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Question

Suppose z is any root of 11z8+20iz7+10iz22=0, where i=1 . Then S=|z|2+|z|+1 satisfies
  1. S3
  2. 3<S<7
  3. S13
  4. 7S<13


Solution

The correct option is B 3<S<7
Let z1,z2,z3,,z8 be the roots of the equation.
Then 8i=1 zi=2
8i=1 |zi|=2     (1) 
Let a=min|zi| and b=max|zi|
Then |a|<2 and |b|>1     [From (1)]
From |a|<2, we get
|a|2+|a|+1<22+2+1=7
From |b|>1, we get
|b|2+|b|+1>12+1+1=3
3<S<7

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