First slide

Algebra of complex numbers

Question

Let $z$ be a complex number such that $|Z| = 2$ then maximum possible value of $|z + \frac{2}{z}|$ is

Moderate

A

1
B

2
C

3
D

4

Solution

Let $z = 2 (\cos θ + i \sin θ)$ .Now
$z + \frac{2}{z} = 2 (\cos θ + i \sin θ) + (\cos θ - i \sin θ)$
$= 3 \cos θ + i \sin θ$
$\Rightarrow {|z + \frac{2}{z}|}^{2} = 9 \cos^{2} θ + \sin^{2} θ$
$= 1 + 4 (1 + \cos 2 θ) \leq 9$
$∴$ maximum possible value of $|z + \frac{2}{z}| is 3$ which
is attained when z = 1

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