Higher November 2020 Paper 2R Q25
25
Diagram NOT accurately drawn
\(OAN\), \(OMB\), \(APB\) and \(MPN\) are straight lines.
\(OA : AN = 1 : 4\)
\(OM : MB = 1 : 1\)
\(\overrightarrow{OA} = 2\mathbf{a}\) \(\overrightarrow{OB} = 2\mathbf{b}\)
By using a vector method, find the ratio \(AP : PB\)
Give your answer in its simplest form.
(5)
| Scheme | Marks |
|---|---|
| \(\overrightarrow{AB} = 2\mathbf{b} - 2\mathbf{a}\) or \(\overrightarrow{BA} = 2\mathbf{a} - 2\mathbf{b}\) \(\overrightarrow{MN} = 10\mathbf{a} - \mathbf{b}\) or \(\overrightarrow{NM} = -10\mathbf{a} + \mathbf{b}\) | M1 |
| eg \(\overrightarrow{MP} = -\mathbf{b} + 2\mathbf{a} + k(2\mathbf{b} - 2\mathbf{a})\) and \(\overrightarrow{MP} = \lambda(10\mathbf{a} - \mathbf{b})\) or eg \(\overrightarrow{MP} = \mathbf{b} + k(2\mathbf{a} - 2\mathbf{b})\) and \(\overrightarrow{MP} = \lambda(10\mathbf{a} - \mathbf{b})\) or eg \(\overrightarrow{PN} = 8\mathbf{a} + k(2\mathbf{a} - 2\mathbf{b})\) and \(\overrightarrow{PN} = \lambda(10\mathbf{a} - \mathbf{b})\) or eg \(\overrightarrow{AP} = 8\mathbf{a} + k(\mathbf{b} - 10\mathbf{a})\) and \(\overrightarrow{AP} = \lambda(2\mathbf{b} - 2\mathbf{a})\) or eg \(\overrightarrow{AP} = -2\mathbf{a} + \mathbf{b} + k(10\mathbf{a} - \mathbf{b})\) and \(\overrightarrow{AP} = \lambda(2\mathbf{b} - 2\mathbf{a})\) or eg \(\overrightarrow{AM} = k(2\mathbf{b} - 2\mathbf{a}) + \lambda(\mathbf{b} - 10\mathbf{a})\) and \(\overrightarrow{AM} = -2\mathbf{a} + \mathbf{b}\) | M2 |
| eg \(2 - 2k = 10\lambda\) and \(-1 + 2k = -\lambda\) (from \(\overrightarrow{MP}\) 1st) or eg \(2k = 10\lambda\) and \(1 - 2k = -\lambda\) (from \(\overrightarrow{MP}\) 2nd) or eg \(8 + 2k = 10\lambda\) and \(-2k = -\lambda\) (from \(\overrightarrow{PN}\)) or eg \(8 - 10k = -2\lambda\) and \(k = 2\lambda\) (from \(\overrightarrow{AP}\) 1st) or eg \(-2 + 10k = -2\lambda\) and \(1 - k = 2\lambda\) (from \(\overrightarrow{AP}\) 2nd) or eg \(-2k - 10\lambda = -2\) and \(2k + \lambda = 1\) (from \(\overrightarrow{AM}\)) | M1 |
| 4 : 5 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for finding \(\overrightarrow{AB}\) or \(\overrightarrow{BA}\) or \(\overrightarrow{MN}\) or \(\overrightarrow{NM}\)
M2: for writing eg \(\overrightarrow{MP}\) or \(\overrightarrow{PN}\) or \(\overrightarrow{AP}\) or \(\overrightarrow{AM}\) in two different ways in terms of a and b
(M1 for writing eg \(\overrightarrow{MP}\) or \(\overrightarrow{PN}\) or \(\overrightarrow{AP}\) or \(\overrightarrow{AM}\) in one way)
These may be written as eg \(\overrightarrow{PM}\) in place of \(\overrightarrow{MP}\)
A1: cao
(corrected from the printed mark scheme: the first pair of equations is printed as \(2 - 2k = 10\lambda\) and \(-1 + k = -\lambda\); the \(\mathbf{b}\) components of \(\overrightarrow{MP}\) give \(-1 + 2k = -\lambda\))