| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard C3/C4 techniques: evaluating a function at given points, applying the trapezium rule with provided values, integration by parts for ∫x cos x dx, and using definite integration. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.08i Integration by parts1.09f Trapezium rule: numerical integration |
| \(x\) | \(\frac { 3 \pi } { 2 }\) | \(\frac { 7 \pi } { 4 }\) | \(2 \pi\) | \(\frac { 9 \pi } { 4 }\) | \(\frac { 5 \pi } { 2 }\) |
| \(y\) | 0 | \(2 \pi\) | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{7\pi}{4\sqrt{2}}\) or equivalent e.g. \(\frac{7\pi\sqrt{2}}{8}\) AND \(\frac{9\pi}{4\sqrt{2}}\) or equivalent e.g. \(\frac{9\pi\sqrt{2}}{8}\) | B1 (1) | Both correct, must be exact not decimal |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}\times\frac{\pi}{4}\times\{......\}\) | B1 oe | \(h=\frac{\pi}{4}\) stated or used |
| \(\frac{1}{2}\times h\times\left\{0+0+2\left(\text{"}\frac{7\pi}{4\sqrt{2}}\text{"}+2\pi+\text{"}\frac{9\pi}{4\sqrt{2}}\text{"}\right)\right\}\) | M1 | Correct structure of bracket in trapezium rule |
| \(= 11.91\) (only) | A1 (3) | 11.91 only; decimal equivalents acceptable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int x\cos x\, dx = [x\sin x] - \int \sin x\, dx\) | M1 | Correct attempt at integration by parts giving form \([\pm x\sin x] - \int \pm\sin x\, dx\) |
| \(= x\sin x + \cos x\ (+c)\) | dM1 A1 (3) | dM1: For \(\pm x\sin x \pm \cos x\). A1: cso. All 3 marks for correct answer with no working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left[x\sin x+\cos x\right]_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} = \frac{5\pi}{2}+\frac{3\pi}{2} = 4\pi\) | M1 A1 (2) | M1: Correct limits \(\frac{5\pi}{2}\) and \(\frac{3\pi}{2}\) substituted into part (c) answer. A1: \(4\pi\) exact, must derive from \(x\sin x + \cos x\) |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{7\pi}{4\sqrt{2}}$ or equivalent e.g. $\frac{7\pi\sqrt{2}}{8}$ **AND** $\frac{9\pi}{4\sqrt{2}}$ or equivalent e.g. $\frac{9\pi\sqrt{2}}{8}$ | B1 (1) | Both correct, must be exact not decimal |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}\times\frac{\pi}{4}\times\{......\}$ | B1 oe | $h=\frac{\pi}{4}$ stated or used |
| $\frac{1}{2}\times h\times\left\{0+0+2\left(\text{"}\frac{7\pi}{4\sqrt{2}}\text{"}+2\pi+\text{"}\frac{9\pi}{4\sqrt{2}}\text{"}\right)\right\}$ | M1 | Correct structure of bracket in trapezium rule |
| $= 11.91$ (only) | A1 (3) | 11.91 only; decimal equivalents acceptable |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int x\cos x\, dx = [x\sin x] - \int \sin x\, dx$ | M1 | Correct attempt at integration by parts giving form $[\pm x\sin x] - \int \pm\sin x\, dx$ |
| $= x\sin x + \cos x\ (+c)$ | dM1 A1 (3) | dM1: For $\pm x\sin x \pm \cos x$. A1: cso. All 3 marks for correct answer with no working |
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[x\sin x+\cos x\right]_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} = \frac{5\pi}{2}+\frac{3\pi}{2} = 4\pi$ | M1 A1 (2) | M1: Correct limits $\frac{5\pi}{2}$ and $\frac{3\pi}{2}$ substituted into part (c) answer. A1: $4\pi$ exact, must derive from $x\sin x + \cos x$ |5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-08_579_1038_258_452}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve $C$ with equation
$$y = x \cos x , \quad x \in \mathbb { R }$$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve $C$ and the $x$-axis for $\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }$
\begin{enumerate}[label=(\alph*)]
\item Complete the table below with the exact value of $y$ corresponding to $x = \frac { 7 \pi } { 4 }$ and with the exact value of $y$ corresponding to $x = \frac { 9 \pi } { 4 }$
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & $\frac { 3 \pi } { 2 }$ & $\frac { 7 \pi } { 4 }$ & $2 \pi$ & $\frac { 9 \pi } { 4 }$ & $\frac { 5 \pi } { 2 }$ \\
\hline
$y$ & 0 & & $2 \pi$ & & 0 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule, with all five $y$ values in the completed table, to find an approximate value for the area of $R$, giving your answer to 4 significant figures.
\item Find
$$\int x \cos x d x$$
\item Using your answer from part (c), find the exact area of the region $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2017 Q5 [9]}}