All the pairs (x, y) that satisfy the inequality
$${2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1$$
also satisfy the equation

The ratio of surface tensions of mercury and water is given to be 7.5 while the ratio of thier densities is 13.6. Their contact angles, with glass, are close to 135° and 0°, respectively. It is observed that mercury gets depressed by an amount h in a capillary tube of radius r₁, while water rises by the same amount h in a capillary tube of radius r₂. The ratio, (r₁/r₂), is then close to :

Two wires A & B are carrying currents I₁ & I₂ as shown in the figure. The separation between them is d. A third wire C carrying a current I is to be kept parallel to them at a distance x from A such that the net force acting on it is zero. The possible values of x are :

A thin disc of mass M and radius R has mass per unit area $$\sigma $$(r) = kr² where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :