Question

The value of $\left|\frac{1+i\sqrt{3}}{{\left(1+\frac{1}{i+1}\right)}^2}\right|$ is

• A. $20$
• B. $9$
• C. $\frac{5}{4}$
• D. $\frac{4}{5}$

Answer 1

Answer: (D)

$\left|\frac{1+i\sqrt{3}}{{\left(1+\frac{1}{i+1}\right)}^2}\right|=\left|\frac{(1+i\sqrt{3})(i+1{)}^2}{(i+2{)}^2}\right|\left(\because i^2=-1\right)$
$=\left|\frac{(1+i\sqrt{3})\left(i^2+1+2i\right)}{\left(i^2+4+4i\right)}\right|$
$=\left|\frac{(1+i\sqrt{3})2i}{(4i+3)}\right|$
$=\left|\frac{-2\sqrt{3}+2i}{3+4i}\right|$
$=2\left|\frac{(-\sqrt{3}+i)(3-4i)}{(3+4i)(3-4i)}\right|$
$=2\left|\frac{-3\sqrt{3}+3i+4\sqrt{3}i+4}{25}\right|$
$=\frac{2}{25}|(4-3\sqrt{3})+(3+4\sqrt{3})i|$
$=\frac{2}{25}\sqrt{(4-3\sqrt{3}{)}^2+(3+4\sqrt{3}{)}^2}$
$=\frac{2}{25}\sqrt{16+27-24\sqrt{3}+9+48+24\sqrt{3}}$
$=\frac{2}{25}\sqrt{100}=\frac{2\times 10}{25}=\frac{4}{5}$