4MA0 Higher June 2013 Paper 3R Q23
23
Diagram NOT accurately drawn
\(OABC\) is a parallelogram.
\(BCD\) is a straight line.
\(BD = 3BC\).
\(M\) is the midpoint of \(OC\).
\(\overrightarrow{OA} = \mathbf{x}\quad \overrightarrow{AB} = \mathbf{y}\)
(a) Find, in terms of \(\mathbf{x}\) and \(\mathbf{y}\),
(i) \(\overrightarrow{AM}\)
(ii) \(\overrightarrow{OD}\)
(2)(b) Use your answers to (a)(i) and (ii) to write down two different geometric facts about the lines \(AM\) and \(OD\). (2)
| Scheme | Marks |
|---|---|
| (i) \(\tfrac{1}{2}\mathbf{y} - \mathbf{x}\) | B1 |
| (ii) \(\mathbf{y} - 2\mathbf{x}\) | B1 |
| (2) |
Notes
B1: or \(-\mathbf{x} + \tfrac{1}{2}\mathbf{y}\) oe eg \(\mathbf{y} - \mathbf{x} - \tfrac{1}{2}\mathbf{y}\)
B1: or \(-2\mathbf{x} + \mathbf{y}\) oe eg \(\mathbf{x} + \mathbf{y} - 3\mathbf{x}\)
| Scheme | Marks |
|---|---|
| \(OD\) is parallel to \(AM\) | B1 |
| OD is twice length of AM oe | B1 |
| (2) |
Notes
both marks dependent on a(i) and a(ii) correct and simplified