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}

KS Comments:

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.

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