4400 Higher June 2005 Paper 4 Q16
16 \(PQR\) is a triangle.
\(M\) and \(N\) are the midpoints of \(PQ\) and \(PR\) respectively.
Diagram NOT accurately drawn
\(\overrightarrow{PM} = \mathbf{a}\) \(\qquad \overrightarrow{PN} = \mathbf{b}\).
(a) Find, in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\),
(i) \(\overrightarrow{MN}\)
(ii) \(\overrightarrow{PQ}\)
(iii) \(\overrightarrow{QR}\) (3)
(b) Use your answers to (a)(i) and (iii) to write down two geometrical facts about the lines \(MN\) and \(QR\). (2)
| Scheme | Marks |
|---|---|
| (i) \(-\mathbf{a} + \mathbf{b}\) oe | B1 |
| (ii) \(2\mathbf{a}\) oe | B1 |
| (iii) \(-2\mathbf{a} + 2\mathbf{b}\) oe | B1 |
| (3) |
Notes
Simplification not required
Allow plain \(a\), \(b\)
| Scheme | Marks |
|---|---|
| Parallel | B1 |
| \(QR = 2MN\) or lines in ratio 1:2 or 2:1 | B1 |
| (2) | |
| (5 marks) |
Notes
(b) marks dep (a)(i)&(iii) correct
Without vector symbols unless “length” stated.