Match the statements/expressions in Column $$I$$ with the open intervals in Column $$II$$.

Column $$I$$
(A)$$\,\,\,$$ Interval contained in the domain of definition of non-zero solutions of the differential equation $${\left( {x + 3} \right)^2} + y' + y = 0$$
(B)$$\,\,\,$$ Interval containing the value of the integral $$\int\limits_1^5 {\left( {x + 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 4} \right)\left( {x - 5} \right)dx} $$
(C)$$\,\,\,$$ Interval in which at least one of the points of local maximum of $${\cos ^2}x + \sin x$$ lies
(D)$$\,\,\,$$ Interval in which $${\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$ is increasing

Column $$II$$
(p) $$\,\,\,\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
(q) $$\,\,\,\left( {0,{\pi \over 2}} \right)$$
(r) $$\,\,\,\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$$
(s) $$\,\,\,\,\left( {0,{\pi \over 8}} \right)$$
(t) $$\,\,\,\,\,\left( { - \pi ,\pi } \right)$$

If length of tangent at any point on the curve $$y=f(x)$$ intecepted between the point and the $$x$$-axis is length $$1.$$ Find the equation of the curve.

A curve $$'C''$$ passes through $$(2,0)$$ and the slope at $$(x,y|)$$ as $$\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$$. Find the equation of the curve. Find the area bounded by curve and $$x$$-axis in fourth quadrant.

A right circular cone with radius $$R$$ and height $$H$$ contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant $$ = k > 0$$. Find the time after which the come is empty.