Question

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. $m_3 < m_1 < m_2 < m_4$

Answer 1

Answer: (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 $m_{4}=\left|z_{4}\right|$
$\Rightarrow m_{1}=\sqrt{1+4^{2}}, m_{2}=\sqrt{3^{2}+1^{2}}$
$m_{3}=\sqrt{1^{2}+1^{2}}$ and $m_{4}=\sqrt{2^{2}+3^{2}}$
$\Rightarrow m_{1}=\sqrt{17}, m_{2}=\sqrt{10}$
$m_{3}=\sqrt{2}$ and $m_{4}=\sqrt{13}$
$\Rightarrow m_{3} < m_{2} < m_{4} < m_{1}$