Question

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

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

Solution

## The correct option is A -4$$z$$ is a non-real complex number.let $$z=re^{i\theta}$$$$\displaystyle \frac{Im z^5}{(Im z)^5}=\frac{r^5\sin 5\theta}{r^5\sin^{5}\theta}$$say, $$f(\theta)=\displaystyle\frac{\sin 5\theta}{\sin^{5}\theta}$$equating first derivative to $$0$$$$f'(\theta)=0$$$$\Rightarrow \sin^{5}\theta \cdot 5\cos 5\theta-\sin 5\theta \cdot 5\sin^{4} \theta \cdot \cos \theta$$$$\Rightarrow \sin \theta \cdot \cos 5\theta = \cos \theta \cdot \sin 5\theta$$$$\Rightarrow \tan \theta = \tan 5\theta$$$$\Rightarrow \theta =0,\pi/4 ...$$but $$f(0)$$ is indeterminate$$f(\pi/4)=-4$$$$\therefore$$ Minimum value of $$\displaystyle \frac{Im z^5}{(Im z)^5}$$ is $$-4$$Hence, option $$B$$.Mathematics

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