Question

If $ i = \sqrt -1$ then $ 4 + 5 \bigg(-\frac{1}{2}+\frac{i\sqrt 3}{2}\bigg)^{334}+3\bigg(-\frac{1}{2}+\frac{i\sqrt 3}{2}\bigg)^{365}$ is equal to

• A. 1-i $\sqrt 3$
• B. -1 + i $\sqrt 3$
• C. i$\sqrt 3$
• D. -i$\sqrt 3$

Answer 1

The Correct Option is C

If in a complex number a + ib, the ratio a : b is 1: \(\sqrt 3\) 
then it always convert the complex number in \(\omega\) 
.Since, \(\omega =-\frac{1}{2}+\frac{\sqrt 3}{2}i\)
\(\therefore \, 4+5 \bigg(-\frac{1}{2}+\frac{i\sqrt 3}{2}\bigg)^{334}+3\bigg(-\frac{1}{2}+\frac{i\sqrt 3}{2}\bigg)^{365}\)
\(=4+5\omega^{334}+3\omega^{365}\)
\(=4+5.(\omega^3)^{111}.\omega +3.(\omega^3)^{121}.\omega^2\)
\(= 4+5\omega+3\omega^2\ , \ [\because\ \omega^3=1]\)
\(=1+3+2\omega+3(1+\omega+\omega^2)=1+2\omega +3 \times 0\)
\([\because 1+\omega+\omega^2=0]\)
\(=1+(-1+\sqrt 3 i)=\sqrt 3 i\)

So, the correct answer is (C): \(i\sqrt 3\)