| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then evaluate integral |
| Difficulty | Standard +0.3 This is a standard C4 harmonic form question combined with a routine integration. Part (i) is textbook R-cos(θ-α) manipulation requiring basic trigonometry. Part (ii) uses the given derivative of tan to integrate a transformed expression - the connection is straightforward once the harmonic form is substituted. The calculation is methodical with no novel insight required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
\begin{enumerate}[label=(\roman*)]
\item 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 $\pi$. [4]
\item Write down the derivative of $\tan\theta$.
Hence show that $\int_0^{\frac{\pi}{3}} \frac{1}{(\cos\theta + \sqrt{3}\sin\theta)^2} \, d\theta = \frac{\sqrt{3}}{4}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 Q9 [8]}}