| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find intersection or crossing points |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with routine parts: (a) finding a normal requires basic differentiation of trig functions, (b) converting to R sin(2x + α) is a textbook procedure using R² = a² + b² and tan α = b/a, (c) solving R sin(2x + α) = 0 is straightforward once part (b) is complete. All parts follow well-practiced algorithms with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 6\cos 2x - 8\sin 2x\) | M1 A1 | |
| \(\left(\frac{dy}{dx}\right)_0 = 6\) | B1 | |
| \(y - 4 = -\frac{1}{6}x\) | M1 A1 | or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R = \sqrt{3^2 + 4^2} = 5\) | M1 A1 | |
| \(\tan\alpha = \frac{4}{3}\), \(\alpha \approx 0.927\) | M1 A1 | awrt 0.927 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin(2x + \text{their } \alpha) = 0\) | M1 | |
| \(x = -2.03, -0.46, 1.11, 2.68\) | A1 A1 A1 | First A1 any correct solution; second A1 a second correct solution; third A1 all four correct to specified accuracy or better. Ignore \(y\)-coordinate. |
# Question 7:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 6\cos 2x - 8\sin 2x$ | M1 A1 | |
| $\left(\frac{dy}{dx}\right)_0 = 6$ | B1 | |
| $y - 4 = -\frac{1}{6}x$ | M1 A1 | or equivalent |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R = \sqrt{3^2 + 4^2} = 5$ | M1 A1 | |
| $\tan\alpha = \frac{4}{3}$, $\alpha \approx 0.927$ | M1 A1 | awrt 0.927 |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin(2x + \text{their } \alpha) = 0$ | M1 | |
| $x = -2.03, -0.46, 1.11, 2.68$ | A1 A1 A1 | First A1 any correct solution; second A1 a second correct solution; third A1 all four correct to specified accuracy or better. Ignore $y$-coordinate. |\begin{enumerate}
\item A curve $C$ has equation
\end{enumerate}
$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$
The point $A ( 0,4 )$ lies on $C$.\\
(a) Find an equation of the normal to the curve $C$ at $A$.\\
(b) Express $y$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.
Give the value of $\alpha$ to 3 significant figures.\\
(c) Find the coordinates of the points of intersection of the curve $C$ with the $x$-axis. Give your answers to 2 decimal places.
\hfill \mbox{\textit{Edexcel C3 2008 Q7 [13]}}