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Question: If z = \(\left( \frac{\sqrt{3} + i}{2} \right)^{5}\)+\(\left( \frac{\sqrt{3} - i}{2} \right)^{5}\), ...

If z = (3+i2)5\left( \frac{\sqrt{3} + i}{2} \right)^{5}+(3−i2)5\left( \frac{\sqrt{3} - i}{2} \right)^{5}, then –

A

Re (z) = 0

B

Im (z) = 0

C

Re(z), Im (z) > 0

D

Re(z) > 0, Im(z) < 0

Answer

Im (z) = 0

Explanation

Solution

Sol. z = (3+i2)5\left( \frac{\sqrt{3} + i}{2} \right)^{5}+(3−i2)5\left( \frac{\sqrt{3} - i}{2} \right)^{5}

= (eiπ/6)5(e^{i\pi/6})^{5} + (e–iπ/6)5(e^{–i\pi/6})^{5}

= ei5π/6e^{i5\pi/6}+ e–i5π/6e^{–i5\pi/6}

= 2cos 5π6\frac{5\pi}{6} = – 2 cos π6\frac{\pi}{6} = –3\sqrt{3}

Thus Im (z) = 0

Hence (2) is the correct answer.