2013 A-level H2 Mathematics (9740) Paper 1 Question 6 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)
Equation of plane OABC: \textbf{r} = \lambda \textbf{a} + \mu \textbf{b}, for \lambda, \mu \in \mathbb{R}
Since \textbf{c} is on plane OABC, \textbf{c} =\lambda \textbf{a} + \mu \textbf{b}  for \lambda, \mu are constants

(ii)
Using the ratio theorem formula, \vec{ON}=\frac{4\mathbf{a}+3\mathbf{c}}{7}

(iii)
Area ONC = Area OMC
\frac{1}{2} |{\frac{4\mathbf{a}+3\mathbf{c}}{7} \times \mathbf{c}}| = \frac{1}{2}|{\frac{1}{2}\mathbf{b} \times \mathbf{c}}|

\frac{1}{7} |{4 \mathbf{a} \times \mathbf{c} + 3\mathbf{c} \times \mathbf{c}}| = \frac{1}{2}|{\mathbf{b} \times (\lambda \mathbf{a} + \mu \mathbf{b})}|

\frac{2}{7} |{4\mathbf{a} \times (\lambda \mathbf{a} + \mu \mathbf{b})}| = |{\mathbf{b} \times \lambda \mathbf{a} + \mathbf{b} \times \mu \mathbf{b}}|

\frac{2}{7} |{4\mathbf{a} \times \lambda \mathbf{a} + 4\mathbf{a} \times \mu \mathbf{b}}| = \frac{1}{2}|{\mathbf{b} \times \lambda \mathbf{a}}|

\frac{8\mu}{7} |{\mathbf{a} \times \mathbf{b}}| = \lambda|{\mathbf{a} \times \mathbf{b}}|

\frac{8\mu}{7} = \lambda

KS Comments:

For (i), some students can rely on the law of parallelogram to show too. For (iii), students must know all their vector product manipulations well to do this.

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