1MA0 Higher November 2013 Paper 1 Q24

EdexcelHigherOld spec6 marksVectors

Diagram NOT accurately drawn

\(OACB\) is a parallelogram.

\(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\)

\(D\) is the point such that \(\overrightarrow{AC} = \overrightarrow{CD}\)

The point \(N\) divides \(AB\) in the ratio \(2:1\)

(a) Write an expression for \(\overrightarrow{ON}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). (3)

*(b) Prove that \(OND\) is a straight line. (3)

Mark scheme (a)

Working	Answer	Mark	Notes
\(\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}\) \(\overrightarrow{ON} = \overrightarrow{OA} + \frac{2}{3}\overrightarrow{AB}\) \(\overrightarrow{ON} = \mathbf{a} + \frac{2}{3}(-\mathbf{a} + \mathbf{b})\) \(= \frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{b}\) OR \(\overrightarrow{ON} = \overrightarrow{OB} + \frac{1}{3}\overrightarrow{BA}\) \(\overrightarrow{ON} = \mathbf{b} + \frac{1}{3}(-\mathbf{b} + \mathbf{a})\) \(= \frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{b}\)	\(\frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{b}\)	3	M1 for correct vector equation involving \(\overrightarrow{ON}\), eg. \(\overrightarrow{ON} = \overrightarrow{OA} + \overrightarrow{AN}\), may be written, partially or fully, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), e.g. (\(\overrightarrow{ON} =\)) \(\mathbf{a} + \frac{2}{3}\overrightarrow{AB}\) M1 for showing answer requires \(\overrightarrow{AN} = \frac{2}{3}\overrightarrow{AB}\) or \(\overrightarrow{BN} = \frac{1}{3}\overrightarrow{BA}\) A1 \(\frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{b}\) oe

Mark scheme (b)

Working	Answer	Mark	Notes
\(\overrightarrow{OD} = \overrightarrow{OA} + \overrightarrow{AC} + \overrightarrow{CD}\) \(= \mathbf{a} + \mathbf{b} + \mathbf{b}\) \(= \mathbf{a} + 2\mathbf{b}\) \(\overrightarrow{OD} = 3(\frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{b})\) \(\overrightarrow{OD} = 3\overrightarrow{ON}\)	Proof	3	M1 for a correct vector statement for \(\overrightarrow{OD}\) or \(\overrightarrow{ND}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), e.g. \(\overrightarrow{OD} = \mathbf{a} + \mathbf{b} + \mathbf{b}\) oe or \(\overrightarrow{ND} = \frac{2}{3}(-\mathbf{b} + \mathbf{a}) + \mathbf{b} + \mathbf{b}\) oe A1 for correct and fully simplified vectors for \(\overrightarrow{ON}\) (may be seen in (a)) and for \(\overrightarrow{OD}\) (\(= \mathbf{a} + 2\mathbf{b}\)) or \(\overrightarrow{ND}\) (\(= \frac{2}{3}\mathbf{a} + \frac{4}{3}\mathbf{b}\)) C1 (dep on A1) for statement that \(\overrightarrow{OD}\) or \(\overrightarrow{ND}\) is a multiple of \(\overrightarrow{ON}\) (+ common point)