A) -1
B) -2
C) -4
D) -5
Correct Answer: C
Solution :
Let\[Z=r\left( \cos \theta +i\sin \theta \right)\] \[\Rightarrow \]\[25=r5\left( \cos 5\theta +i\sin 5\theta \right)\] \[\Rightarrow \]\[\frac{\operatorname{Im}{{Z}^{5}}}{{{(\operatorname{Im}Z)}^{5}}}=\frac{\sin 5\theta }{{{\sin }^{5}}\theta }\]Let\[Z=\frac{\sin 5\theta }{{{\sin }^{5}}\theta }\]\[\frac{d}{d\theta }=\frac{{{\sin }^{5}}\theta \cos 5\theta -\sin 5\theta 5{{\sin }^{4}}\theta \cos \theta }{{{\left( {{\sin }^{5}}\theta \right)}^{2}}}\] \[\Rightarrow \]\[5{{\sin }^{4}}\theta \left( \sin \theta \cos 5\theta -\cos \theta \sin 5\theta \right)=0\] \[\Rightarrow \]\[\sin \theta =0\]or\[\sin \left( -4\theta \right)=0\] \[\Rightarrow \]\[\theta =n\pi \]or\[\theta =\frac{n\pi }{4}\] \[\theta =-\frac{\pi }{4}\]\[\Rightarrow \]\[{{Z}_{\min }}=-4\]
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