If `z_1` is a root of the equation `a_0z^n+a_1z^(n-1)++a_(n-1)z+a_n=3,w h e r e|a_i|<2fori=0,1, ,n ,t h e n` `|z|=3/2` b. `|z|<1/4` c.`|z|>1/4` d. `|z|<1/3`

Options

A. `|z_(1)|gt(1)/(2)`
B. `|z_(1)|lt(1)/(2)`
C. `|z_(1)|gt(1)/(4)`
D. `|z| lt(1)/(2)`

Answer

A

Explanation

`a_(0)z^(n)+a_(1)z^(n-1)+a_(2)z^(n-2)+...+a_(n-1)z+a_(n)=3`
or `|3|=|a_(0)z^(n)+a_(1)z^(n-1)+...+a_(n-1)z+a_(n)|`
or `3le|a_(0)||z|^(n)+|a_(1)||z|^(n-1)+... +|a_(n-1)||z|+|a_(n)|`
or `3lt2(|z|^(n)+|z|^(n-1)+... |z|+1)" " (therefore |a_(1)|lt2)`
or `1+|z|+|z|^(2)+...+|z|^(n)gt(3)/(2)`
If `|z|ge 1`, the inequality is clearly satisfied. For `|z|lt1`, we have,
`(1-|z|^(n+1))/(1-|z|)gt(3)/(2)`
or `2-2|z|^(n-1)gt3-3|z|`
`or 2|z|^(n+1)lt3|z|-1`
`or 3|z|-1gt0`
or `|z|gt(1)/(3)`