Question

All the roots of the equation $a_1{z}^3+a_2{z}^2+a_{3}z+a_4=3$, where $\left|a_i\right|\le 1,i-1,2,3,4$ lie outside the circle with center origin and radius

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. None of these

Answer 1

Answer: (B)

$\frac{2}{3}$

We have, $\left|3\right|=|a_1{z}^3+a_2{z}^2+a_{3}z+a_4|$

$\Rightarrow 3\le \left|a_1{z}^3\right|+\left|a_2{z}^2\right|+\left|a_{3}z\right|+\left|a_4\right|$

$\Rightarrow 3\le \left|a_1\right|\left|z^3\right|+\left|a_2\right|\left|z^2\right|+\left|a_3\right|\left|z\right|+\left|a_4\right|$

$\Rightarrow 3\le \left|z^3\right|+\left|z^2\right|+\left|z\right|+1(\because \left|a_i\right|\le 1)$

$\Rightarrow 3<1+\left|z\right|+{\left|z\right|}^2+{\left|z\right|}^3<1+\left|z\right|+{\left|z\right|}^2+{\left|z\right|}^3+...\infty$

$\Rightarrow 3<\frac{1}{1-\left|z\right|}$ [By using $\frac{a}{1-r}$ ]

$\Rightarrow 1-\left|z\right|<\frac{1}{3}\Rightarrow \left|z\right|>\frac{2}{3}$