The value of $1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots$ up to $\infty$ is

$\int\limits_0^{\frac{\pi }{4}} {\sin 2x} \,dx$ is equal to

$\frac{d}{dx}\left( \frac{1}{x^4 \sec x} \right) = $

The equation of a curve is given as $y = x^2 - 3x + 2$. The curve intersects the $x$-axis at

If $\tan \theta = \frac{1}{\sqrt{5}}$ and $\theta$ lies in the first quadrant,the value of $\cos \theta$ is

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