Question

If |z| < √3 - 1, then |z² + 2z cos α| is
1. less than 2
2. √3 + 1
3. √3 - 1
4. None of these T₁ T₂

Answer 1

Correct Answer - Option 1 : less than 2

Concept:

Triangle inequality:

| z1 + z₂  | \(\leq\)| z₁ | + | Z₂| 

| cos α | \(\leq\) 1

Calculations:

Given, | z2 + 2z cos α | \(\leq\)| z2 | + | 2z cos α |

We know that, | cos α | \(\leq\) 1

⇒ | z2 + 2z cos α | \(\leq\)| z2 | + | 2z |

Put the value of |z| as  √3 - 1

⇒ | z2 + 2z cos α | < (√3 - 1)² + 2 (√3 - 1)

⇒ | z2 + 2z cos α | < 2 

Hence, If |z| < √3 - 1, then |z2 + 2z cos α| is less than 2