If $A(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), B(\lambda \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}), C(-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ and $D(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ are four points in space such that $\mathbf{A B}=x \mathbf{A C}+y \mathbf{A D}$ for some real number $x \neq 0, y \neq 0$, then $17(\lambda+9)=$

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \cdot \mathbf{b}=1$, $\cos (\mathbf{a} \cdot \mathbf{b})=\frac{1}{3}$ and the components of $\mathbf{b}$ w.r.t. $(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}})$ be integers. Then, the number of possible vectors that represent $\mathbf{b}$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $\mathbf{a}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$, $|\mathbf{b}|=7, \mathbf{a} \cdot \mathbf{b}=4$ and $|\mathbf{a} \times \mathbf{b}|=5 \sqrt{17}$, then $p=$

In a $\triangle A B C, D$ and $E$ divide the sides $B C$ and $C A$ in the ratio $2: 1$ respectively. If $P$ is the point of intersection of $A D$ and $B E$, then the ratio in which $P$ divides $A D$ is