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 :
$${m_3} < {m_2} < {m_4} < {m_1}$$
Solution :
$$\eqalign{
& {\text{Let, }}{z_1} = 1 + 4i,{z_2} = 3 + i,{z_3} = 1 - i{\text{ and }}{z_4} = 2 - 3i \cr
& \therefore {m_1} = \left| {{z_1}} \right|,{m_2} = \left| {{z_2}} \right|,{m_3} = \left| {{z_3}} \right|{\text{ and }}{m_4} = \left| {{z_4}} \right| \cr
& \Rightarrow {m_1} = \sqrt {17} ,{m_2} = \sqrt {10} ,{m_3} = \sqrt 2 {\text{ and }}{m_4} = \sqrt {13} \cr
& \Rightarrow {m_3} < {m_2} < {m_4} < {m_1}. \cr} $$