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Question
If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x2 + y2 =
Options
0
1
100
none of these
Solution
100
\[\because x + iy = (1 + i)(1 + 2i)(1 + 3i)\]
\[\text { Taking modulus on both the sides }: \]
\[\left| x + iy \right| = \left| (1 + i)(1 + 2i)(1 + 3i) \right|\]
\[ \Rightarrow \left| x + iy \right| = \left| 1 + i \right| \times \left| 1 + 2i \right| \times \left| 1 + 3i \right|\]
\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{1^2 + 1^2}\sqrt{1^2 + 2^2}\sqrt{1^2 + 3^2}\]
\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{2}\sqrt{5}\sqrt{10} \]
\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{100}\]
\[\text { Squaring both the sides }, \]
\[ \Rightarrow x^2 + y^2 = 100\]
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