Higher June 2024 Paper 2 Q24
24
Diagram NOT accurately drawn
The diagram shows a quadrilateral \(OACB\) in which
\(\overrightarrow{OA} = 4\mathbf{a} \qquad \overrightarrow{OB} = 3\mathbf{b} \qquad \overrightarrow{BC} = 2\mathbf{a} + \mathbf{b}\)
Give your answer in its simplest form. (2)
The point \(P\) lies on \(AC\) such that \(AP : PC = 3 : 2\)
The point \(Q\) is such that \(OPQ\) and \(BCQ\) are straight lines.
Give your answer in its simplest form.
Show your working clearly. (4)
| Scheme | Marks |
|---|---|
| eg \(\overrightarrow{AC} = \overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{BC}\) or eg \(-4\mathbf{a} + 3\mathbf{b} + 2\mathbf{a} + \mathbf{b}\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(4\mathbf{b} - 2\mathbf{a}\) | A1 |
| (2) |
Notes
A1: oe but must be simplified
eg \(-2\mathbf{a} + 4\mathbf{b}\), \(2(2\mathbf{b} - \mathbf{a})\)
| Scheme | Marks |
|---|---|
eg \(\overrightarrow{OP} = 4\mathbf{a} + \dfrac{3}{5}\text{``}{(4\mathbf{b} - 2\mathbf{a})}\text{''}\left(= \dfrac{14}{5}\mathbf{a} + \dfrac{12}{5}\mathbf{b} \text{ or } 2.8\mathbf{a} + 2.4\mathbf{b}\right)\) oe or eg \(\overrightarrow{OP} = 3\mathbf{b} + 2\mathbf{a} + \mathbf{b} - \dfrac{2}{5}\text{``}{(4\mathbf{b} - 2\mathbf{a})}\text{''}\left(= \dfrac{14}{5}\mathbf{a} + \dfrac{12}{5}\mathbf{b}\right)\) oe | M1ft |
eg \(\overrightarrow{OQ} = \lambda\text{``}{\left(\dfrac{14}{5}\mathbf{a} + \dfrac{12}{5}\mathbf{b}\right)}\text{''}\) or eg \(\overrightarrow{OQ} = 3\mathbf{b} + \mu(2\mathbf{a} + \mathbf{b})\) or eg \(\overrightarrow{OQ} = 4\mathbf{b} + 2\mathbf{a} + \omega(2\mathbf{a} + \mathbf{b})\) or eg \(\overrightarrow{PQ} = k(2.8\mathbf{a} + 2.4\mathbf{b})\) or eg \(\overrightarrow{PQ} = \tfrac{2}{5}(4\mathbf{b} - 2\mathbf{a}) + m(2\mathbf{a} + \mathbf{b})\) | M1ft |
eg \(\overrightarrow{OQ} = \lambda\text{``}{\left(\dfrac{14}{5}\mathbf{a} + \dfrac{12}{5}\mathbf{b}\right)}\text{''}\) and eg \(\overrightarrow{OQ} = 3\mathbf{b} + \mu(2\mathbf{a} + \mathbf{b})\) or \(4\mathbf{b} + 2\mathbf{a} + \omega(2\mathbf{a} + \mathbf{b})\) or eg \(\overrightarrow{PQ} = k(2.8\mathbf{a} + 2.4\mathbf{b})\) and eg \(\overrightarrow{PQ} = \tfrac{2}{5}(4\mathbf{b} - 2\mathbf{a}) + m(2\mathbf{a} + \mathbf{b})\) | M1ft |
working required Answer: \(\dfrac{42}{5}\mathbf{a} + \dfrac{36}{5}\mathbf{b}\) | A1 |
| (4) | |
| (6 marks) |
Notes
M1ft: For \(\overrightarrow{OP}\) (could be part of another vector equation)
ft their \(\overrightarrow{AC}\)
M1ft: ft their \(\overrightarrow{AC}\)
(This mark can be awarded without the previous mark awarded)
a correct expression for \(\overrightarrow{OQ}\) or \(\overrightarrow{PQ}\) oe
M1ft: ft their \(\overrightarrow{AC}\)
2 correct expressions for \(\overrightarrow{OQ}\) or \(\overrightarrow{PQ}\) oe
ft dep on previous M1
A1: oe dep on M2
\(8.4\mathbf{a} + 7.2\mathbf{b}\)