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