Twenty metres of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is:

If $(2, a)$ and $(b, 19)$ are two stationary points of the curve $y=2x^3-15x^2+36x+c$,then $a+b+c=$

Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi-vertical angle $\alpha$ is one-third that of the cone and the greatest volume of the cylinder is $\frac{4}{27} \pi h^{3} \tan^{2} \alpha$.

The sixth term of an $A.P.$ is equal to $2$. The value of the common difference $x$ of the $A.P.$ that makes the product $a_1 a_4 a_5$ least is given by:

Find the maximum profit that a company can make,if the profit function is given by $p(x) = 41 - 72x - 18x^{2}$. (in $units$)

If $g(x) = 2f(2x^3 - 3x^2) + f(6x^2 - 4x^3 - 3)$,$\forall x \in R$ and $f''(x) > 0$,$\forall x \in R$,then $g'(x) > 0$ for $x$ belonging to