Question

If z is a non - real complex number, then the minimum value of $\frac{Im\left(z^5\right)}{{\left(Im\right(z\left)\right)}^5}$ is

• A. -1
• B. -2
• C. -4
• D. -5

Answer 1

The correct option is C -4

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$ $a^2{b}^3{i}^3+$ ${}^5{C}_4{i}^5{b}^5$
Im $z^5=5a^{4}b-10a^2{b}^3+b^5$
$y=\frac{Im{z}^5}{{Im}^{5}z}=5{\left(\frac{a}{b}\right)}^4-10{\left(\frac{a}{b}\right)}^2+1$
Let ${\left(\frac{a}{b}\right)}^2=x\left(say\right),x\in R^{+}$
$y={5x}^2-10x+1=5[x^2-2x]+1$
$=5\left[{(x-1)}^2\right]-4$
Hence $y_{min}=-4$