Higher November 2024 Paper 1 Q24
24 \(OAB\) is a triangle.
Diagram NOT accurately drawn
\(\overrightarrow{OA} = 10\mathbf{a}\) \(\overrightarrow{OB} = 10\mathbf{b}\)
\(ARQ\) and \(ORP\) are straight lines.
\(\overrightarrow{AP} = \dfrac{1}{4}\overrightarrow{AB}\) and \(\overrightarrow{OQ} = \dfrac{1}{5}\overrightarrow{OB}\)
Write the following vectors in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Simplify your answers.
| Scheme | Marks |
|---|---|
| \(-10\mathbf{a} + 2\mathbf{b}\) | B1 |
| (1) |
Notes
B1: or \(2\mathbf{b} - 10\mathbf{a}\)
Must be simplified
| Scheme | Marks |
|---|---|
| \(\dfrac{15}{2}\mathbf{a} + \dfrac{5}{2}\mathbf{b}\) | B1 |
| (1) |
Notes
| Scheme | Marks |
|---|---|
eg \(\left(\overrightarrow{OR} = \text{“}k\text{”}\overrightarrow{OP} =\right)\;\text{“}k\text{”}\left(\text{“}\dfrac{15}{2}\mathbf{a} + \dfrac{5}{2}\mathbf{b}\text{”}\right)\) oe | M1 |
eg \(\left(\overrightarrow{OR} = \overrightarrow{OA} + \overrightarrow{AR} =\right)\;10\mathbf{a} + \text{“}\lambda\text{”}(\text{“}{-10}\mathbf{a} + 2\mathbf{b}\text{”})\;\left(= (10 - 10\text{“}\lambda\text{”})\mathbf{a} + 2\text{“}\lambda\text{”}\mathbf{b}\right)\) oe or \(\left(\overrightarrow{OR} = \overrightarrow{OQ} + \overrightarrow{QR} =\right)\;2\mathbf{b} - \text{“}\mu\text{”}(\text{“}{-10}\mathbf{a} + 2\mathbf{b}\text{”})\;\left(= 10\text{“}\mu\text{”}\mathbf{a} + (2 - 2\text{“}\mu\text{”})\mathbf{b}\right)\) oe | M1 |
eg \(\dfrac{15}{2}\text{“}k\text{”} = 10 - 10\text{“}\lambda\text{”}\) oe and \(\dfrac{5}{2}\text{“}k\text{”} = 2\text{“}\lambda\text{”}\) oe or \(\text{“}\lambda\text{”} = \dfrac{5}{8}\) oe or \(\dfrac{5}{2}\text{“}k\text{”} = 2 - 2\text{“}\mu\text{”}\) and \(\dfrac{15}{2}\text{“}k\text{”} = 10\text{“}\mu\text{”}\) oe or “\(k\)” = 0.5 oe | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(\dfrac{15}{4}\mathbf{a} + \dfrac{5}{4}\mathbf{b}\) | A1 |
| (4) | |
| (6 marks) |
Notes
M1: ft from part (ii) for \(\overrightarrow{OR}\)
\(\overrightarrow{OP}\) must be in terms of \(\mathbf{a}\) and \(\mathbf{b}\) (lower case)
M1: ft from part (i) for another path for \(\overrightarrow{OR}\)
\(\overrightarrow{AQ}\) must be in terms of \(\mathbf{a}\) and \(\mathbf{b}\) (lower case)
M1: for correct equations
(not followed through equations)
A1: oe eg \(3\dfrac{3}{4}\mathbf{a} + 1\dfrac{1}{4}\mathbf{b}\) or \(3.75\mathbf{a} + 1.25\mathbf{b}\)