SPM 2021 Add Maths Trial Paper (Selangor) - Paper 1 (Question 8)

Question 8:

\begin{array}{l} Diagram 3 shows a parallelogram O A B C . Point P is the midpoint of A B and the \\ ratio of \vec{Q P} : \vec{O P} = 2 : 5. \\ Given that \vec{O A} = 3 \underset{˜}{i} + 4 \underset{˜}{j} and \vec{O C} = 6 \underset{˜}{i} + \underset{˜}{j} \end{array}

Diagram 3

\begin{array}{l} Express in the terms of \underset{˜}{i} and \underset{˜}{j}, \\ (a) \vec{O P} (1 mark) \\ (b) \vec{A Q} (3 marks) \end{array}

Solution:
(a)

\begin{array}{l} \vec{O P} = \vec{O A} + \vec{A P} \\ = 3 \underset{˜}{i} + 4 \underset{˜}{j} + \frac{1}{2} (\vec{A B}) \\ = 3 \underset{˜}{i} + 4 \underset{˜}{j} + \frac{1}{2} (\vec{O C}) \\ = 3 \underset{˜}{i} + 4 \underset{˜}{j} + \frac{1}{2} (6 \underset{˜}{i} + \underset{˜}{j}) \\ = 3 \underset{˜}{i} + 4 \underset{˜}{j} + 3 \underset{˜}{i} + \frac{1}{2} \underset{˜}{j} \\ = 6 \underset{˜}{i} + \frac{9}{2} \underset{˜}{j} \end{array}

(b)

\begin{array}{l} \vec{Q P} : \vec{O P} = 2 : 5 \\ \frac{\vec{Q P}}{\vec{O P}} = \frac{2}{5} \\ \vec{Q P} = \frac{2}{5} \times \vec{O P} \\ \vec{A Q} = \vec{A O} + \vec{O Q} \\ = - \vec{O A} + (\frac{3}{5} \times \vec{O P}) \\ = - (3 \underset{˜}{i} + 4 \underset{˜}{j}) + \frac{3}{5} (6 \underset{˜}{i} + \frac{9}{2} \underset{˜}{j}) \\ = - 3 \underset{˜}{i} - 4 \underset{˜}{j} + \frac{18}{5} \underset{˜}{i} + \frac{27}{10} \underset{˜}{j} \\ = \frac{3}{5} \underset{˜}{i} - \frac{13}{10} \underset{˜}{j} \end{array}

SPM 2021 Add Maths Trial Paper (Selangor) – Paper 1 (Question 8)