Higher June 2025 Paper 3 Q23
23 \(OACB\) is a quadrilateral.
\(ACP\) is a straight line.
\(M\) is the midpoint of \(OA\).
\(N\) is the point on \(BC\) such that \(BN : NC = 5 : 3\)
\(\overrightarrow{OA} = \mathbf{a} \qquad \overrightarrow{OB} = 3\mathbf{b} \qquad \overrightarrow{AC} = 2\mathbf{b}\)
\(\overrightarrow{CP} = k \times \overrightarrow{AC}\) where \(k\) is a scalar.
Given that \(MNP\) is a straight line, find the value of \(k\).
You must show all your working. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{15}{4}\) | P1 | for process to find \(\overrightarrow{BC}\) or \(\overrightarrow{CB}\), eg \(\overrightarrow{BC} = -3\mathbf{b} + \mathbf{a} + 2\mathbf{b}\) oe (\(= \mathbf{a} - \mathbf{b}\)) or \(\overrightarrow{CB} = -2\mathbf{b} - \mathbf{a} + 3\mathbf{b}\) oe (\(= -\mathbf{a} + \mathbf{b}\)) |
| P1 | for process that uses the ratio 5 : 3, eg \(\overrightarrow{BN} = \dfrac{5}{8}(-3\mathbf{b} + \mathbf{a} + 2\mathbf{b})\ \left(= \dfrac{5}{8}\mathbf{a} - \dfrac{5}{8}\mathbf{b}\right)\) or \(\overrightarrow{NB} = -\dfrac{5}{8}\mathbf{a} + \dfrac{5}{8}\mathbf{b}\) oe or \(\overrightarrow{CN} = \dfrac{3}{8}(-2\mathbf{b} - \mathbf{a} + 3\mathbf{b})\ \left(= -\dfrac{3}{8}\mathbf{a} + \dfrac{3}{8}\mathbf{b}\right)\) or \(\overrightarrow{NC} = \dfrac{3}{8}\mathbf{a} - \dfrac{3}{8}\mathbf{b}\) oe | |
| P1 | for a process to find an expression, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), for \(\overrightarrow{MN}\) or \(\overrightarrow{MP}\) or \(\overrightarrow{NP}\) eg \(\overrightarrow{MN} = -\dfrac{1}{2}\mathbf{a} + 3\mathbf{b} + \dfrac{5}{8}(\mathbf{a} - \mathbf{b})\) oe \(\left(= \dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) or \(\overrightarrow{MN} = \dfrac{1}{2}\mathbf{a} + 2\mathbf{b} + \dfrac{3}{8}(-\mathbf{a} + \mathbf{b})\) oe \(\left(= \dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) or \(\overrightarrow{MP} = \dfrac{1}{2}\mathbf{a} + 2\mathbf{b} + k(2\mathbf{b})\) oe or \(\overrightarrow{NP} = \dfrac{3}{8}\mathbf{a} - \dfrac{3}{8}\mathbf{b} + k(2\mathbf{b})\) oe | |
| P1 | for a process to find a correct expression, in terms of \(\mathbf{a}\) and \(\mathbf{b}\) for the same vector eg \(\overrightarrow{MP}\) or \(\overrightarrow{NP}\) or parallel vectors eg \(\overrightarrow{MP}\) and \(\overrightarrow{MN}\) or \(\overrightarrow{NP}\) and \(\overrightarrow{MN}\) eg \(\overrightarrow{MP} = \dfrac{1}{2}\mathbf{a} + 2\mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{MP} = \lambda\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe or \(\overrightarrow{NP} = \dfrac{3}{8}\mathbf{a} - \dfrac{3}{8}\mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{NP} = \lambda\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe or \(\overrightarrow{MP} = \dfrac{1}{2}\mathbf{a} + 2\mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{MN} = \dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\) oe or \(\overrightarrow{NP} = \dfrac{3}{8}\mathbf{a} - \dfrac{3}{8}\mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{MN} = \dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\) oe OR for a process to find a correct expression in terms of \(\mathbf{a}\) and \(\mathbf{b}\), for \(\overrightarrow{AP}\) or \(\overrightarrow{CP}\) using \(\overrightarrow{MN}\) \(\overrightarrow{AP} = -\dfrac{1}{2}\mathbf{a} + \mu\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe or \(\overrightarrow{CP} = -\dfrac{3}{8}\mathbf{a} + \dfrac{3}{8}\mathbf{b} + \mu\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe OR for a process to find a correct expression in terms of \(\mathbf{a}\) and \(\mathbf{b}\) for the same vector \(\overrightarrow{BP}\) or \(\overrightarrow{OP}\) \(\overrightarrow{BP} = \mathbf{a} - \mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{BP} = \dfrac{5}{8}\mathbf{a} - \dfrac{5}{8}\mathbf{b} + \mu\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe \(\overrightarrow{OP} = \mathbf{a} + 2\mathbf{b} + k(2\mathbf{b})\) oe and \(\overrightarrow{OP} = \dfrac{1}{2}\mathbf{a} + \mu\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\) oe | |
| A1 | for \(\dfrac{15}{4}\) oe |
Additional guidance
Vectors must be unambiguously identified
Accept \(BC\) for \(\overrightarrow{BC}\) etc throughout
Allow a for \(\mathbf{a}\) and b for \(\mathbf{b}\) throughout
Vectors could be written on the diagram and may be in either direction.
Throughout, do not condone missing brackets unless recovered
Follow through candidates \(\overrightarrow{BC}\) or \(\overrightarrow{CB}\) provided full method to find subsequent vectors is clearly shown
A correct expression for \(\overrightarrow{BN}\) or \(\overrightarrow{NB}\) or \(\overrightarrow{CN}\) or \(\overrightarrow{NC}\) implies the previous P mark
Follow through candidates \(\overrightarrow{BN}\) or \(\overrightarrow{NB}\) or \(\overrightarrow{CN}\) or \(\overrightarrow{NC}\) provided full method to find subsequent vectors is clearly shown
May use \(\overrightarrow{NM}\) or \(\overrightarrow{PM}\) or \(\overrightarrow{PN}\)
Allow equivalent vectors throughout
eg \(\overrightarrow{MP} = \dfrac{1}{2}\mathbf{a} + 2\mathbf{b} + \mu(\mathbf{b})\)
This mark may be awarded without the previous mark being awarded
Vectors do not have to be simplified
May use \(\overrightarrow{NM}\) or \(\overrightarrow{PM}\) or \(\overrightarrow{PN}\)
Condone use of same variable for equivalent vector journeys
Condone lack of labelling if vector journeys are correctly equated
May use \(\overrightarrow{PA}\) or \(\overrightarrow{PC}\)
NB: \(\overrightarrow{CP} = -\dfrac{1}{2}\mathbf{a} - 2\mathbf{b} + \mu\left(\dfrac{1}{8}\mathbf{a} + \dfrac{19}{8}\mathbf{b}\right)\)
May use \(\overrightarrow{PB}\) or \(\overrightarrow{PO}\)
Award 0 marks for a correct answer with no supportive working