4MA0 Higher June 2015 Paper 3 Q24

EdexcelHigherOld spec4 marksVectors

Diagram NOT accurately drawn

In triangle \(OPQ\), \(\overrightarrow{OP} = 6\mathbf{a}\) and \(\overrightarrow{OQ} = 6\mathbf{b}\)
\(X\) is the midpoint of \(PQ\).

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

\(Y\) is the point on \(OX\) such that \(OY : YX = 2 : 1\)

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

Mark scheme (a)

Scheme	Marks
\(\overrightarrow{PQ} = 6\mathbf{b} - 6\mathbf{a}\) or \(\overrightarrow{QP} = 6\mathbf{a} - 6\mathbf{b}\) or \((\overrightarrow{OX}) = \overrightarrow{OP} + \overrightarrow{PX}\) oe or \((\overrightarrow{OX}) = \overrightarrow{OQ} + \overrightarrow{QX}\) oe or \(6\mathbf{a} + \dfrac{1}{2}(6\mathbf{b} - 6\mathbf{a})\) or \(6\mathbf{b} + \dfrac{1}{2}(6\mathbf{a} - 6\mathbf{b})\)	M1
\(3\mathbf{a} + 3\mathbf{b}\)	A1
	(2)

M1: NB: \(\overrightarrow{OX}\) may be partially in terms of a and/or b

A1: or \(3(\mathbf{a} + \mathbf{b})\)

Mark scheme (b)

Scheme	Marks
eg. \((\overrightarrow{QY} =)\;\overrightarrow{QO} + \dfrac{2}{3}\overrightarrow{OX}\) or \((\overrightarrow{QY} =)\;-6\mathbf{b} + \dfrac{2}{3}(3\mathbf{a} + 3\mathbf{b})\)	M1
\(2\mathbf{a} - 4\mathbf{b}\) or \(2(\mathbf{a} - 2\mathbf{b})\)	A1ft
	(2)

M1: for a complete method ft from (a)

A1ft: from (a)