| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Difficulty | Standard +0.3 Part (a) is a routine proof of a standard double angle formula. Part (b) is a standard trigonometric substitution integral with given limits, requiring careful execution but following a predictable method. Part (c) involves parametric differentiation with sec and ln functions, which is slightly more involved than basic parametric questions but still follows standard C4 techniques. Overall, this is a straightforward multi-part question testing standard C4 content with no novel insights required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07s Parametric and implicit differentiation1.08h Integration by substitution |
\begin{enumerate}[label=(\alph*)]
\item Use the identity for $\cos(A + B)$ to prove that $\cos 2A = 2\cos^2 A - 1$.
[2]
\item Use the substitution $x = 2\sqrt{2} \sin \theta$ to prove that
$$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$
[7]
\end{enumerate}
A curve is given by the parametric equations
$$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find an equation of the tangent to the curve at the point where $\theta = \frac{\pi}{3}$.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [14]}}