\(\overrightarrow{CP} = k \times \overrightarrow{AC}\) where \(k\) is a scalar.
Given that \(MNP\) is a straight line, find the value of \(k\). You must show all your working. (5)
Mark scheme
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
\((\boldsymbol{BC} =)\ 5\mathbf{a} - 2\mathbf{b} - 3\mathbf{a} - \mathbf{b}\) or \(2\mathbf{a} - 3\mathbf{b}\) or \((\boldsymbol{CD} =)\ 3\mathbf{a} + \mathbf{b} + 3\mathbf{a} - 9\mathbf{b}\) or \(6\mathbf{a} - 8\mathbf{b}\) or \((\boldsymbol{BD} =)\ 5\mathbf{a} - 2\mathbf{b} + 3\mathbf{a} - 9\mathbf{b}\) or \(8\mathbf{a} - 11\mathbf{b}\)
M1
oe eg \((\boldsymbol{CB} =)\ 3\mathbf{a} + \mathbf{b} - 5\mathbf{a} + 2\mathbf{b}\) or \(-2\mathbf{a} + 3\mathbf{b}\) or \((\boldsymbol{DC} =)\ -3\mathbf{a} + 9\mathbf{b} - 3\mathbf{a} - \mathbf{b}\) or \(-6\mathbf{a} + 8\mathbf{b}\) or \((\boldsymbol{DB} =)\ -3\mathbf{a} + 9\mathbf{b} - 5\mathbf{a} + 2\mathbf{b}\) or \(-8\mathbf{a} + 11\mathbf{b}\) Allow with brackets eg \((\boldsymbol{BC} =)\ 5\mathbf{a} - 2\mathbf{b} - (3\mathbf{a} + \mathbf{b})\)
Correct expressions for any two of \(\boldsymbol{BC}\), \(\boldsymbol{CD}\) and \(\boldsymbol{BD}\)
M1dep
oe eg1 correct expressions for \(\boldsymbol{BC}\) and \(\boldsymbol{DB}\) eg2 correct expressions for \(\boldsymbol{CB}\) and \(\boldsymbol{DC}\) Allow with brackets eg \((\boldsymbol{BC} =)\ 5\mathbf{a} - 2\mathbf{b} - (3\mathbf{a} + \mathbf{b})\) and \((\boldsymbol{DB} =)\ -(3\mathbf{a} - 9\mathbf{b}) - (5\mathbf{a} - 2\mathbf{b})\)
Correct simplified expressions for any two of \(\boldsymbol{BC}\), \(\boldsymbol{CD}\) and \(\boldsymbol{BD}\) and valid explanation and No
A1
oe eg correct expressions for \(\boldsymbol{BC}\) and \(\boldsymbol{DB}\) and valid explanation and No eg \(\boldsymbol{BC} = 2\mathbf{a} - 3\mathbf{b}\) and \(\boldsymbol{CD} = 6\mathbf{a} - 8\mathbf{b}\) and \(3(2\mathbf{a} - 3\mathbf{b}) = 6\mathbf{a} - 9\mathbf{b}\) and No or \(\boldsymbol{DC} = -6\mathbf{a} + 8\mathbf{b}\) and \(\boldsymbol{BD} = 8\mathbf{a} - 11\mathbf{b}\) and \(\boldsymbol{DC}\) is not a multiple of \(\boldsymbol{BD}\) and not straight
Additional guidance
Award marks for correct expressions, ignoring any incorrect ones unless contradictions of correct ones
\(\boldsymbol{BAD}\) means \(\boldsymbol{BD}\)
\(\boldsymbol{BD} = 5\mathbf{a} - 2\mathbf{b} + 3\mathbf{a} - 9\mathbf{b}\) or \(8\mathbf{a} - 11\mathbf{b}\) and \(\boldsymbol{BAD} =\) their \(\boldsymbol{BC} +\) their \(\boldsymbol{CD}\) and answer not \(8\mathbf{a} - 11\mathbf{b}\) Do not take \(\boldsymbol{BAD}\) to be a contradiction to \(\boldsymbol{BD}\)
Two correct simplified expressions used for a valid explanation and saying No with any incorrect non-contradictory expressions seen
M2A1
Condone absence of vector notation eg Condone CD to mean the vector from C to D
\(\overrightarrow{CD}\) means the vector from C to D and \(\overleftarrow{CD}\) means the vector from D to C
Do not allow any misreads
Missing brackets may be recovered
Allow for up to M2 expressions like \((\boldsymbol{BC} =)\ 5\mathbf{a} - 2\mathbf{b} + -3\mathbf{a} + -\mathbf{b}\)
Valid explanations: eg1 \(\;\boldsymbol{BC} = 2\mathbf{a} - 3\mathbf{b}\) and \(\boldsymbol{CD} = 6\mathbf{a} - 8\mathbf{b}\) and \(3(2\mathbf{a} - 3\mathbf{b}) = 6\mathbf{a} - 9\mathbf{b}\) is acceptable as there is a matching coefficient of \(\mathbf{a}\) eg2 \(\;\boldsymbol{CD} = 6\mathbf{a} - 8\mathbf{b}\) and \(\boldsymbol{BD} = 8\mathbf{a} - 11\mathbf{b}\) and \(2(6\mathbf{a} - 8\mathbf{b}) = 12\mathbf{a} - 16\mathbf{b}\) is not acceptable because there is no matching coefficient of \(\mathbf{a}\) or \(\mathbf{b}\) eg3 \(\;\boldsymbol{BC} = 2\mathbf{a} - 3\mathbf{b}\) and \(\boldsymbol{CD} = 6\mathbf{a} - 8\mathbf{b}\) and \(6\mathbf{a} - 8\mathbf{b} = 3(2\mathbf{a} - 2.6\mathbf{b})\) is acceptable because there is a matching coefficient of \(\mathbf{a}\) and no error in factorisation (just a truncation) eg4 \(\;\boldsymbol{BC} = 2\mathbf{a} - 3\mathbf{b}\) and \(\boldsymbol{CD} = 6\mathbf{a} - 8\mathbf{b}\) and \(3(2\mathbf{a} - 3\mathbf{b}) = 6\mathbf{a} - 10\mathbf{b}\) is not acceptable because there is an error in expansion
Allow not parallel or not same gradient for No
Allow \(\boldsymbol{DC}\) is not a factor of \(\boldsymbol{BD}\) as a valid explanation
Do not allow \(\boldsymbol{DC}\) is not a scalar of \(\boldsymbol{BD}\) as a valid explanation
Look for decision in working lines if answer line is blank
Note that \(\;\boldsymbol{BD} = \boldsymbol{BC} + \boldsymbol{CD}\;\) is a fact but is not a valid explanation
