If a circle is inscribed in an equilateral triangle of side $a$, then the area of any square (in sq units) inscribed in this circle is

Match the items of List I with those of List II (here, $\Delta$ denotes the area of $\triangle A B C$ )

List I		List II
(A)	$$ \sum \cot A $$	(i)	$$ (a+b+c)^2 \frac{1}{4 \Delta} $$
(B)	$$ \sum \cot \frac{A}{2} $$	(ii)	$$ \left(a^2+b^2+c^2\right) \frac{1}{4 \Delta} $$
(C)	If $\tan A: \tan B: \tan C=1: 2: 3$, then $\sin A: \sin B: \sin C=$	(iii)	$$ 8: 6: 5 $$
(D)	$$ \begin{aligned} &\text { If } \cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}=3: 7: 9\\ &\text { then } a: b: c= \end{aligned} $$	(iv)	$$ 12: 5: 13 $$
		(v)	$$ \sqrt{5}: 2 \sqrt{2}: 3 $$
		(vi)	$$ 4 \Delta $$

$$ \text { Then, the correct match is } $$

Let $O(\mathbf{O}), A(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}), B(-2 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}), C(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})$ and $D(4 \hat{\mathbf{k}})$ are position vectors of the points $O, A, B, C$ and $D$. If a line passing through $A$ and $B$ intersects the plane passing through $O, C$ and $D$ at the point $R$, then position vector of $R$ is

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors. If $\alpha \mathbf{d}=\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $\beta \mathbf{a}=\mathbf{b}+\mathbf{c}+\mathbf{d}$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}|=$