{\sqrt{9+1}}{\sqrt{4+4}}=\frac{\sqrt{5}}{2}=\frac{|\mathrm{z}|}{2}\end{aligned}\)

Hence, the correct answer is option: 'A'.

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