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Question

If α is a complex constant such that αz2+z+¯¯¯¯α=0 has a real root, then

A
α+¯¯¯¯α=1
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B
α+¯¯¯¯α=0
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C
α+¯¯¯¯α=1
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D
The absolute value of the real root is 1
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Solution

The correct options are
A α+¯¯¯¯α=1
C α+¯¯¯¯α=1
D The absolute value of the real root is 1
Let z=c be a real root. Then,
ac2+c+¯¯¯a=0 (1)
Putting α=p+iq, we have
(p+iq)c2+c+piq=0
pc2+c+p=0 and qc2q=0c=±1(q0)
(1)α±1+¯¯¯¯α=0
Also, |c|=1

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