Higher January 2022 Paper 1 Q22
22 The diagram shows triangle \(OAB\)
Diagram NOT accurately drawn
\(\overrightarrow{OA} = 8\mathbf{a}\) \(\overrightarrow{OB} = 6\mathbf{b}\)
\(M\) is the point on \(OB\) such that \(OM : MB = 1 : 2\)
\(N\) is the midpoint of \(AB\)
\(P\) is the point of intersection of \(ON\) and \(AM\)
Using a vector method, find \(\overrightarrow{OP}\) as a simplified expression in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
Show your working clearly.
(5)
| Scheme | Marks |
|---|---|
eg \(\overrightarrow{ON} = 8\mathbf{a} + \dfrac{1}{2}(6\mathbf{b} - 8\mathbf{a})\;(= 3\mathbf{b} + 4\mathbf{a})\) or \(\overrightarrow{ON} = 6\mathbf{b} + \dfrac{1}{2}(-6\mathbf{b} + 8\mathbf{a})\;(= 3\mathbf{b} + 4\mathbf{a})\) or \(\overrightarrow{NO} = \dfrac{1}{2}(8\mathbf{a} - 6\mathbf{b}) - 8\mathbf{a}\;(= -4\mathbf{a} - 3\mathbf{b})\) or \(\overrightarrow{NO} = -6\mathbf{b} + \dfrac{1}{2}(6\mathbf{b} - 8\mathbf{a})\;(= -3\mathbf{b} - 4\mathbf{a})\) or \(\overrightarrow{AM} = -8\mathbf{a} + \dfrac{1}{3}(6\mathbf{b})\;(= 2\mathbf{b} - 8\mathbf{a})\) or \(\overrightarrow{AM} = -8\mathbf{a} + 6\mathbf{b} - \dfrac{2}{3}(6\mathbf{b})\;(= 2\mathbf{b} - 8\mathbf{a})\) or \(\overrightarrow{MA} = 8\mathbf{a} - \dfrac{1}{3}(6\mathbf{b})\;(= 8\mathbf{a} - 2\mathbf{b})\) or \(\overrightarrow{MA} = \dfrac{2}{3}(6\mathbf{b}) + 8\mathbf{a} - 6\mathbf{b}\;(= 8\mathbf{a} - 2\mathbf{b})\) | M1 |
| \(\overrightarrow{OP} = \mu(3\mathbf{b} + 4\mathbf{a})\) and one of eg \(\overrightarrow{OP} = 8\mathbf{a} + x(2\mathbf{b} - 8\mathbf{a})\;(= (8 - 8x)\mathbf{a} + 2x\mathbf{b})\) or \(\overrightarrow{OP} = 2\mathbf{b} + y(8\mathbf{a} - 2\mathbf{b})\;(= (2 - 2y)\mathbf{b} + 8y\mathbf{a})\) | M2 |
eg \(\dfrac{4}{3} = \dfrac{8y}{2 - 2y}\) or \(\dfrac{4}{3} = \dfrac{8 - 8x}{2x}\) oe or \(3\mu = 2x\) and \(4\mu = 8 - 8x\) or \(3\mu = 2 - 2y\) and \(4\mu = 8y\) | M1 |
Working required Answer: \(2\mathbf{a} + \dfrac{3}{2}\mathbf{b}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: a correct expression for \(\overrightarrow{ON}\) or \(\overrightarrow{NO}\) or \(\overrightarrow{AM}\) or \(\overrightarrow{MA}\)
M2: oe
(M1 for one correct expression for \(\overrightarrow{OP}\))
(where \(\mu\), \(x\), \(y\) are scalars)
M1: A correct expression to find the position of \(P\) along \(ON\) or two correct simultaneous equations coming from the expressions for \(\overrightarrow{OP}\)
A1: dep on M3, oe eg \(2\mathbf{a} + 1.5\mathbf{b}\)