A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³ /min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

If the tangent to the curve $$y = {x \over {{x^2} - 3}}$$ , $$x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)$$, at a point ($$\alpha $$, $$\beta $$) $$ \ne $$ (0, 0) on it is parallel to the line 2x + 6y – 11 = 0, then :

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$. Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-

If ƒ(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set
S = {x $$ \in $$ R : ƒ(x) = ƒ(0)}
Contains exactly :