1MA0 Higher June 2012 Paper 2 Q26

EdexcelHigherOld spec4 marksVectors

Diagram NOT accurately drawn

\(OAB\) is a triangle.

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

(a) Find \(\overrightarrow{AB}\) in terms of a and b. (1)

\(P\) is the point on \(AB\) such that \(AP : PB = 3 : 1\)

(b) Find \(\overrightarrow{OP}\) in terms of a and b.
Give your answer in its simplest form. (3)

Answer	Mark	Notes
\(\mathbf{b} - \mathbf{a}\)	1	B1 for \(\mathbf{b} - \mathbf{a}\) or \(-\mathbf{a} + \mathbf{b}\)

Mark scheme (b)

Working	Answer	Mark	Notes
\(\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{AP}\) \(\overrightarrow{AP} = \frac{3}{4} \times (\mathbf{b} - \mathbf{a})\) \(\overrightarrow{OP} = \mathbf{a} + \frac{3}{4} \times (\mathbf{b} - \mathbf{a})\) OR \(\overrightarrow{OP} = \overrightarrow{OB} + \overrightarrow{BP}\) \(\overrightarrow{BP} = \frac{1}{4} \times (\mathbf{a} - \mathbf{b})\) \(\overrightarrow{OP} = \mathbf{b} + \frac{1}{4} \times (\mathbf{a} - \mathbf{b})\)	\(\frac{1}{4}(\mathbf{a} + 3\mathbf{b})\)	3	B1 for \(\frac{3}{4} \times\) ‘\((\mathbf{b} - \mathbf{a})\)’ M1 for \(\overrightarrow{OA} + \overrightarrow{AP}\) (\(= \overrightarrow{OP}\)) or \(\overrightarrow{OA} + \frac{3}{4}\overrightarrow{AB}\) (\(= \overrightarrow{OP}\)) or \(\mathbf{a} \pm \frac{3}{4} \times\) ‘\((\mathbf{b} - \mathbf{a})\)’ A1 for \(\frac{1}{4}(\mathbf{a} + 3\mathbf{b})\) or \(\frac{1}{4}\mathbf{a} + \frac{3}{4}\mathbf{b}\) OR B1 for \(\frac{1}{4} \times\) ‘\((\mathbf{a} - \mathbf{b})\)’ M1 for \(\overrightarrow{OB} + \overrightarrow{BP}\) (\(= \overrightarrow{OP}\)) or \(\overrightarrow{OB} + \frac{1}{4}\overrightarrow{BA}\) (\(= \overrightarrow{OP}\)) or \(\mathbf{b} \pm \frac{1}{4} \times\) ‘\((\mathbf{a} - \mathbf{b})\)’ A1 for \(\frac{1}{4}(\mathbf{a} + 3\mathbf{b})\) or \(\frac{1}{4}\mathbf{a} + \frac{3}{4}\mathbf{b}\)