C4 June 2010 Q4
4. A curve \(C\) has parametric equations \[x = \sin^2 t, \qquad y = 2\tan t, \qquad 0 \leqslant t < \frac{\pi}{2}\]
(a) Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). (4)
The tangent to \(C\) at the point where \(t = \dfrac{\pi}{3}\) cuts the \(x\)-axis at the point \(P\).
(b) Find the \(x\)-coordinate of \(P\). (6)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = 2\sin t\cos t,\quad \dfrac{\mathrm{d}y}{\mathrm{d}t} = 2\sec^2 t\) | B1 B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\sec^2 t}{\sin t\cos t}\quad \left(= \dfrac{1}{\sin t\cos^3 t}\right)\) or equivalent | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| At \(t = \dfrac{\pi}{3}\), \(x = \dfrac{3}{4}\), \(y = 2\sqrt{3}\) | B1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\sec^2\frac{\pi}{3}}{\sin\frac{\pi}{3}\cos\frac{\pi}{3}} = \dfrac{16}{\sqrt{3}}\) | M1 A1 |
| \(y - 2\sqrt{3} = \dfrac{16}{\sqrt{3}}\left(x - \dfrac{3}{4}\right)\) | M1 |
| \(y = 0 \Rightarrow x = \dfrac{3}{8}\) | M1 A1 |
| (6) | |
| (10 marks) |