If `Z^5` is a non-real complex number, then find the minimum value of `(Imz^5)/(Im^5z)`.

Options

A. `-1`
B. `-2`
C. `-4`
D. `-5`

Answer

C

Explanation

`(c )` Let `Z=a+ib`, `b ne 0` where `Im Z=b`
`Z^(5)=(a+ib)^(5)`
`=a^(5)+^(5)C_(1)a^(4)bi+^(5)C_(2)a^(3)b^(2)i^(2)+^(5)C_(3)a^(2)b^(3)i^(3)+^(5)C_(4)ab^(4)i^(4)+i^(5)b^(5)`
`ImZ^(5)=5a^(4)b-10a^(2)b^(3)+b^(5)`
`y=(ImZ^(5))/(Im^(5)Z)=5((a)/(b))^(4)-10((a)/(b))^(2)+1`
Let `((a)/(b))^(2)=x("say")`, `x in R^(+)`
`y=5x^(2)-10x+1`
`=5[x^(2)-2x]+1`
`=5[(x-1)^(2)]-4`
Hence,`y_(min)=-4`