| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Trigonometric substitution: direct evaluation |
| Difficulty | Standard +0.8 This is a structured integration by substitution question requiring multiple steps: proving the derivative relationship, setting up the integral with correct limits, substituting to get a trigonometric integral, and simplifying to an exact answer. While the substitution is given and part (a) scaffolds the derivative, students must handle the square root simplification (√(9sec²θ - 9) = 3tanθ), change limits correctly, and integrate sec θ or equivalent. This requires solid technique and careful algebra, making it moderately challenging but still within standard C3/C4 scope. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08h Integration by substitution |
11.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-36_601_1140_242_402}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item By writing $\sec \theta$ as $\frac { 1 } { \cos \theta }$, show that when $x = 3 \sec \theta$,
$$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = 3 \sec \theta \tan \theta$$
Figure 3 shows a sketch of part of the curve $C$ with equation
$$y = \frac { \sqrt { x ^ { 2 } - 9 } } { x } \quad x \geqslant 3$$
The finite region $R$, shown shaded in Figure 3, is bounded by the curve $C$, the $x$-axis and the line with equation $x = 6$
\item Use the substitution $x = 3 \sec \theta$ to find the exact value of the area of $R$. [Solutions based entirely on graphical or numerical methods are not acceptable.]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2018 Q11 [9]}}