Higher June 2017 Paper 2 Q20
20. The vectors OK, OL and OM are shown in the diagram.
Diagram not drawn to scale
(a) Find each of the following vectors in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
Give each answer in its simplest form.
Give each answer in its simplest form.
(i) LM [2]
(ii) KL [2]
(iii) What do your answers to (i) and (ii) tell you about the following? [2]
The lengths of the lines LM and KL.
The points K, L and M.
The lengths of the lines LM and KL.
The points K, L and M.
(b) The point \(Q\) is the midpoint of the line \(OL\).
Find MQ in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
Give your answer in the form \(x\mathbf{a} + y\mathbf{b}\). [3]
Find MQ in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
Give your answer in the form \(x\mathbf{a} + y\mathbf{b}\). [3]
| Scheme | Marks | Comments |
|---|---|---|
| (i) \(\mathbf{LM} = -\mathbf{OL} + \mathbf{OM}\) or \(\mathbf{LM} = -(4\mathbf{a}+3\mathbf{b}) + (18\mathbf{a} - 3\mathbf{b})\) | M1 | Accept intention of brackets |
| \(= 14\mathbf{a} - 6\mathbf{b}\) | A1 | CAO. ISW. Must be simplified form |
| (ii) \(\mathbf{KL} = -\mathbf{OK} + \mathbf{OL}\) or \(\mathbf{KL} = -(-3\mathbf{a}+6\mathbf{b}) + (4\mathbf{a} + 3\mathbf{b})\) | M1 | Accept intention of brackets |
| \(= 7\mathbf{a} - 3\mathbf{b}\) | A1 | CAO. Must be simplified form In (a) if both A0 as not in simplest form, also award SC1 for correct unsimplified vectors (ii) \(3\mathbf{a} - 6\mathbf{b} + 4\mathbf{a} + 3\mathbf{b}\) and (i) \(-4\mathbf{a} - 3\mathbf{b} + 18\mathbf{a} - 3\mathbf{b}\) |
| (iii) Reasonable statement, e.g. ‘LM is twice the length of KL’ | B1 | Allow for sight of LM = 2× KL |
| States or implies collinear, or exact description of collinear | B1 | Accept, e.g. ‘all points on the same straight line’ Allow, e.g. ‘parts of a straight line’ |
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(-\tfrac12(4\mathbf{a} + 3\mathbf{b})\) or \(\tfrac12(4\mathbf{a} + 3\mathbf{b})\) | B1 | May be simplified at a later stage |
| \(\mathbf{MQ} = -\mathbf{OM} + \tfrac12\mathbf{OL}\) or \(\mathbf{MQ} = -(18\mathbf{a} - 3\mathbf{b}) + \tfrac12(4\mathbf{a} + 3\mathbf{b})\) | M1 | Intention of brackets \((= -18\mathbf{a} + 3\mathbf{b} + 2\mathbf{a} + 1.5\mathbf{b})\) |
| \(= -16\mathbf{a} + 4.5\mathbf{b}\) | A1 | CAO. |
| (9) |