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

A firm is manufacturing 2000 items. It is estimated that the rate of change of production $$P$$ with respect to additional number of workers $$x$$ is given by $$\frac{\mathrm{d} P}{\mathrm{~d} x}=100-12 \sqrt{x}$$. If the firm employs 25 more workers, then the new level of production of items is

If the normal to the curve $$y=f(x)$$ at the point $$(3,4)$$ makes an angle $$\left(\frac{3 \pi}{4}\right)^c$$ with positive $$X$$-axis, then $$f^{\prime}(3)$$ is equal to