If $$\bar{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \bar{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}, \bar{c}=2 \hat{i}-\hat{j}+4 \hat{k}$$, then a vector $$\overline{\mathrm{d}}$$ which is parallel to vector $$\overline{\mathrm{a}} \times \overline{\mathrm{b}}$$ and which $$\overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=15$$, is

If the vertices of a triangle are $$(-2,3),(6,-1)$$ and $$(4,3)$$, then the co-ordinates of the circumcentre of the triangle are

The solution of $$\frac{\mathrm{d} x}{\mathrm{~d} y}+\frac{x}{y}=x^2$$ is

If $$\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} \mathrm{~d} x=\mathrm{A} \cos 8 x+\mathrm{c}$$, where $$\mathrm{c}$$ is an arbitrary constant, then the value of $$\mathrm{A}$$ is