Question

The distance of the roots of the equation $|\sin {\theta }_1|z^3+|\sin {\theta }_2|z^2+|\sin {\theta }_3|z+|\sin {\theta }_4|=3,$ from $z=0$ is

• A. greater than $\frac{2}{3}$
• B. less than $\frac{2}{3}$
• C. greater than $|\sin {\theta }_1|+|\sin {\theta }_2|+|\sin {\theta }_3|+|\sin {\theta }_4|$
• D. less than $|\sin {\theta }_1|+|\sin {\theta }_2|+|\sin {\theta }_3|+|\sin {\theta }_4|$

Answer 1

The correct option is A greater than $\frac{2}{3}$

$||\sin {\theta }_1|z^3+|\sin {\theta }_2|z^2+|\sin {\theta }_3|z+|\sin {\theta }_4||=|3||3|\le |z{|}^3+|z{|}^2+|z|+1(\because |\sin {\theta }_i|\le 1|)$
Case 1 : $|z|<1$
$3<1+|z|+|z{|}^2+|z{|}^3+......+\infty$
$3<\frac{1}{1-|z|}$
$|z|>\frac{2}{3}$
Case 2 : $|z|\ge 1$
$\Rightarrow |z|\ge 1>\frac{2}{3}$