What is the value of $\alpha$ if $\alpha = \frac{25}{50}$?

The sum of the series $1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots \infty$ is

The value of $\frac{d}{dx}(\log_e x)$ is:

Which of the following relation$(s)$ is/are correct?
$(A)\ \sin (90^{\circ}+\theta)=\cos (-\theta)$
$(B)\ \sin (180^{\circ}-\theta)=\cos (90^{\circ}-\theta)$
$(C)\ \sin (360^{\circ}-\theta)=\cos (360^{\circ}-\theta)$
$(D)\ \sin (180^{\circ}+\theta)=\cos (90^{\circ}-\theta)$

If $y = x + \frac{1}{x}$,then

If $xy = c^2$ where $c$ is a constant,then $\frac{dy}{dx}$ is equal to