| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Integration using harmonic form |
| Difficulty | Standard +0.8 This question combines harmonic form (standard C4 technique) with a non-routine integration requiring recognition that the derivative of tan relates to sec². The key insight is substituting u = θ - α and recognizing the integral becomes ∫sec²u du = tan u. While harmonic form is routine, the integration step requires genuine problem-solving and careful manipulation of limits, making it moderately challenging but within reach of strong C4 students. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y=10\) | B1 | |
| Substitute for \(y\) in (4): \(V = \frac{1}{1000}\int_0^{100}375\,dx\) | M1 | |
| \(V = \frac{1}{1000}\times37500 = 37.5\) * | E1 [18] |
## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=10$ | B1 | |
| Substitute for $y$ in (4): $V = \frac{1}{1000}\int_0^{100}375\,dx$ | M1 | |
| $V = \frac{1}{1000}\times37500 = 37.5$ * | E1 [18] | |6 (i) Express $\cos \theta + \sqrt { 3 } \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $\alpha$ is acute, expressing $\alpha$ in terms of $\pi$.\\
(ii) Write down the derivative of $\tan \theta$.
Hence show that $\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }$.
\hfill \mbox{\textit{OCR MEI C4 2009 Q6 [8]}}