2015 A-level H2 Mathematics (9740) Paper 1 Question 7 Suggested Solutions

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

(i)
$2 \vec{OC} = 3 \vec{CA} = 3 (\vec{OA} - \vec{OC})$

$5 \vec{OC} = 3 \textbf{a}$

$\vec{OC} = \frac{3}{5} \textbf{a}$

$11 \vec{OD} = 5 \vec{OB}$

$\vec{OD} = \frac{5}{11} \textbf{b}$

(ii)
$\vec{BC} = \frac{3}{5}\textbf{a} - \textbf{b}$

$l_{BC}: \textbf{r} = \textbf{b} + \lambda (\frac{3}{5} \textbf{a} - \textbf{b})$

$l_{BC}: \textbf{r} = \frac{3}{5} \lambda \textbf{a} + (1-\lambda) \textbf{b}, \lambda \in \mathbb{R}$

$\vec{AD} = \frac{3}{5} \textbf{a} - \textbf{b}$

$l_{AD}: \textbf{r} = \textbf{a} + \mu (\frac{5}{11} \textbf{b} - \textbf{a})$

$l_{AD}: \textbf{r} = \frac{5}{11} \mu \textbf{b} + (1-\mu) \textbf{a}, \mu \in \mathbb{R}$

(iii)
At E, $\frac{5}{11} \mu \textbf{b} + (1 - \mu)a = \frac{3}{5} \lambda \textbf{a} + (1 - \lambda) \textbf{b}$

$\frac{5}{11} \mu = 1 - \lambda \rightarrow(1)$

$\frac{3}{5} \lambda = 1 - \mu \rightarrow(1)$

Solving with GC, $\lambda = \frac{3}{4}$ and $\mu = \frac{11}{20}$

$\vec{OC} = \frac{9}{20} \textbf{a} + \frac{1}{4} \textbf{b}$

$\vec{AE} = \frac{1}{4} \textbf{b} - \frac{11}{20} \textbf{a}$

$\vec{ED} = \frac{9}{44} \textbf{b} - \frac{9}{20} \textbf{a}$

$= \frac{9}{11}(\frac{1}{4} \textbf{b} - \frac{11}{20} \textbf{a})$

$\vec{ED} = \frac{9}{11} \vec{AE}$

$\therefore, AE: ED = 11: 9$

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KS Comments:

Students that faced difficulties with ratio (like me) might struggle a bit here. But if you guys do listen in class, you should know the above method I use, should help a lot.
(ii) should be manageable. For (iii), students need to be careful if they use a GC to solve the question. Its like statistics, solving $\mu$ and $\lambda$ .

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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2015 A-level H2 Mathematics (9740) Paper 1 Question 7 Suggested Solutions

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