If $$\bar{a}=\hat{i}+2 \hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+\hat{k}, \bar{c}=\hat{i}+\hat{j}-\hat{k}$$, then a vector in the plane of $$\bar{a}$$ and $$\bar{b}$$, whose projection on $$\overline{\mathrm{c}}$$ is $$\frac{1}{\sqrt{3}}$$, is

The number of solutions of $$\tan x+\sec x=2 \cos x$$ in $$[0,2 \pi]$$ are

If $$\int_\limits0^{\frac{1}{2}} \frac{x^2}{\left(1-x^2\right)^{\frac{3}{2}}} \mathrm{~d} x=\frac{\mathrm{k}}{6}$$, then the value of $$\mathrm{k}$$ is

General solution of the differential equation $$\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0$$ is