C4 January 2007 Q5
5. A set of curves is given by the equation \(\sin x + \cos y = 0.5\).
For \(-\pi \lt x \lt \pi\) and \(-\pi \lt y \lt \pi\),
| Scheme | Marks |
|---|---|
| \(\sin x + \cos y = 0.5\) (eqn \(\ast\)) | |
| \(\cos x - \sin y\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) (eqn #) | M1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \underline{\dfrac{\cos x}{\sin y}}\) | A1 cso |
| (2) |
Notes
M1 Differentiates implicitly to include \(\pm\sin y\dfrac{\mathrm{d}y}{\mathrm{d}x}\). (Ignore \(\left(\frac{\mathrm{d}y}{\mathrm{d}x} = \right)\).)
A1 cso \(\underline{\frac{\cos x}{\sin y}}\)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0 \Rightarrow \dfrac{\cos x}{\sin y} = 0 \Rightarrow \cos x = 0\) | M1ft |
| giving \(\underline{x = -\frac{\pi}{2}}\) or \(\underline{x = \frac{\pi}{2}}\) | A1 |
| When \(x = -\frac{\pi}{2}\), \(\sin\left(-\frac{\pi}{2}\right) + \cos y = 0.5\) When \(x = \frac{\pi}{2}\), \(\sin\left(\frac{\pi}{2}\right) + \cos y = 0.5\) | M1 |
| \(\Rightarrow \cos y = 1.5 \Rightarrow y\) has no solutions \(\Rightarrow \cos y = -0.5 \Rightarrow y = \frac{2\pi}{3}\) or \(-\frac{2\pi}{3}\) | A1 |
| In specified range \((x, y) = \underline{\left(\frac{\pi}{2}, \frac{2\pi}{3}\right)}\) and \(\underline{\left(\frac{\pi}{2}, -\frac{2\pi}{3}\right)}\) | A1 |
| (5) | |
| (7 marks) |
Notes
M1ft Candidate realises that they need to solve ‘their numerator’ \(= 0\) …or candidate sets \(\frac{\mathrm{d}y}{\mathrm{d}x} = 0\) in their (eqn #) and attempts to solve the resulting equation.
A1 both \(\underline{x = -\frac{\pi}{2}, \frac{\pi}{2}}\) or \(\underline{x = \pm 90^\circ}\) or awrt \(\underline{x = \pm 1.57}\) required here
M1 Substitutes either their \(x = \frac{\pi}{2}\) or \(x = -\frac{\pi}{2}\) into eqn \(\ast\)
A1 Only one of \(y = \frac{2\pi}{3}\) or \(-\frac{2\pi}{3}\) or \(\underline{120^\circ}\) or \(\underline{-120^\circ}\) or awrt \(\underline{-2.09}\) or awrt \(\underline{2.09}\)
A1 Only exact coordinates of \(\left(\frac{\pi}{2}, \frac{2\pi}{3}\right)\) and \(\left(\frac{\pi}{2}, -\frac{2\pi}{3}\right)\). Do not award this mark if candidate states other coordinates inside the required range.