4400 Higher November 2010 Paper 4 Q21

EdexcelHigherOld spec5 marksVectors

Trapezium PQRS with PQ = 3a, SR = a, PS = b; T is a point on SQ

Diagram NOT accurately drawn

\(PQRS\) is a trapezium with \(PQ\) parallel to \(SR\).
\(\overrightarrow{SR} = \mathbf{a}\) \(\quad \overrightarrow{PQ} = 3\mathbf{a}\) \(\quad \overrightarrow{PS} = \mathbf{b}\)

\(T\) is the point on \(SQ\) such that \(ST = \frac{1}{4}SQ\).

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

(i) \(\overrightarrow{PR}\)

(ii) \(\overrightarrow{SQ}\)

(iii) \(\overrightarrow{PT}\)

(3)

(b) \(\overrightarrow{PT} = k\,\overrightarrow{PR}\) where \(k\) is a fraction.

(i) What does this result tell you about the points \(P\), \(T\) and \(R\)?

(ii) Find the value of \(k\).

(2)

Mark scheme (a)

Scheme	Marks
(i) Answer: \(\mathbf{a} + \mathbf{b}\)	B1
(ii) Answer: \(3\mathbf{a} - \mathbf{b}\)	B1
(iii) \(\frac{3}{4}\mathbf{a} + \frac{3}{4}\mathbf{b}\) or \(\mathbf{b} + \frac{1}{4}(3\mathbf{a} - \mathbf{b})\) or \(3\mathbf{a} - \frac{3}{4}(3\mathbf{a} - \mathbf{b})\) oe	B1
	(3)

Scheme	Marks
(i) Answer: collinear, in a (straight) line oe	B1
(ii) Answer: \(\frac{3}{4}\)	B1
	(2)
	(5 marks)

Notes