The centroid of tetrahedron with vertices at $$\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)$$ and $$\mathrm{D}(-1,-2,4)$$ is

Two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ are given by $$\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}$$ and $$\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$$. The side $$\mathrm{AD}$$ is rotated by an acute angle $$\alpha$$ in the plane of parallelogram so that $$\mathrm{AD}$$ becomes $$\mathrm{AD}^{\prime}$$. If $$\mathrm{AD}^{\prime}$$ makes a right angle with side AB, then the cosine of the angle $$\alpha$$ is given by

The values of $$a$$ and $$b$$, so that the function

$$f(x)=\left\{\begin{array}{l} x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x, \frac{\pi}{2} < x \leq \pi \end{array}\right.$$

is continuous for $$0 \leq x \leq \pi$$, are respectively given by

For a feasible region OCDBO given below, the maximum value of the objective function $$z=3 x+4 y$$ is