4MA0 Higher January 2014 Paper 3 Q21

EdexcelHigherOld spec5 marksVectors

Trapezium ABCD with AB = 12a, AD = 3b and DC = 18a; E on the diagonal DB

Diagram NOT accurately drawn

\(ABCD\) is a trapezium.
\(AB\) is parallel to \(DC\).

\(\overrightarrow{AB} = 12\mathbf{a}\)
\(\overrightarrow{AD} = 3\mathbf{b}\)
\(\overrightarrow{DC} = 18\mathbf{a}\)

\(E\) is the point on the diagonal \(DB\) such that \(DE = \dfrac{1}{3}DB\).

(a) Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),

(i) \(\overrightarrow{DB}\)

(ii) \(\overrightarrow{DE}\)

(iii) \(\overrightarrow{AE}\)

(3)

(b) Show by a vector method that \(BC\) is parallel to \(AE\). (2)

Mark scheme (a)

Scheme	Marks
(i) Answer: \(12\mathbf{a} - 3\mathbf{b}\) oe	B1
(ii) Answer: \(4\mathbf{a} - \mathbf{b}\) oe	B1
(iii) Answer: \(4\mathbf{a} + 2\mathbf{b}\) oe	B1
	(3)

Notes

B1: Accept correct, unsimplified expression

Mark scheme (b)

Scheme	Marks
\(\overrightarrow{BC} = 6\mathbf{a} + 3\mathbf{b}\) oe	M1
\(\overrightarrow{BC} = \frac{3}{2}\overrightarrow{AE}\)	A1
	(2)
	(5 marks)

Notes

M1: Accept correct, unsimplified expression eg. \(-12\mathbf{a} + 3\mathbf{b} + 18\mathbf{a}\)

A1: Also award A1 if this relationship is clearly implied by expressions for \(\overrightarrow{BC}\) and \(\overrightarrow{AE}\) eg \(\overrightarrow{AE} = 2(\mathbf{b} + 2\mathbf{a})\) and \(\overrightarrow{BC} = 3(\mathbf{b} + 2\mathbf{a})\)
NB Correct expressions for \(\overrightarrow{BC}\) and \(\overrightarrow{AE}\) must be given