4400 Higher June 2009 Paper 4 Q18

EdexcelHigherOld spec5 marksVectors

18 The diagram shows a parallelogram, \(ABCD\).
\(M\) is the midpoint of \(BC\).
\(N\) is the midpoint of \(AD\).

Parallelogram ABCD with AB marked x, AD marked y, M the midpoint of BC and N the midpoint of AD

Diagram NOT accurately drawn

\(\overrightarrow{AB} = \mathbf{x}\)
\(\overrightarrow{AD} = \mathbf{y}\)

Find, in terms of \(\mathbf{x}\) and/or \(\mathbf{y}\), the vectors

(a) \(\overrightarrow{MN}\) (1)

(b) \(\overrightarrow{AC}\) (1)

\(P\) is the point such that \(\overrightarrow{CP} = \mathbf{y} - \frac{1}{2}\mathbf{x}\)

(c) Find, in terms of \(\mathbf{x}\) and/or \(\mathbf{y}\), the vector \(\overrightarrow{PA}\)
Simplify your answer as much as possible. (3)

Scheme	Marks
\(-\mathbf{x}\) oe	B1
	(1)

Scheme	Marks
\(\mathbf{x} + \mathbf{y}\) oe	B1
	(1)

Mark scheme (c)

Scheme	Marks
Unsimplified expression in terms of \(x\) and \(y\) for \(PA\) or \(AP\) (either correct or ft from b) e.g. \((AP =)\) “\(\mathbf{x} + \mathbf{y}\)” \(+ \mathbf{y} - \frac{1}{2}\mathbf{x}\) or \((PA =)\) \(\frac{1}{2}\mathbf{x} - \mathbf{y} -\) “\(\mathbf{x} + \mathbf{y}\)”	B2
\(-0.5\mathbf{x} - 2\mathbf{y}\)	B1
	(3)
	(5 marks)

Notes

B2: B1 Correct vector statement with at least 3 terms including \(AP\) or \(PA\) e.g. \(PA = PC + CA\) or \(AP = AC + CP\) can include \(x\) and/or \(y\)

B1: cao