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Question

If α and β are the roots of the equation x22x+4=0, such that αn+βn=2kcosnπ3, then value of k is

A
n1
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B
n
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C
n+1
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D
2n
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Solution

The correct option is C n+1
x22x+4=0
x=2±4162=1±i3
Let α=1+i3=2[cosπ3+isinπ3]
and β=1i3=2[cosπ3isinπ3]
So, αn+βn=2n[cosπ3+isinπ3]n+2n[cosπ3isinπ3]n
=2n[cosnπ3+isinnπ3+cosnπ3isinnπ3]
=2n2cosnπ3=2n+1cosnπ3
k=n+1

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