C3 January 2009 Q4
4. Find the equation of the tangent to the curve \(x = \cos(2y + \pi)\) at \(\left(0, \dfrac{\pi}{4}\right)\).
Give your answer in the form \(y = ax + b\), where \(a\) and \(b\) are constants to be found. (6)
| Scheme | Marks |
|---|---|
| \(x = \cos(2y + \pi)\) \(\dfrac{\mathrm{d}x}{\mathrm{d}y} = -2\sin(2y + \pi)\) | M1 A1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{1}{2\sin(2y + \pi)}\) Follow through their \(\dfrac{\mathrm{d}x}{\mathrm{d}y}\) before or after substitution | A1ft |
| At \(y = \dfrac{\pi}{4}\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{1}{2\sin\frac{3\pi}{2}} = \dfrac{1}{2}\) | B1 |
| \(y - \dfrac{\pi}{4} = \dfrac{1}{2}x\) | M1 |
| \(y = \dfrac{1}{2}x + \dfrac{\pi}{4}\) | A1 |
| (6) | |
| (6 marks) |