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Question

If a is a complex number such that |a|=1 and az2+z+1=0 has one purely imaginary root, then cos(arg(a)) is

A
212
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B
3+12
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C
512
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D
712
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Solution

The correct option is C 512
az2+z+1=0 (1)
Taking conjugate of both sides,
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯az2+z+1=¯0
¯a(¯z)2+¯z+¯1=0
¯az2z+1=0 (2)
(since ¯z=z as z is purely imaginary)
Eliminating z from equations (1) and (2), we get
(¯aa)2+2(a+¯a)=0 (3)
Let,
a=cosθ+isinθ (|a|=1)
From equation (3) we get,
(2isinθ)2+2(2cosθ)=0
4cos2θ+4cosθ4=0
cos2θ+cosθ1=0
cosθ=1±1+42
θ=arg(a)
cos(arg(a))=512

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