If $5 \sin \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 3$ lies between $\alpha$ and $\beta$ (including $\alpha, \beta$ also),then $(\alpha - \beta)(\alpha + \beta - 6) = $

The maximum value of the expression $E = \sin \theta + \cos \theta + \sin 2\theta$ is-

Let $a = \max_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\}$ and $\beta = \min_{x \in R} \{8^{2 \sin 3x} \cdot 4^{4 \cos 3x}\}$. If $8x^2 + bx + c = 0$ is a quadratic equation whose roots are $\alpha^{1/5}$ and $\beta^{1/5}$,then the value of $c - b$ is equal to:

If $(\cot \alpha_1)(\cot \alpha_2) \ldots (\cot \alpha_n) = 1$ where $0 < \alpha_1, \alpha_2, \ldots, \alpha_n < \pi/2$,then the maximum value of $(\cos \alpha_1)(\cos \alpha_2) \ldots (\cos \alpha_n)$ is given by

If $10 \sin^4 \alpha + 15 \cos^4 \alpha = 6$,then $16 \tan^6 \alpha + 27 \cot^6 \alpha =$

The maximum value of $\cos \alpha_1 \cdot \cos \alpha_2 \cdots \cos \alpha_n$ under the restrictions $0 \le \alpha_1, \alpha_2, \dots, \alpha_n \le \frac{\pi}{2}$ and $\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1$ is