Question

The modulus and amplitude of $\frac{1+2i}{1-(1-i{)}^2}$ are respectively

• A. $1,\frac{\pi }{3}$
• B. $\sqrt{2},\frac{\pi }{6}$
• C. $1,0$
• D. $\sqrt{3},0$

Answer 1

Answer: (C)

Let $Z=\frac{1+2i}{1-(1-i{)}^2}$
$=\frac{1+2i}{1-\left(1+i^2-2i\right)}=\frac{1+2i}{1+2i}=1\Rightarrow Z=1+0i$
Now, put $1=r\cos \theta ,0=r\sin \theta$
$r=\sqrt{1+0}=1\Rightarrow \left|Z\right|=\sqrt{a^2+b^2}=\sqrt{1^2+0}=1$
$\therefore \tan \theta =0\Rightarrow \theta =0$
$\therefore$ Modulus is 1 and amplitude is 0.