Higher June 2023 Paper 5 Q20
20 OABC is a parallelogram.
Not to scale
\(\overrightarrow{\mathrm{OA}}\) = \(\mathbf{a}\) and \(\overrightarrow{\mathrm{OC}}\) = \(\mathbf{c}\).
The point N lies on line AB such that AN : NB = 3 : 5.
(a) Find the following vectors in terms of a and c.
Give your answers in their simplest form.
Give your answers in their simplest form.
(i) \(\overrightarrow{\mathrm{OB}}\) [1]
(ii) \(\overrightarrow{\mathrm{ON}}\) [2]
(b) Line CN is extended to reach point P, such that \(\overrightarrow{\mathrm{CP}} = \frac{8}{5}\overrightarrow{\mathrm{CN}}\).
Show, using vectors, that OAP is a straight line. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (a)(i) | |||
| \(\mathbf{a} + \mathbf{c}\) | 1 | Not A + C, but if use of capitals in other parts of question, penalise the first occurrence only Accept \(\mathbf{c} + \mathbf{a}\) | |
| (a)(ii) | |||
| \(\mathbf{a} + \frac{3}{8}\mathbf{c}\) final answer | 2 | M1 for correct route or for \(\overrightarrow{\mathrm{AN}} = \frac{3}{8}\mathbf{c}\) or \(\overrightarrow{\mathrm{BN}} = -\frac{5}{8}\mathbf{c}\) | M1 for e.g. OA + AN, OC + \(\mathbf{a}\) – NB Condone poor vector notation for method Could be written on diagram |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\overrightarrow{\mathrm{CN}} = \mathbf{a} - \frac{5}{8}\mathbf{c}\) oe | M1 | or FT (their (a)(ii)) – \(\mathbf{c}\) must be vector route in terms of \(\mathbf{a}\) and/or \(\mathbf{c}\) | Condone omission of vector arrows etc throughout question Allow \(\overrightarrow{\mathrm{CN}}\), \(\overrightarrow{\mathrm{CP}}\) and \(\overrightarrow{\mathrm{NP}}\) both unsimplified and isw attempts to simplify NP may be embedded in working leading to OP when e.g. they do ON + NP |
| \(\overrightarrow{\mathrm{CP}} = \frac{8}{5}\mathbf{a} - \mathbf{c}\) oe or \(\overrightarrow{\mathrm{NP}} = \frac{3}{5}\mathbf{a} - \frac{3}{8}\mathbf{c}\) oe | M1 | or FT \(\overrightarrow{\mathrm{CP}} = \frac{8}{5}\)(their \(\overrightarrow{\mathrm{CN}}\)) must be vector route in terms of \(\mathbf{a}\) and/or \(\mathbf{c}\) or FT \(\overrightarrow{\mathrm{NP}} = \frac{3}{5}\)(their \(\overrightarrow{\mathrm{CN}}\)) must be vector route in terms of \(\mathbf{a}\) and/or \(\mathbf{c}\) Alt method (using similar triangles NCB and NPA) M2 for \(\overrightarrow{\mathrm{AP}} = \frac{3}{5}\mathbf{a}\) oe | |
| \(\overrightarrow{\mathrm{OP}} = \frac{8}{5}\mathbf{a}\) oe or \(\overrightarrow{\mathrm{AP}} = \frac{3}{5}\mathbf{a}\) | B1 | Award B1 for e.g. OP = \(\mathbf{c} + \frac{8}{5}\mathbf{a} - \mathbf{c}\) Accept OAP = \(\frac{8}{5}\mathbf{a}\) oe | |
| Correct conclusion \(\overrightarrow{\mathrm{OP}} = \frac{8}{5}\overrightarrow{\mathrm{OA}}\) oe or \(\overrightarrow{\mathrm{OP}}\) is a multiple of \(\overrightarrow{\mathrm{OA}}\) oe | A1 | Dep on M2B1 | Accept correct equivalent vector conclusions involving AP and OP or OA |