Question

$i^{1 + 3 + 5 + ... + (2n + 1)}$ will be purely imaginary, if

• A. $x + \frac{1}{x} = 2\cos\theta,$
• B. $\cos\theta + i\sin\theta$
• C. $\cos\theta - i\sin\theta$
• D. None of these [Where $\cos\theta \pm i\sin\theta$ is an integer]

Answer 1

Answer: (C)

$\cos\theta - i\sin\theta$

Answer 2

$(x + iy) + (1 - i) = 0$ will be purely imaginary, if the real part vanishes, i.e., $x + 1 = 0$

⇒ $\mathbf{y = 1}$ (only if $\therefore$ be real)

⇒ $\left( 2\sin\frac{\theta}{2} \right)^{- 1}\left\{ \sin\frac{\theta}{2} + i.2\cos\frac{\theta}{2} \right\}$

⇒ $\because$