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The number of integral values of $k$ for which the equation $3 \sin x + 4 \cos x = k + 1$ has a solution,where $k \in R$,is:

If $A + B + C = 180^o$,then the value of $\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2}$ is equal to

If $\sin \theta = \frac{1}{2} \left( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} \right)$,where $x, y \in \mathbb{R} - \{0\}$. Then:

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

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