Question

If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x² + y² =

Options

• 0

• 1

• 100

• none of these

Answer 1

100

\[\because x + iy = (1 + i)(1 + 2i)(1 + 3i)\]

\[\text { Taking modulus on both the sides }: \]

\[\left| x + iy \right| = \left| (1 + i)(1 + 2i)(1 + 3i) \right|\]

\[ \Rightarrow \left| x + iy \right| = \left| 1 + i \right| \times \left| 1 + 2i \right| \times \left| 1 + 3i \right|\]

\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{1^2 + 1^2}\sqrt{1^2 + 2^2}\sqrt{1^2 + 3^2}\]

\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{2}\sqrt{5}\sqrt{10} \]

\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{100}\]

\[\text { Squaring both the sides }, \]

\[ \Rightarrow x^2 + y^2 = 100\]