$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$

The number of symmetric matrices of order 3, with all the entries from the set $$\{0,1,2,3,4,5,6,7,8,9\}$$ is :

Let $$m_{1}$$ and $$m_{2}$$ be the slopes of the tangents drawn from the point $$\mathrm{P}(4,1)$$ to the hyperbola $$H: \frac{y^{2}}{25}-\frac{x^{2}}{16}=1$$. If $$\mathrm{Q}$$ is the point from which the tangents drawn to $$\mathrm{H}$$ have slopes $$\left|m_{1}\right|$$ and $$\left|m_{2}\right|$$ and they make positive intercepts $$\alpha$$ and $$\beta$$ on the $$x$$-axis, then $$\frac{(P Q)^{2}}{\alpha \beta}$$ is equal to __________.

Let the mean of the data

be 5. If $$m$$ and $$\sigma^{2}$$ are respectively the mean deviation about the mean and the variance of the data, then $$\frac{3 \alpha}{m+\sigma^{2}}$$ is equal to __________