The direction of the angular velocity vector is along

If $\frac{1 - \cos x}{\cos x(1 + \cos x)} = \frac{\sin \alpha}{\cos x} - \frac{2}{1 + \cos x}$,then $\alpha = $

$A$ point moves in the $xy$-plane such that the sum of its distances from two mutually perpendicular lines is always equal to $5$ units. The area (in sq units) enclosed by the locus of the point is

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-y^2=8$ to its asymptotes is

Glucose and mannose are

Smoke is an example of