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

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y = -2x^2 + 54$ at points $(x, y)$ and $(-x, y)$ where $y > 0$ is:

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is

$A$ tank with a rectangular base and rectangular sides,open at the top is to be constructed so that its depth is $4 \ m$ and volume is $36 \ m^3$. If building of the tank costs $₹ 100$ per square meter for the base and $₹ 50$ per square meter for the sides,then the cost of the least expensive tank is:

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ is :-

On the interval $[0,1]$,the function $f(x) = x^{25}(1-x)^{75}$ takes its maximum value at the point