October 2021 Paper 2 Q6
6 You are given that \(\mathbf{v} = 2\mathbf{a} + 3\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are the position vectors
\(\mathbf{a} = \begin{pmatrix}5\\3\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}-1\\6\end{pmatrix}\).
(a) Determine the magnitude of \(\mathbf{v}\). [3]
(b) Determine the angle between \(\mathbf{v}\) and the vector \(\begin{pmatrix}1\\0\end{pmatrix}\). [2]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix}2 \times 5 + 3 \times (-1)\\2 \times 3 + 3 \times 6\end{pmatrix}\) soi | M1 | 3.1a |
| \(\sqrt{7^2 + 24^2}\) | M1 | 1.1 |
| 25 | A1 | 1.1 |
| [3] |
Notes
M1: may be implied if one component fully correct
M1: dependent on award of first M1
| Scheme | Marks | AO |
|---|---|---|
| \(\tan^{-1}\left(\frac{24}{7}\right)\) oe | M1 | 1.1a |
| \(74^\circ\) or \(73.7^\circ\) or awrt \(73.74^\circ\) or 1.3 or 1.29 or awrt 1.287 | A1 | 1.1 |
| [2] |
Notes
M1: or \(\cos^{-1}\left(\frac{7}{25}\right)\) or \(\sin^{-1}\left(\frac{24}{25}\right)\) FT their 7, 24 or 25