All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Two direction vectors on the plane are $\begin{pmatrix}6\\4\\{-5}\end{pmatrix}$ and $\begin{pmatrix}0\\{-2}\\{1}\end{pmatrix}$

Normal vector of plane, $\underset{\sim}{n} = \begin{pmatrix}6\\4\\{-5}\end{pmatrix} \times \begin{pmatrix}0\\{-2}\\1\end{pmatrix} = \begin{pmatrix}6\\6\\12\end{pmatrix}$

$\Rightarrow p: \underset{\sim}{r} \cdot \begin{pmatrix}1\\1\\{2}\end{pmatrix} = -3$

$\therefore, x + y + 2z = -3$

(ii)
$l_1 : \underset{\sim}{r} = \begin{pmatrix}1\\2\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}2\\{-4}\\{1}\end{pmatrix}, \lambda \in \mathbb{R}$

$l_2 : \underset{\sim}{r} = \begin{pmatrix}-2\\1\\3\end{pmatrix} + \mu \begin{pmatrix}1\\5\\k\end{pmatrix}, \mu \in \mathbb{R}$

Since both lines intersect, $\begin{pmatrix}1\\2\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}2\\{-4}\\{1}\end{pmatrix} = \begin{pmatrix}-2\\1\\3\end{pmatrix} + \mu \begin{pmatrix}1\\5\\k\end{pmatrix} \mathrm{~for~some~} \lambda, \mu$

$2\lambda - \mu = -3 \rightarrow (1)$

$-4\lambda - 5\mu = -1 \rightarrow (2)$

$\lambda - k\mu = 6 \rightarrow (3)$

Using GC, $\lambda = -1, \mu = 1, k = -7$

(iii)
Since $\begin{pmatrix}2\\{-4}\\1\end{pmatrix} \cdot \begin{pmatrix}1\\1\\2\end{pmatrix} = 0, l_1$ is parallel to plane p.

Since $\begin{pmatrix}1\\2\\-3\end{pmatrix} \cdot \begin{pmatrix}1\\1\\2\end{pmatrix} = -3, \begin{pmatrix}1\\2\\-3\end{pmatrix}$ lies on plane p.

Thus, $l_1$ lies in p.

$[\begin{pmatrix}-2\\1\\3\end{pmatrix} + \mu \begin{pmatrix}1\\5\\k\end{pmatrix}] \cdot \begin{pmatrix}1\\1\\2\end{pmatrix} = -3$

$\mu = 1$

Required coordinate $= (-1, 6, -4)$

(iv)
Required acute angle $=sin^{-1} \frac{\begin{pmatrix}1\\5\\-7\end{pmatrix}] \cdot \begin{pmatrix}1\\1\\2\end{pmatrix}}{\sqrt{75} \sqrt{6}} = 22.2^{\circ}$

This question was well attempted. For (iii), some students might find it easier to plug the entire equation of $l_1$ into the equation of plane p and show. Aside from being careless in the vector manipulations, there were not much mistakes.