Suppose `A` is a complex number and `n in N ,` such that `A^n=(A+1)^n=1,` then the least value of `n` is `3` b. `6` c. `9` d. `12`

Options

A. `3`
B. `6`
C. `9`
D. `12`

Answer

B

Explanation

`(b)` Let `A=x+iy`,
`A^(n)=(A+1)^(n)=1`
`implies|A|=1`
`implies x^(2)+y^(2)=1` ………`(i)`
And `|A+1|=1`
`implies(x+1)^(2)+y^(2)=1` …………`(ii)`
`impliesx=-(1)/(2)` and `y=+-(sqrt(3))/(2)`
`impliesA=omega` or `omega^(2)`
`implies (omega)^(n)=(1+omega)^(n)=(-omega^(2))^(n)`
`impliesn` must be even and divisible by `3`.
`implies` Least value of `n` is `6`