Let $\left(-2-\frac{1}{3} \mathrm{i}\right)^3=\frac{x+\mathrm{i} y}{27}, \mathrm{i}=\sqrt{-1}$, where $x$ and $y$ are real numbers, then $(y-x)$ has the value

The shaded region in the following figure is the solution set of the inequations

Let $\bar{p}$ and $\bar{q}$ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $|\vec{p}|=p,|\vec{q}|=q$. The points $R$ and $S$ divide PQ internally and externally in the ratio $2: 3$ respectively. If OR and $O S$ are perpendiculars, then

The value of a for which the volume of parallelepiped formed by $\hat{i}+a \hat{j}+\hat{k}, \hat{j}+a \hat{k}$ and $a \hat{i}+\hat{k}$ becomes minimum is