| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Integrate using double angle |
| Difficulty | Standard +0.3 This is a standard C4 harmonic form question with routine integration. Part (i) is textbook R-cos(θ-α) manipulation yielding R=2, α=π/3. Part (ii) combines the standard derivative sec²θ with substitution u=tan θ, but the connection is straightforward once the harmonic form is recognized. The integration requires seeing that 1/(2cos(θ-π/3))² becomes ¼sec²(θ-π/3), making it slightly above average difficulty due to the multi-step nature, but all techniques are standard C4 material with no novel insight required. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.08c 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 $\int_0^{\frac{\pi}{6}} \frac{1}{(\cos \theta + \sqrt{3} \sin \theta)^2} d\theta = \frac{\sqrt{3}}{4}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 Q2 [8]}}