Higher November 2017 Paper 3 Q21
21
\(OAN\), \(OMB\) and \(APB\) are straight lines.
\(AN = 2OA\).
\(M\) is the midpoint of \(OB\).
\(\overrightarrow{OA} = \mathbf{a}\) \(\overrightarrow{OB} = \mathbf{b}\)
\(\overrightarrow{AP} = k\overrightarrow{AB}\) where \(k\) is a scalar quantity.
Given that \(MPN\) is a straight line, find the value of \(k\). (5)
| Answer | Mark | Notes |
|---|---|---|
| \(\dfrac{2}{5}\) | P1 | for process to find \(\overrightarrow{AB}\ (= \mathbf{b} - \mathbf{a})\) or \(\overrightarrow{BA}\ (= \mathbf{a} - \mathbf{b})\) |
| P1 | for process to find \(\overrightarrow{MN}\ \left(= -\tfrac{1}{2}\mathbf{b} + \mathbf{a} + 2\mathbf{a}\right)\) or \(\overrightarrow{PN}\ (= -k(\mathbf{b} - \mathbf{a}) + 2\mathbf{a})\) or \(\overrightarrow{MP}\) (= \(-\tfrac{1}{2}\mathbf{b} + \mathbf{a} + k(\mathbf{b} - \mathbf{a})\) or \(\tfrac{1}{2}\mathbf{b} + (1 - k)(\mathbf{a} - \mathbf{b})\)) | |
| P1 | for process to find two of \(\overrightarrow{MN}\), \(\overrightarrow{PN}\) and \(\overrightarrow{MP}\) | |
| P1 | for process to find \(k\), using \(\overrightarrow{MN}\) as a multiple of \(\overrightarrow{PN}\) or using \(\overrightarrow{MN}\) as a multiple of \(\overrightarrow{MP}\) or using \(\overrightarrow{PN}\) as a multiple of \(\overrightarrow{MP}\) | |
| A1 | for \(\dfrac{2}{5}\) oe |