Let $\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be three vectors. A vector $\bar{v}$ in the plane of $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$, whose projection on $\overline{\mathrm{c}}$ is $\frac{1}{\sqrt{3}}$, is given by

The differential equation, having general solution as $A x^2+B y^2=1$, where $A$ and $B$ are arbitrary constants, is

Let $z$ be a complex number such that $|z|+z=2+i$, where $i=\sqrt{-1}$, then $|z|$ is equal to

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are two unit vectors such that $5 \overline{\mathrm{a}}+4 \overline{\mathrm{~b}}$ and $\overline{\mathrm{a}}-2 \overline{\mathrm{~b}}$ are perpendicular to each other, then the angle between $\bar{a}$ and $\bar{b}$ is