Complex Numbers Ques 2
- If $i=\sqrt{-1}$, then $4+5\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}$ is equal to
(1999, 2M)
(a) $1-i \sqrt{3}$
(b) $-1+i \sqrt{3}$
(c) $i \sqrt{3}$
(d) $-i \sqrt{3}$
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Answer:
Correct Answer: 2.(c)
Solution: (c) We know that,
$ \begin{aligned} & \omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} i \\ & \therefore 4+5\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365} \\ & =4+5 \omega^{334}+3 \omega^{365} \\ & =4+5 \cdot\left(\omega^3\right)^{111} \cdot \omega+3 \cdot\left(\omega^3\right)^{121} \cdot \omega^2 \\ & =4+5 \omega+3 \omega^2 \quad\left[\because \omega^3=1\right] \\ & =1+3+2 \omega+3 \omega+3 \omega^2 \\ & =1+2 \omega+3\left(1+\omega+\omega^2\right)=1+2 \omega+3 \times 0 \\ & {\left[\because 1+\omega+\omega^2=0\right]} \\ & =1+(-1+\sqrt{3} i)=\sqrt{3} i \\ \end{aligned} $
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