Higher June 2017 Paper 2 Q23
23
Not drawn accurately
Is BCD a straight line?
Show working to support your answer. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((\boldsymbol{BC} =)\ 5\mathbf{a} - 2\mathbf{b} - 3\mathbf{a} - \mathbf{b}\) 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 |