AP EAMCET · Maths · Complex Number
If \(z=\left(\frac{\sqrt{3}+i}{2}\right)^5+\left(\frac{\sqrt{3}-i}{2}\right)^5\), then
- A \(\operatorname{Re}(z)>0, \operatorname{Im}(z) < 0\)
- B \(\operatorname{Re}(z)>0, \operatorname{Im}(z)>0\)
- C \(\operatorname{Re}(z)=0\)
- D \(\operatorname{Im}(z)=0\)
Answer & Solution
Correct Answer
(D) \(\operatorname{Im}(z)=0\)
Step-by-step Solution
Detailed explanation
\( \frac{\sqrt{3}+i}{2} = e^{i\frac{\pi}{6}} \) \( \left(\frac{\sqrt{3}+i}{2}\right)^5 = e^{i\frac{5\pi}{6}} \) \( \left(\frac{\sqrt{3}-i}{2}\right)^5 = e^{-i\frac{5\pi}{6}} \) \( z = e^{i\frac{5\pi}{6}} + e^{-i\frac{5\pi}{6}} \) \( z = 2\cos\left(\frac{5\pi}{6}\right) \)…
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