Let $\vec{a}=\sqrt{7} \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+2 \hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}+\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{r} \cdot \vec{a}=0$, then $|3 \vec{r}|^2$ is equal to:

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0$ and $\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})$ from the origin is :

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 27,1 \leq y \leq x^2\right\}$ is equal to :

The product of all possible values of $\alpha$, for which

$\lim \limits_{x \rightarrow 0}\left(\frac{1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x)}{\sin ^2((\alpha+1) x)}\right)=2$, is :