If $\sin \theta + \sqrt{3} \cos \theta = 6x - x^2 - 11$,where $x \in R$ and $0 \le \theta \le 2\pi$,then the equation has a solution for:

The maximum value of $12 \sin x - 5 \cos x + 3$ is

Let $f(x) = \cos 5x + A \cos 4x + B \cos 3x + C \cos 2x + D \cos x + E$,and $T = f(0) - f\left(\frac{\pi}{5}\right) + f\left(\frac{2\pi}{5}\right) - f\left(\frac{3\pi}{5}\right) + \dots + f\left(\frac{8\pi}{5}\right) - f\left(\frac{9\pi}{5}\right)$. Then,$T$

If $A+B+C=270^{\circ}$,then $\cos 2A + \cos 2B + \cos 2C + 4 \sin A \sin B \sin C =$

What is the minimum value of $2^{((x^2 - 3)^3 + 27)}$?

If $A + B = \frac{\pi}{2},$ the maximum value of $\cos A \cos B$ is