Find two positive numbers $x$ and $y$ such that $x+y=60$ and $x y^{3}$ is maximum.

If $A = \{x \in R : \frac{\pi}{4} \leq x \leq \frac{\pi}{3}\}$ and $f(x) = \sin x - x$,then $f(A)$ is equal to

$20 \ m$ of wire is available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area,then the radius of the circle is (in $m$)

If the local maximum value of the function $f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}, \quad x \in\left(0, \frac{\pi}{2}\right)$,is $\frac{k}{e}$,then $\left(\frac{ k }{ e }\right)^8+\frac{ k ^8}{ e ^5}+ k ^8$ is equal to

$P(x) = x^4 + ax^3 + bx^2 + cx + d$ is such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

Divide $10$ into two parts such that the sum of double of the first and the square of the second is minimum.