Question

Find $\left|\right(1+i\left)\frac{(2+i)}{(3+i)}\right|$

Answer 1

$\left|\right(1+i\left)\frac{(2+i)}{(3+i)}\right|$

$|1+i|\frac{|2+i|}{|3+i|}$ $[\because \left|\frac{z_1\cdot z_2}{z_3}\right|=\frac{|z_1|\cdot |z_2|}{|z_3|}]$

$=\left(\sqrt{(1{)}^2+(1{)}^2}\right)\frac{\sqrt{(2{)}^2+(1{)}^2}}{\sqrt{(3{)}^2+(1{)}^2}}$

$[\because |x+iy|=\sqrt{x^2+y^2}]$

$=\sqrt{2}\cdot \frac{\sqrt{5}}{\sqrt{10}}$

$=1$

Hence, the value of $\left|\right(1+i\left)\frac{(2+i)}{(3+i)}\right|$ is $1$.