Higher June 2025 Paper 2R Q18
18 \(OAB\) is a triangle.
Diagram NOT accurately drawn
\(\overrightarrow{OA} = 4\mathbf{a}\)
\(\overrightarrow{OB} = 4\mathbf{b}\)
\(P\) is the point on \(AB\) such that \(AP : PB = 1 : 3\)
(a) Write down \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) (1)
(b) Express \(\overrightarrow{OP}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Give your answer in its simplest form. (2)
Give your answer in its simplest form. (2)
| Scheme | Marks |
|---|---|
| \(-4\mathbf{a} + 4\mathbf{b}\) | B1 |
| (1) |
Notes
B1: oe eg \(4(\mathbf{b} - \mathbf{a})\)
| Scheme | Marks |
|---|---|
\(4\mathbf{a} + \dfrac{1}{4}(-4\mathbf{a} + 4\mathbf{b})\) oe or \(4\mathbf{a} + \dfrac{1}{4}[\overrightarrow{AB}]\) or \(4\mathbf{b} - \dfrac{3}{4}(-4\mathbf{a} + 4\mathbf{b})\) oe or \(4\mathbf{b} - \dfrac{3}{4}[\overrightarrow{AB}]\) | M1ft |
| Correct answer only scores full marks (unless from obviously incorrect working) Answer: \(3\mathbf{a} + \mathbf{b}\) | A1 |
| (2) | |
| (3 marks) |
Notes
M1ft: for a correct expression ft their (a)
where \([\overrightarrow{AB}]\) is their answer to (a) of the form \(m\mathbf{a} + n\mathbf{b}\), \(m, n \ne 0\)
A1: allow \(\mathbf{b} + 3\mathbf{a}\)