In the graph of the function $\sqrt{3} \sin x + \cos x$,the maximum distance of a point from the $x$-axis is:

If $u = \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta} + \sqrt{a^2 \sin^2 \theta + b^2 \cos^2 \theta}$,then what is the difference between the maximum and minimum values of $u^2$?

If the range of $f(\theta) = \frac{\sin^4 \theta + 3 \cos^2 \theta}{\sin^4 \theta + \cos^2 \theta}$,$\theta \in R$ is $[\alpha, \beta]$,then the sum of the infinite $G.P.$,whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$,is equal to...........

What is the minimum value of $a \sec x + b \csc x$ for $0 < a < b$ and $0 < x < \pi/2$?

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta)=4(\sin^{4}(\frac{7\pi}{2}-\theta)+\sin^{4}(11\pi+\theta)) - 2(\sin^{6}(\frac{3\pi}{2}-\theta)+\sin^{6}(9\pi-\theta))$, $\theta \in R$. Then $\alpha+2\beta$ is equal to:

The maximum value of the function $f(x) = 3 \sin^{12} x + 4 \cos^{16} x$ is