All the roots of the equation `a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)=3` where `|a_(i)| lt 1, i=1,2,3,4,` lie outside the circle with center at the origin and radius equal to

Options

A. 1
B. `1//3`
C. `2//3`
D. none of these

Answer

C

Explanation

We have,
`a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)=3`
`rArr |a_(1)z^(3)+a_(2)z^(2)+a_(3)z+a_(4)|=3`
`rArr 3 le |a_(1)z^(3)|+|a_(2)z^(2)|+|a_(3)z|+|a_(4)|`
`rArr 3 le |a_(1)||z|^(3)|+|z_(2)||z^(2)|+|z_(3)||z|+|z_(4)|`
`rArr 3 lt |z|^(3) +|z|^(2)+|z|+1` `[|therefore a_(i)| lt 1, i=1,2,3,4]`
`rArr 3 lt (1-|z|^(4))/(1-|z|)`
`rArr 3 lt 1/(1-|z|) rArr 1-|z| lt 1/3 rArr 2/3 lt |z| rArr |z| gt 2/3`