Higher June 2022 Paper 2 Q21
21
Diagram NOT accurately drawn
\(OAB\) is a triangle.
\(Q\) is the point on \(AB\) such that \(OQP\) is a straight line.
\(\overrightarrow{OA} = 4\mathbf{a}\) \(\overrightarrow{OB} = 6\mathbf{b}\) \(\overrightarrow{AP} = 2\mathbf{a} + 8\mathbf{b}\)
Using a vector method, find the ratio \(AQ : QB\)
(5)
| Scheme | Marks |
|---|---|
| \(\overrightarrow{OP} = 4\mathbf{a} + 2\mathbf{a} + 8\mathbf{b}\;(= 6\mathbf{a} + 8\mathbf{b})\) oe OR \(\overrightarrow{PO} = -6\mathbf{a} - 8\mathbf{b}\) oe or \(\overrightarrow{AB} = 6\mathbf{b} - 4\mathbf{a}\) oe OR \(\overrightarrow{BA} = 4\mathbf{a} - 6\mathbf{b}\) oe or \(\overrightarrow{BP} = 6\mathbf{a} + 2\mathbf{b}\) oe OR \(\overrightarrow{PB} = -6\mathbf{a} - 2\mathbf{b}\) oe | M1 |
| \(\overrightarrow{OQ} = 4\mathbf{a} + \lambda(6\mathbf{b} - 4\mathbf{a})\) oe OR \(6\mathbf{b} + \mu(4\mathbf{a} - 6\mathbf{b})\) oe OR \(x(6\mathbf{a} + 8\mathbf{b})\) oe or \(\overrightarrow{BQ} = \mu(4\mathbf{a} - 6\mathbf{b})\) oe OR \(-6\mathbf{b} + \lambda(6\mathbf{a} + 8\mathbf{b})\) oe OR \(4\mathbf{a} - 6\mathbf{b} + x(6\mathbf{b} - 4\mathbf{a})\) oe or \(\overrightarrow{AQ} = y(6\mathbf{b} - 4\mathbf{a})\) oe OR \(-4\mathbf{a} + x(6\mathbf{a} + 8\mathbf{b})\) oe OR \(6\mathbf{b} - 4\mathbf{a} + \mu(4\mathbf{a} - 6\mathbf{b})\) oe OR \(2\mathbf{a} + 8\mathbf{b} + m(6\mathbf{a} + 8\mathbf{b})\) oe or \(\overrightarrow{QP} = \lambda(6\mathbf{a} + 8\mathbf{b})\) oe OR \(\mu(4\mathbf{a} - 6\mathbf{b}) + 2\mathbf{a} + 8\mathbf{b}\) oe | M1 |
| M1 | |
| eg \(\lambda = \dfrac{8}{17}\) or \(\mu = \dfrac{9}{17}\) or \(AQ:QB = \dfrac{4x}{3} : \dfrac{3x}{2}\) oe | A1 |
| Working required Answer: 8 : 9 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: oe for one of \(\overrightarrow{OP}\) or \(\overrightarrow{PO}\) or \(\overrightarrow{AB}\) or \(\overrightarrow{BA}\) or \(\overrightarrow{BP}\) or \(\overrightarrow{PB}\) (may be seen as part of another vector calculation)
M1: for one of \(\overrightarrow{OQ}\) or \(\overrightarrow{QO}\) or \(\overrightarrow{BQ}\) or \(\overrightarrow{QB}\) or \(\overrightarrow{AQ}\) or \(\overrightarrow{QA}\) or \(\overrightarrow{QP}\) or \(\overrightarrow{PQ}\)
A1: oe
A1: oe