Question:
If $x+i y=(1+i)(1+2 i)(1+3 i)$, then $x^{2}+y^{2}=$
(a) 0
(b) 1
(c) 100
(d) none of these
Solution:
(c) 100
$\because x+i y=(1+i)(1+2 i)(1+3 i)$
Taking modulus on both the sides:
$|x+i y|=|(1+i)(1+2 i)(1+3 i)|$
$\Rightarrow|x+i y|=|1+i| \times|1+2 i| \times|1+3 i|$
$\Rightarrow \sqrt{x^{2}+y^{2}}=\sqrt{1^{2}+1^{2}} \sqrt{1^{2}+2^{2}} \sqrt{1^{2}+3^{2}}$
$\Rightarrow \sqrt{x^{2}+y^{2}}=\sqrt{2} \sqrt{5} \sqrt{10}$
$\Rightarrow \sqrt{x^{2}+y^{2}}=\sqrt{100}$
Squaring both the sides,
$\Rightarrow x^{2}+y^{2}=100$