The height of a right circular cone of maximum volume inscribed in a sphere of diameter $a$ is

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ unit}$ is (in square unit)

Let $f(x) = x^3 e^{-3x}, x > 0$. Then the maximum value of $f(x)$ is

Let $f(x) = x^{2025} - x^{2000}$, $x \in [0, 1]$ and the minimum value of the function $f(x)$ in the interval $[0, 1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

Let $f, g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}$,$g(x)=x e^{x^2}+e^{-x^2}$ and $h(x)=x^2 e^{x^2}+e^{-x^2}$. If $a, b$ and $c$ denote,respectively,the absolute maximum of $f, g$ and $h$ on $[0,1]$,then

Suppose the cubic $x^3 - px + q$ has three distinct real roots where $p > 0$ and $q > 0$. Then which one of the following holds?