If $a \sin^2 \theta + b \cos^2 \theta = c$,then $\tan^2 \theta$ is equal to

The logic behind the $NOR$ gate is that it gives

Every planet revolves around the sun in an elliptical orbit. Consider the following statements:
$A.$ The gravitational force acting on a planet is inversely proportional to the square of the distance from the sun.
$B.$ The gravitational force acting on a planet is inversely proportional to the product of the masses of the planet and the sun.
$C.$ The centripetal force acting on the planet is directed away from the sun.
$D.$ The square of the time period of revolution of a planet around the sun is directly proportional to the cube of the semi-major axis of the elliptical orbit.
Choose the correct statements from the options given above:

If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

In the following reaction,$X$ and $Y$ respectively are:
$C_2H_5OH \xrightarrow{KMnO_4 / H^{+}} X$
$X \xrightarrow[H_2SO_4 / \Delta]{Y} CH_3COOC_2H_5$

The component of vector $\overrightarrow{A} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$ along the direction of $\hat{i} - \hat{j}$ is