If $\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k} \quad$ and $\bar{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are non-zero non-coplanar vectors and $m$ is non-zero scalar such that $[\mathrm{m} \overline{\mathrm{a}}+\overline{\mathrm{b}} \quad \mathrm{m} \overline{\mathrm{b}}+\overline{\mathrm{c}} \mathrm{m} \overline{\mathrm{c}}+\overline{\mathrm{a}}]=28[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$, then the value of $m$ is equal to
The area of the region, bounded by the parabola $y=x^2+2$ and the lines $y=x, x=0$ and $x=3$, is
If $0< x < 1$, then
$$\sqrt{1+x^2}\left[\left\{x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1\right]^{\frac{1}{2}}=$$
For the probability distribution
| $\mathrm{X:}$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|
| $\mathrm{p}(x):$ | 0.1 | 0.2 | 0.2 | 0.3 | 0.15 | 0.05 |
Then the $\operatorname{Var}(\mathrm{X})$ is
(Given : $$\left.(0.25)^2=0.0625,(0.35)^2=0.1225,(0.45)^2=0.2025\right)$$