AP EAMCET · Maths · Complex Number
\((1+\sqrt{5}+\mathrm{i} \sqrt{10-2 \sqrt{5}})^5=\)
- A \(1024\)
- B \(-1024\)
- C \(512\)
- D \(-512\)
Answer & Solution
Correct Answer
(B) \(-1024\)
Step-by-step Solution
Detailed explanation
\(1+\sqrt{5} = 4\cos(\pi/5)\) \(\sqrt{10-2\sqrt{5}} = 4\sin(\pi/5)\) \((1+\sqrt{5}+\mathrm{i} \sqrt{10-2 \sqrt{5}})^5 = (4\cos(\pi/5) + \mathrm{i} 4\sin(\pi/5))^5\) \(= (4(\cos(\pi/5) + \mathrm{i} \sin(\pi/5)))^5\) \(= 4^5(\cos(5 \cdot \pi/5) + \mathrm{i} \sin(5 \cdot \pi/5))\)…
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