Complex Numbers And Quadratic Equations question 664
Question: If $ f(z)=\frac{7-z}{1-z^{2}}, $ where $ z=1+2i, $ then $ |f(z)| $ is equal to:
Options:
A) $ \frac{|z|}{2} $
B) $ |z| $
C) $ 2|z| $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ z=1+2i\Rightarrow | z |=\sqrt{1+4}=\sqrt{5} $
$ \therefore f(z)=\frac{7-z}{1-z^{2}}=\frac{7-1-2i}{1-{{(1+2i)}^{2}}} $ $ =\frac{6-2i}{1-(1-4+4i)}=\frac{6-2i}{4-4i}=\frac{3-i}{2-2i} $
$ \Rightarrow | f(z) |=| \frac{3-i}{2-2i} |=\frac{| 3-i |}{| 2-2i |} $ $ =\frac{\sqrt{9+1}}{\sqrt{4+4}}=\frac{\sqrt{5}}{2}=\frac{| z |}{2} $
BETA
