Question

For any two complex numbers Z₁ and Z₂ with |Z₁| ¹ |Z₂|

|$\sqrt{2}$Z₁ + i$\sqrt { 3 }$ ${\bar{Z}}_{2}$|² + |$\sqrt { 3 }$ ${\bar{Z}}_{1}$+ i$\sqrt{2}$Z₂|² is –

• A. Less than 5 $\left( |Z_{1}|^{2} + |Z_{2}|^{2} \right)$
• B. Greater than 10 |Z₁Z₂|
• C. Equal to 2 |Z₁|² + 3|Z₂|²
• D. Zero

Answer 1

Answer: (B)

Greater than 10 |Z₁Z₂|

Answer 2

Sol.$\sqrt{2}$Z₁ + i$\sqrt { 3 }$ ${\bar{Z}}_{2}$|² + | $\sqrt { 3 }$ ${\bar{Z}}_{1}$+ i$\sqrt{2}$Z₂|²

= ($\sqrt{2}$Z₁ + i$\sqrt { 2 }$ ${\bar{Z}}_{1}$– i$\sqrt{3}$Z₂) + ($\sqrt { 2 }$ ${\bar{Z}}_{2}$)

= 5$\left( |Z_{1}|^{2} + |Z_{2}|^{2} \right)$ > 5. 2 $\sqrt{\left| Z_{1} \right|^{2}\left| Z_{2} \right|^{2}}$

= 10|Z₁Z₂|, since AM > GM for |Z₁| ¹ |Z₂|