Q. If $m_{1},m_{2},m_{3}$ and $m_{4}$ respectively denote the moduli of the complex numbers $1 + 4i,\ 3 + i,\ 1 - i$ and $2 - 3i$, then the correct one, among the following is
A) $m_{1} < m_{2} < m_{3}{< m}_{4}$
B) $m_{4} < m_{3} < m_{2} < m_{1}$
C) $m_{3} < m_{2} < m_{4} < m_{1}$
D) $m_{3} < m_{1} < m_{2} < m_{4}$
Explanation
(c)
Let $z_{1} = 1 + 4i,\ \ \ z_{2} = 3 + i,\ \ z_{3} = 1 - i$ and $z_{4} = 2 - 3i$
$\therefore\ \ \ \ m_{1}\left| z_{1} \right|,\ \ \ \ m_{2} = \left| z_{2} \right|,\ \ m_{3} = \left| z_{3} \right|$ and $m_{4} = |z_{4}$|
$\Rightarrow \ \ \ m_{1} = \sqrt{1 + 4^{2}} = \sqrt{17},\ \ \ \ \ \ \ m_{2} = \sqrt{3^{2} + 1^{2}} = \sqrt{10}$,
$m_{3} = \sqrt{1^{2} + 1^{2}} = \sqrt{2}$ and $m_{4} = \sqrt{2^{2} + 3^{2}} = \sqrt{13}$
$\Rightarrow \ \ \ \ m_{3} < m_{2} < m_{4} < m_{1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.