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

Let a function $f(x) = \begin{cases} -\ln(3x - [3x]) & ; 3x \neq n, n \in N \\ \ln(\operatorname{sgn}(3x)) & ; 3x = n, n \in N \end{cases}$,where $[.]$ and $\operatorname{sgn}(x)$ denote the greatest integer function and signum function respectively. Then the number of points where $f(x)$ is minimum in $x \in (0, 5)$ is:

$36$ is factorized into two factors such that the sum of the factors is minimum. What are the factors?

The maximum value of $x^4 y^4$ given the constraint $a^2 x^4 + b^2 y^4 = c^6$ is

Let $k$ and $K$ be the minimum and the maximum values of the function $f(x) = \frac{(1 + x)^{0.6}}{1 + x^{0.6}}$ in $[0, 1]$ respectively,then the ordered pair $(k, K)$ is equal to

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: