The number that exceeds its square by the greatest amount is

The sum of the lengths of the hypotenuse and one side of a right-angled triangle is constant. The area of the triangle will be maximum if the angle between them is:

If $y = a \log x + b x^2 + x$ has its extreme values at $x = -1$ and $x = 2$,then the value of $\left(\frac{a}{b} + \frac{b}{a}\right)$ is

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

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

Let $f(x) = 2 \cos^{-1} x + 4 \cot^{-1} x - 3x^2 - 2x + 10$,where $x \in [-1, 1]$. If $[a, b]$ is the range of the function,then $4a - b$ is equal to: