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If $m_{1}, m_{2}, m_{3}$ and $m_{4}$ respectively denote the moduli of the complex numbers $1 + 4 i, 3 + i, 1 - i$ and $2 - 3 i$ then the correct one, among the following is

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$m_{3} < m_{1} < m_{2} < m_{4}$

$m_{1} < m_{2} < m_{3} < m_{4}$

$m_{3} < m_{2} < m_{4} < m_{1}$

$m_{4} < m_{3} < m_{2} < m_{1}$

answer is C.

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Detailed Solution

$z_{1} = 1 + 4 i, z_{2} = 3 + i, z_{3} = 1 - i and z_{4} = 2 - 3 i$

$\begin{aligned} ∴ m_{1} = |z_{1}|, m_{2} = |z_{2}|, m_{3} = |z_{3}| and m_{4} = |z_{4}| \end{aligned}$

$\Rightarrow m_{1} = \sqrt{1 + 4^{2}} = \sqrt{17}, m_{2} = \sqrt{3^{2} + 1^{2}} = \sqrt{10}$

$m_{3} = \sqrt{1^{2} + 1^{2}} = \sqrt{2} and m_{4} = \sqrt{2^{2} + 3^{2}} = \sqrt{13}$

$\Rightarrow m_{3} < m_{2} < m_{4} < m_{1}$

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