4400 Higher June 2004 Paper 4 Q22
22
\(PQRS\) is a parallelogram.
\(X\) is the midpoint of \(QR\) and \(Y\) is the midpoint of \(SR\).
\(\overrightarrow{PQ} = \mathbf{a}\) and \(\overrightarrow{PS} = \mathbf{b}\).
(a) Write down, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), expressions for
(i) \(\overrightarrow{PX}\)
(ii) \(\overrightarrow{PY}\)
(iii) \(\overrightarrow{QS}\) (3)
(b) Use a vector method to show that \(XY\) is parallel to \(QS\) and that \(XY = \frac{1}{2}QS\). (2)
| Scheme | Marks |
|---|---|
| (i) Answer: \(\mathbf{a} + \frac{1}{2}\mathbf{b}\) oe | B1 |
| (ii) Answer: \(\frac{1}{2}\mathbf{a} + \mathbf{b}\) oe | B1 |
| (iii) Answer: \(\mathbf{b} - \mathbf{a}\) oe | B1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\frac{1}{2}\mathbf{a} + \mathbf{b} - \mathbf{a} - \frac{1}{2}\mathbf{b}\) or \(\frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a}\) | B1 |
| \(\overrightarrow{XY} = \frac{1}{2}\overrightarrow{QS}\) | B1 |
| (2) | |
| (5 marks) |
Notes
B1: Or equivalent. Must use vector not’n dep on 1st B1