JEE PYQ: Complex Numbers Question 39
Question 39 - 2019 (10 Jan Shift 2)
Let $z = \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^5 + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^5$. If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$, then:
(1) $I(z) = 0$
(2) $R(z) > 0$ and $I(z) > 0$
(3) $R(z) < 0$ and $I(z) > 0$
(4) $R(z) = -3$
Show Answer
Answer: (1)
Solution
$z = (e^{i\pi/6})^5 + (e^{-i\pi/6})^5 = 2\cos\frac{5\pi}{6}$. Purely real, so $I(z) = 0$.
BETA
