Two positive numbers $x$ and $y$ are such that $(x+y)=60$ and $x y^3$ is maximum. Then the numbers $x$ and $y$ are respectively:

The maximum area of a right-angled triangle with hypotenuse $h$ is

An open metallic tank is to be constructed with a square base and vertical sides,having a volume of $500 \,m^3$. The dimensions of the tank for the minimum area of the sheet metal used in its construction are:

Suppose $x_1$ and $x_2$ are the point of maximum and the point of minimum respectively of the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If the equality $x_1^2 = x_2$ holds true,then the value of $a$ must be:

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

Point $A$ lies on the curve $y = e^{-x^2}$ and has the coordinates $(x, e^{-x^2})$ where $x > 0$. Point $B$ has the coordinates $(x, 0)$. If $O$ is the origin,then the maximum area of the triangle $AOB$ is