2011 A-level H2 Mathematics (9740) Paper 1 Question 11 Suggested Solutions

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

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

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

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)

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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