The locus of the point $P(\mathbf{r})$ which encloses a triangle $A B P$ of area 1 sq. unit with the fixed points $A(\hat{\mathbf{i}})$ and $B(\hat{\mathbf{j}})$ is

The shortest distance between the skew-lines $\mathbf{r}=(-\hat{\mathbf{i}}+3 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})$ and $\mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+s(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is

In a discrete data $\frac{1 \text { th }}{4}$ of the observations are equal to $a$, another $\frac{1 \text { th }}{4}$ of the observations are equal to $-a$. Out of the remaining, half of them are equal to $b$ and the rest are equal to $-b$. If the variance of all the observations is $(a b)$, then

For the following distribution, the mean deviation about the median is

$$ \begin{array}{cccccccc} \hline x_i & 6 & 12 & 18 & 24 & 30 & 36 & 42 \\ \hline f_i & 4 & 7 & 9 & 18 & 15 & 10 & 5 \\ \hline \end{array} $$