Question

If z is a complex number satisfying $|z^3+z^{-3}|\le 2$,then the maximum possible value of $|z+z^{-1}|$ is

• A. 2
• B.
• C.
• D. 1

Answer 1

The correct option is A 2

$|z^3+\frac{1}{z^3}|\le 2$
But we know that $|z^3+\frac{1}{z^3}|\le |z{|}^3+\frac{1}{|z{|}^3}$
So when |z|=1 by $AM\ge GM$ $\text{Since}\frac{|z^3|+\frac{1}{|z^3|}}{2}\ge \sqrt{(|z^3|\times \frac{1}{|z^3|})}$
$|z{|}^3+\frac{1}{|z{|}^3}=2Now|z+\frac{1}{z}|\le |z|+\frac{1}{|z|}=2$
Maximum value =2