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If alpha, beta in C are distinct roots o...

If `alpha, beta in C` are distinct roots of the equation `x^2+1=0` then `alpha^(101)+beta^(107)` is equal to

A

2

B

`-1`

C

0

D

1

Text Solution

Verified by Experts

The correct Answer is:
D
`x^(2) - x + 1 =0`
`rArr x = (1 pm sqrt(-3))/(2)`
`= (1pm isqrt(3))/(2)`
`=-[(-1 pm isqrt(3))/(2)]`
`therefore x = - omega, -omega^(2)` where `omega` is imaginary cube root of unity.
Let `alpha = - omega and beta= - omega^(2)`
` therefore alpha^(101) + beta^(107)= (-omega )^(101) + (-omega^(2))^(107)`
`= -[omega^(101) + omega^(214)]`
` =-[omega^(99) omega^(2) + omega^(213)omega ]`
`=-[omega^(2) + omega]`
` =-[omega^(2) + omega]`
`=- (-1) = 1" "("as" 1+ omega+omega^(2) =0)`
If we consider `alpha = - omega^(2) and beta = -omega`, we get the same results.
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