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Question

If α and β are the roots of the equation x22x+4=0, then the value of αn+βn will be:

A
i2n+1sin(nπ/3)
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B
2n+1cos(nπ/3)
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C
i2n1sin(nπ/3)
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D
2n1cos(nπ/3)
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Solution

The correct option is B 2n+1cos(nπ/3)
Since, α and β are the roots of x22x+4=0

α+β=2 and αβ=4

Now, (αβ)=(α+β)24αβ

=416=23i

On solving, we get

2α=2+23i

α=2(12+32i)

=2(cosπ3+isinπ3)

and β=223i2=2(cosπ3isinπ3)

αn+βn=[2(cosπ3+isinπ3)]n+[2(cosπ3isinπ3)]n

=2n[2cosnπ3]=2n+1cosnπ3

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