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

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 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 __________

Let $$w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}$$. Let $$\operatorname{Re}(w)=0$$ be the circle $$\mathrm{C}$$ of radius 1 in the first quadrant touching the line $$y=1$$ and the $$y$$-axis. If the curve $$\operatorname{Im}(w)=0$$ intersects $$\mathrm{C}$$ at $$\mathrm{A}$$ and $$\mathrm{B}$$, then $$30(A B)^{2}$$ is equal to __________