2012 A-level H2 Mathematics (9740) Paper 1 Question 5 Suggested Solutions

By KS Teng

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

(i)
\vec{OC} = \lambda \vec{OA} + \mu \vec{OB}

=\lambda \begin{pmatrix}1\\-1\\1\end{pmatrix} + \mu \begin{pmatrix}1\\2\\0\end{pmatrix}

Area = \frac{1}{2}|\vec{OA} \times \vec{OC}|

= \frac{1}{2} |\begin{pmatrix}1\\-1\\1\end{pmatrix} \times \begin{pmatrix}{\lambda + \mu}\\{-\lambda + 2 \mu}\\{\lambda}\end{pmatrix}|

= \frac{1}{2} |\begin{pmatrix}{-2\mu}\\{\mu}\\{3\mu}\end{pmatrix}|

= \frac{1}{2} \mu \sqrt{14}

\Rightarrow \frac{1}{2} \mu \sqrt{14} = \sqrt{126}

\mu = 6

(ii)
|\vec{OC}| = |\begin{pmatrix}{\lambda + 4}\\{8-\lambda}\\{\lambda}\end{pmatrix}| = 5 \sqrt{3}

3 \lambda^2 - 8 \lambda + 5 = 0

\lambda = 1 \mathrm{~or~} \frac{5}{3}

Thus required coordinates are (5, 7, 1) and (\frac{17}{3}, \frac{19}{3}, \frac{5}{3})

KS Comments:

Students just got to be careful when doing the cross product since there are unknowns.

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