Let p,q∈R and (1−i3)200=2199(p+iq),i=−1, Then (p,q)=

(1−i3)200=2200(12−i322)200=2200(cosπ3×200−isinπ3×200)=2200(−12−i32)=2199(−1−i3)=2199(P+iq)P=−1,q=−3

Let p,q∈R and (1−i3)200=2199(p+iq),i=−1, Then (p,q)=

(1−i3)200=2200(12−i322)200=2200(cosπ3×200−isinπ3×200)=2200(−12−i32)=2199(−1−i3)=2199(P+iq)P=−1,q=−3

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Let $p, q \in R$ and ${(1 - i \sqrt{3})}^{200} = 2^{199} (p + i q), i = \sqrt{- 1},$ Then $(p, q) =$

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$(- 1, \sqrt{3)}$

$(- 1, - \sqrt{3)}$

$(1, - \sqrt{3)}$

$(1, \sqrt{3)}$

answer is B.

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Detailed Solution

$\begin{array}{l} {(1 - i \sqrt{3})}^{200} = 2^{200} {(\frac{1}{2} - i \frac{\sqrt{3} 2}{2})}^{200} \\ = 2^{200} (c o s \frac{π}{3} \times 200 - i \sin \frac{π}{3} \times 200) \\ = 2^{200} (\frac{- 1}{2} - i \frac{\sqrt{3}}{2}) \\ = 2^{199} (- 1 - i \sqrt{3}) \\ = 2^{199} (P + i q) \\ P = - 1, q = - \sqrt{3} \end{array}$