The maximum value of xy when x + 2y = 8 is

An object is moving in the clockwise direction around the unit circle $$x^2+y^2=1$$. As it passes through the point $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$, its $$y$$-co-ordinate is decreasing at the rate of 3 units per sec. The rate at which the $$x$$-co-ordinate changes at this point is

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is $$3 \mathrm{~mm}$$ and 1 hour later it reduces to $$2 \mathrm{~mm}$$, then the expression for the radius $$R$$ of the raindrop at any time $$t$$ is

If the function $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$a \in I R$$ is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equation $$\frac{f(x)-14}{(x-1)^2}=0(x \neq 1)$$ is