C4 June 2009 Q5
5.
Figure 2 shows a sketch of the curve with parametric equations \[x = 2\cos 2t, \qquad y = 6\sin t, \qquad 0 \leqslant t \leqslant \frac{\pi}{2}\]
(a) Find the gradient of the curve at the point where \(t = \dfrac{\pi}{3}\). (4)
(b) Find a cartesian equation of the curve in the form \[y = \mathrm{f}(x), \quad -k \leqslant x \leqslant k,\] stating the value of the constant \(k\). (4)
(c) Write down the range of \(\mathrm{f}(x)\). (2)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = -4\sin 2t,\quad \dfrac{\mathrm{d}y}{\mathrm{d}t} = 6\cos t\) | B1, B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{6\cos t}{4\sin 2t}\quad \left(= -\dfrac{3}{4\sin t}\right)\) | M1 |
| At \(t = \dfrac{\pi}{3}\), \(m = -\dfrac{3}{4 \times \frac{\sqrt{3}}{2}} = -\dfrac{\sqrt{3}}{2}\) accept equivalents, awrt \(-0.87\) | A1 |
| (4) |
Alternatives to (a) where the parameter is eliminated
①
| \(y = (18 - 9x)^{\frac{1}{2}}\) | |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{2}(18 - 9x)^{-\frac{1}{2}} \times (-9)\) | B1 |
| At \(t = \dfrac{\pi}{3}\), \(x = \cos\dfrac{2\pi}{3} = -1\) | B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{2} \times \dfrac{1}{\sqrt{(27)}} \times -9 = -\dfrac{\sqrt{3}}{2}\) | M1 A1 |
| (4) |
②
| \(y^2 = 18 - 9x\) | |
| \(2y\dfrac{\mathrm{d}y}{\mathrm{d}x} = -9\) | B1 |
| At \(t = \dfrac{\pi}{3}\), \(y = 6\sin\dfrac{\pi}{3} = 3\sqrt{3}\) | B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{9}{2 \times 3\sqrt{3}} = -\dfrac{\sqrt{3}}{2}\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Use of \(\cos 2t = 1 - 2\sin^2 t\) | M1 |
| \(\cos 2t = \dfrac{x}{2},\ \sin t = \dfrac{y}{6}\) | |
| \(\dfrac{x}{2} = 1 - 2\left(\dfrac{y}{6}\right)^2\) | M1 |
| Leading to \(y = \sqrt{(18 - 9x)}\quad \left(= 3\sqrt{(2 - x)}\right)\) cao | A1 |
| \(-2 \leqslant x \leqslant 2\) \(k = 2\) | B1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(0 \leqslant \mathrm{f}(x) \leqslant 6\) either \(0 \leqslant \mathrm{f}(x)\) or \(\mathrm{f}(x) \leqslant 6\) | B1 |
| Fully correct. Accept \(0 \leqslant y \leqslant 6\), \([0, 6]\) | B1 |
| (2) | |
| (10 marks) |