4MA0 Higher January 2017 Paper 3R Q23

EdexcelHigherOld spec6 marksVectors

23 The diagram shows trapezium \(ABCD\).

Trapezium ABCD with BC parallel to AD; vector b along AB and vector c along BC; diagonal AC drawn

Diagram NOT accurately drawn

\(BC\) is parallel to \(AD\)
\(AD = 3BC\)
\(\overrightarrow{AB} = \mathbf{b}, \quad \overrightarrow{BC} = \mathbf{c}\)

(a) Find, in terms of \(\mathbf{b}\) and \(\mathbf{c}\), the vector \(\overrightarrow{CD}\)
Give your answer in its simplest form. (2)

The point \(P\) lies on the line \(AC\) such that \(AP : PC = 2 : 1\)

(b) Is \(BPD\) a straight line?
Show your working clearly. (4)

Scheme	Marks
\(\overrightarrow{CD} = \overrightarrow{CB} + \overrightarrow{BA} + \overrightarrow{AD}\) or \(-\mathbf{c} - \mathbf{b} + 3\mathbf{c}\)	M1
\(2\mathbf{c} - \mathbf{b}\)	A1
	(2)

Mark scheme (b)

Scheme	Marks
\(\overrightarrow{BP} = \overrightarrow{BA} + \dfrac{2}{3}\overrightarrow{AC}\) or \(\overrightarrow{PD} = \dfrac{1}{3}\overrightarrow{AC} + \overrightarrow{CD}\)	M1ft
\(\overrightarrow{BP} = -\mathbf{b} + \dfrac{2}{3}(\mathbf{b} + \mathbf{c})\left(= \dfrac{2}{3}\mathbf{c} - \dfrac{1}{3}\mathbf{b}\right)\) or \(\overrightarrow{PD} = \dfrac{1}{3}(\mathbf{b} + \mathbf{c}) + 2\mathbf{c} - \mathbf{b}\left(= \dfrac{7}{3}\mathbf{c} - \dfrac{2}{3}\mathbf{b}\right)\)	M1ft
\(\overrightarrow{BP} = -\mathbf{b} + \dfrac{2}{3}(\mathbf{b} + \mathbf{c})\left(= \dfrac{2}{3}\mathbf{c} - \dfrac{1}{3}\mathbf{b}\right)\) AND \(\overrightarrow{PD} = \dfrac{1}{3}(\mathbf{b} + \mathbf{c}) + 2\mathbf{c} - \mathbf{b}\left(= \dfrac{7}{3}\mathbf{c} - \dfrac{2}{3}\mathbf{b}\right)\) OR \(\overrightarrow{BP} = -\mathbf{b} + \dfrac{2}{3}(\mathbf{b} + \mathbf{c})\left(= \dfrac{2}{3}\mathbf{c} - \dfrac{1}{3}\mathbf{b}\right)\) AND \(\overrightarrow{BD} = -\mathbf{b} + 3\mathbf{c}\) OR \(\overrightarrow{PD} = \dfrac{1}{3}(\mathbf{b} + \mathbf{c}) + 2\mathbf{c} - \mathbf{b}\left(= \dfrac{7}{3}\mathbf{c} - \dfrac{2}{3}\mathbf{b}\right)\) AND \(\overrightarrow{BD} = -\mathbf{b} + 3\mathbf{c}\)	M1
No with correct appropriate vectors and reason	A1
	(4)
	(6 marks)

Notes

M1ft: Ft their \(\overrightarrow{CD}\)

M1: or \(\overrightarrow{BP} = \dfrac{1}{3}(2\mathbf{c} - \mathbf{b})\) and

\(\overrightarrow{CD} = 2\mathbf{c} - \mathbf{b}\)

A1: E.g. \(\overrightarrow{BP} = \dfrac{1}{3}(2\mathbf{c} - \mathbf{b})\) and \(\overrightarrow{CD} = 2\mathbf{c} - \mathbf{b}\) are parallel and therefore not in a straight line

Correct simplified vectors for two of \(BP\), \(BD\), \(PD\) with explanation that vectors are not a multiple of each other