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Question

All the roots of the equation a1z3+a2z2+a3z+a4=3, where |ai|1,i1,2,3,4 lie outside the circle with center origin and radius

A
13
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B
23
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C
1
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D
None of these
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Solution

The correct option is C 23
We have, |3|=a1z3+a2z2+a3z+a4

3a1z3+a2z2+|a3z|+|a4|

3|a1|z3+|a2|z2+|a3||z|+|a4|

3z3+z2+|z|+1(|ai|1)

3<1+|z|+|z|2+|z|3<1+|z|+|z|2+|z|3+...


3<11|z| [By using a1r ]

1|z|<13|z|>23

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