Higher June 2023 Paper 1 Q19

EdexcelHigherCurrent spec5 marksVectors

Triangle OAB with M on OA and N on OB; straight lines MB and NA cross at P; vector a is marked along OM and vector b along ON

Diagram NOT accurately drawn

\(OMA\), \(ONB\), \(MPB\) and \(NPA\) are straight lines.

\(M\) is the midpoint of \(OA\)

\(ON : NB = 1 : 5\)

\(\overrightarrow{OM} = \mathbf{a} \qquad \overrightarrow{ON} = \mathbf{b}\)

(a) Find in terms of \(\mathbf{a}\) and \(\mathbf{b}\) the vector \(\overrightarrow{AN}\) (1)

(b) Use a vector method to find the ratio \(AP : PN\) (4)

Scheme	Marks
\(-2\mathbf{a} + \mathbf{b}\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
eg \(\overrightarrow{OP} = \mathbf{a} + y(-\mathbf{a} + 6\mathbf{b})\) and \(\overrightarrow{OP} = 2\mathbf{a} + x(\text{``}{-2\mathbf{a} + \mathbf{b}}\text{''})\) \(\overrightarrow{AP} = -\mathbf{a} + y(-\mathbf{a} + 6\mathbf{b})\) and \(\overrightarrow{AP} = x(\text{``}{-2\mathbf{a} + \mathbf{b}}\text{''})\) \(\overrightarrow{AB} = x(\text{``}{-2\mathbf{a} + \mathbf{b}}\text{''}) + y(-\mathbf{a} + 6\mathbf{b})\) and \(\overrightarrow{AB} = -2\mathbf{a} + 6\mathbf{b}\) \(\overrightarrow{NP} = -\mathbf{b} + \mathbf{a} + y(-\mathbf{a} + 6\mathbf{b})\) and \(\overrightarrow{NP} = x(\text{``}{2\mathbf{a} - \mathbf{b}}\text{''})\) \(\overrightarrow{MP} = x(-\mathbf{a} + 6\mathbf{b})\) and \(\overrightarrow{MP} = \mathbf{a} + y(\text{``}{-2\mathbf{a} + \mathbf{b}}\text{''})\)	M2
eg \(x = 6y\) and \(2 - 2x = 1 - y\) (from \(\overrightarrow{OP}\)) \(x = 6y\) and \(-2x = -1 - y\) (from \(\overrightarrow{AP}\)) \(6 = x + 6y\) and \(-2 = -2x - y\) (from \(\overrightarrow{AB}\)) \(2x = 1 - y\) and \(-x = -1 + 6y\) (from \(\overrightarrow{NP}\)) \(-x = 1 - 2y\) and \(6x = y\) (from \(\overrightarrow{MP}\))	M1
Vector method required Answer: 6 : 5	A1
	(4)
	(5 marks)

Notes

M2: ft from (a), for writing eg \(\overrightarrow{OP}\) or \(\overrightarrow{AP}\) or \(\overrightarrow{AB}\) or \(\overrightarrow{NP}\) or \(\overrightarrow{MP}\) or similar in two different ways in terms of \(\mathbf{a}\) and \(\mathbf{b}\)
(M1 for writing eg \(\overrightarrow{OP}\) or \(\overrightarrow{AP}\) or \(\overrightarrow{AB}\) or \(\overrightarrow{NP}\) or \(\overrightarrow{MP}\) or similar in one way in terms of \(\mathbf{a}\) and \(\mathbf{b}\))

These may be written as eg \(\overrightarrow{PO}\) in place of \(\overrightarrow{OP}\)

M1: dep M2 for writing a pair of equations using their variables

\(\overrightarrow{OP}\) leads to \(x = \dfrac{6}{11}, y = \dfrac{1}{11}\)

\(\overrightarrow{AP}\) leads to \(x = \dfrac{6}{11}, y = \dfrac{1}{11}\)

\(\overrightarrow{AB}\) leads to \(x = \dfrac{6}{11}, y = \dfrac{10}{11}\)

\(\overrightarrow{NP}\) leads to \(x = \dfrac{5}{11}, y = \dfrac{1}{11}\)

\(\overrightarrow{MP}\) leads to \(x = \dfrac{1}{11}, y = \dfrac{6}{11}\)

A1: dep on M2, oe eg \(\dfrac{6}{11} : \dfrac{5}{11}\)