C3 January 2008 Q7
7. A curve \(C\) has equation\[y = 3\sin 2x + 4\cos 2x, \quad -\pi \leqslant x \leqslant \pi.\]The point \(A(0, 4)\) lies on \(C\).
(a) Find an equation of the normal to the curve \(C\) at \(A\). (5)
(b) Express \(y\) in the form \(R\sin(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \dfrac{\pi}{2}\).
Give the value of \(\alpha\) to 3 significant figures. (4)
Give the value of \(\alpha\) to 3 significant figures. (4)
(c) Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places. (4)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6\cos 2x - 8\sin 2x\) | M1 A1 |
| \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)_0 = 6\) | B1 |
| \(y - 4 = -\dfrac{1}{6}x\) or equivalent | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(R = \sqrt{(3^2 + 4^2)} = 5\) | M1 A1 |
| \(\tan\alpha = \dfrac{4}{3},\ \alpha \approx 0.927\) awrt 0.927 | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\sin(2x + \text{their }\alpha) = 0\) | M1 |
| \(x = -2.03, -0.46, 1.11, 2.68\) | A1 A1 A1 |
| (4) | |
| (13 marks) |
Notes
First A1 any correct solution; second A1 a second correct solution; third A1 all four correct and to the specified accuracy or better.
Ignore the \(y\)-coordinate.