If $x=\operatorname{acos}^3 \theta y=\operatorname{asin}^3 \theta$

Then $\sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2}=$

If the difference between the maximum and minimum values of the objective function $\mathrm{z}=7 x-8 y$, subject to the constraints $x+y \leqslant 20, y \geqslant 5, x, y \geqslant 0$ is $5 \mathrm{k}+200$, then the value of k is

The number of solutions of $16^{\sin ^2 x}+16^{\cos ^2 x}=10$ in $0 \leqslant x \leqslant 2 \pi$ are

Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lie in the plane $x+3 y-\alpha z+\beta=0$, then the value of $(\beta-\alpha)$ is equal to