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If alpha and beta are roots of the equat...

If `alpha and beta` are roots of the equation `x^(2)-2x+2=0,` then the least value of n for which `((alpha)/beta)^(n)=1` is

A

2

B

5

C

4

D

3

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `alpha and beta` are the roots of the quadratic equation `x^(2)-2x+2=0`
`implies(x-1)^(2)+1=0`
`implies(x-1)^(2)=-1`
`implies x-a=+-i" "["where i"=sqrt-1]`
`impliesx=(1+i)or (1-i)`
Clearly, if `alpha =1 +i, then beta=1-i`
According to the equation `((alpha)/(beta))^(n)=1`
`implies((1+i)/(1-i))^(n)=1`
`implies(((1+i)(1+i))/((1-i)(1-i)))^(n)=1" "["by rationalization"]`
`implies((1+i^(2)+2i)/(1-i^(2)))=1implies((2i)/(2))^(n)=1impliesi^(n)=1`
So, minimum value of n is `4." "[becausei^(4)=1]`
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