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

If $a^2 x^4 + b^2 y^4 = c^6$,then the maximum value of $xy$ is:

If $a$ and $b$ are positive numbers such that $a > b$,then the minimum value of $a \sec \theta - b \tan \theta$ for $0 < \theta < \frac{\pi}{2}$ is

If $P$ is a point on the segment $AB$ of length $12 \text{ cm}$,then the position of $P$ for $AP^{2} + BP^{2}$ to be minimum is such that

What are the minimum and maximum values of the function $f(x) = x^5 - 5x^4 + 5x^3 - 10$?

$A$ helicopter is flying along the curve given by $y = x^{3/2} + 7, (x \geq 0)$. $A$ soldier positioned at the point $(1/2, 7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is