If $S$ is the circumentre of a $\triangle A B C, a=5, b=6, c=9$ and $S B=\frac{27}{4 \sqrt{2}}$, then $\sin 2 C=$

In a $\triangle A B C$, if $\frac{r}{r_1}=\frac{1}{2}$, then $4 \tan \frac{A}{2}\left(\tan \frac{B}{2}+\tan \frac{C}{2}\right)=$

If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ are the position vectors of the points $\mathbf{A}$ and $\mathbf{B}$ respectively, $\mathbf{C}$ divides $\mathbf{A B}$ in the ratio $2: 3$ and $\mathbf{M}$ is the mid-point of $A B$, then 5 (position vector of $\mathbf{C})-2($ position vector of $\mathbf{M})=$

30. If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the non-coplanar vectors and $\mathbf{a}-2 \mathbf{b}+3 \mathbf{c},-4 \mathbf{a}+5 \mathbf{b}-6 \mathbf{c}, x \mathbf{a}-9 \mathbf{b}+z \mathbf{c}$ are collinear points, then $2 x-z=$