October 2021 Paper 3 Q5
5 A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds \(P\) has velocity \(v\,\mathrm{m\,s^{-1}}\), where
\(v = 12\cos t + 5\sin t.\)
(a) Express \(v\) in the form \(R\cos(t - \alpha)\), where \(R \gt 0\) and \(0 \lt \alpha \lt \frac{1}{2}\pi\). Give the value of \(\alpha\) correct to 4 significant figures. [3]
(b) Hence find the two smallest positive values of \(t\) for which \(P\) is moving, in either direction, with a speed of \(3\,\mathrm{m\,s^{-1}}\). [3]
| Scheme | Marks | AO |
|---|---|---|
| \(R = 13\) | B1 | 1.1 |
| \(\left.\begin{aligned}R\cos\alpha &= 12\\ R\sin\alpha &= 5\end{aligned}\right\}\tan\alpha = \dfrac{5}{12}\) | M1 | 1.1 |
| \(\alpha = 0.3948\) | A1 | 1.1 |
| [3] |
Notes
B1: B0 for \(\sqrt{169}\) or for \(\pm 13\)
M1: M1 for \(\tan\alpha = k\) where \(k = \pm\frac{5}{12}, \pm\frac{12}{5}\)
or for \(\cos\alpha = \pm\frac{12}{R}\), \(\sin\alpha = \pm\frac{5}{R}\)
A1: A0 if in degrees (must be stated to exactly 4 significant figures at some point)
For reference: 0.3947911197…
| Scheme | Marks | AO |
|---|---|---|
| \(13\cos(t - 0.3948) = \pm 3 \Rightarrow t = 0.3948 + \arccos\left(\dfrac{\pm 3}{13}\right)\) | M1 | 1.1 |
| 1.73 | A1 | 1.1 |
| 2.20 | A1 | 1.1 |
| [3] |
Notes
M1: Sets \(R\cos(t - \alpha)\) (with their \(R\) and \(\alpha\)) equal to either 3 or \(-3\), and attempt to solve (with correct order of operations)
M1 only if in degrees
A1: awrt 1.73
1.7327192…
A1: awrt 2.20 (condone 2.2) – ignore other values that are greater than 2.20
2.1984556…