Question

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

• A. $\frac{|z|}{2}$
• B. $| z |$
• C. $2| z |$
• D. None of these

Answer 1

Answer: (A)

Given $f(z)=\frac{7-z}{1-z^{2}}$
where $z=1+2 i$
$\Rightarrow |z|=\sqrt{1^{2}+2^{2}}=\sqrt{5}$
$\Rightarrow f(z)=\frac{y-z}{1-z^{2}}$
$\Rightarrow \frac{y-(1+2 i)}{1-(1+2 i)^{2}}$
$=\frac{7-1-2 i}{1-1-4 i^{2}-4 i}$
$\Rightarrow \frac{6-2 i}{4-4 i}$
$=\frac{3-i}{2-2 i}$
$=\frac{3-i}{2-2 i} \times \frac{2+2 i}{2+2 i}$
$=\frac{6+6 i-2 i-2 i^{2}}{4-4 i}$
$=\frac{6+4 i+2}{4+4} $
$\Rightarrow \frac{8+4 i}{8}$
$ \Rightarrow 1+1 / 2 i$
$\Rightarrow |f(z)|=\sqrt{1^{2}+(1 / 2)^{2}}$
$ \Rightarrow \sqrt{1+1 / 4}$
$=\frac{\sqrt{5}}{2}$
$ \Rightarrow \frac{|z|}{2}$