JEE Mains · Maths · STD 11 - 4.1 complex nubers
If \((\sqrt{3}+\mathrm{i})^{100}=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})\), then \(\mathrm{p}\) and \(\mathrm{q}\) are roots of the equation :
- A \(x^{2}-(\sqrt{3}-1) x-\sqrt{3}=0\)
- B \(x^{2}+(\sqrt{3}+1) x+\sqrt{3}=0\)
- C \(x^{2}+(\sqrt{3}-1) x-\sqrt{3}=0\)
- D \(x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0\)
Answer & Solution
Correct Answer
(A) \(x^{2}-(\sqrt{3}-1) x-\sqrt{3}=0\)
Step-by-step Solution
Detailed explanation
\(\left(2 e^{\mathrm{i} \pi / 6}\right)^{100}=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})\) \(2^{100}\left(\cos \frac{50 \pi}{3}+\mathrm{i} \sin \frac{50 \pi}{3}\right)=2^{99}(\mathrm{p}+\mathrm{i} \mathrm{q})\)…
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