June 2023 Paper 2 Q5
5 You are given that \(\overrightarrow{OA} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}\).
Determine the exact length of \(AB\). [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\left[\begin{pmatrix} 5 \\ -3 \end{pmatrix} - \begin{pmatrix} 3 \\ -1 \end{pmatrix} =\right]\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) or \(\left[\begin{pmatrix} 3 \\ -1 \end{pmatrix} - \begin{pmatrix} 5 \\ -3 \end{pmatrix} =\right]\begin{pmatrix} -2 \\ 2 \end{pmatrix}\) | B1 | 2.1 |
| \(\sqrt{(\pm 2)^2 + (\pm 2)^2}\) oe | M1 | 1.1 |
| \(\sqrt{8}\) or \(2\sqrt{2}\) isw | A1 | 1.1 |
| [3] |
Notes
B1: may be in coordinate form or may see distances identified or on diagram;
may be implied by \(\sqrt{(\pm 2)^2 + (\pm 2)^2}\) oe
M1: or FT their evaluation of \(\begin{pmatrix} 5 \\ -3 \end{pmatrix} - \begin{pmatrix} 3 \\ -1 \end{pmatrix}\)
may be implied by correct answer
A1: if B0M0; allow SC1 for \(\sqrt{80}\) or \(4\sqrt{5}\) (from addition of vectors) if supported by Pythagoras;
if B0M0 allow SC1 for \(\sqrt{8}\) or \(2\sqrt{2}\) unsupported