The maximum value of $3\cos \theta - 4\sin \theta$ is

If $A, B, C$ are angles of a $\triangle ABC$,then $\tan 2A + \tan 2B + \tan 2C =$

If $\left| a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta - \frac{1}{2}(a + c) \right| \le \frac{1}{2}k,$ then $k^2$ is equal to

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:

Let $x, y$ be positive real numbers and $m, n$ be positive integers. The maximum value of the expression $\frac{x^m y^n}{(1 + x^{2m})(1 + y^{2n})}$ is

What is the minimum value of $(1 + a_1 + a_1^2)(1 + a_2 + a_2^2)(1 + a_3 + a_3^2) \dots (1 + a_n + a_n^2)$ given that $a_1 a_2 a_3 \dots a_n = 1$ and $a_i > 0$ for all $i = 1, 2, \dots, n$?