| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.3 This is a straightforward trapezium rule application requiring table completion by substitution, then two standard calculations with different strip numbers. The function evaluation is routine (calculator work), and the trapezium rule formula is a standard C4 technique with no conceptual challenges or problem-solving required. Slightly easier than average due to its purely procedural nature. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac { \pi } { 12 }\) | \(\frac { \pi } { 6 }\) | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 3 }\) |
| \(y\) | 1.3229 | 1.2973 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(y\left(\frac{\pi}{6}\right) \approx 1.2247\), \(y\left(\frac{\pi}{4}\right) = 1.1180\) | B1 B1 |
| \(I \approx \frac{\pi}{12}(1.3229 + 2 \times 1.2247 + 1) \approx 1.249\) | B1 M1 | B1 for \(\frac{\pi}{12}\), cao |
| (b)(i) | \(I \approx \frac{\pi}{12}(1.3229 + 2 \times 1.2247 + 1) \approx 1.249\) | B1 M1 |
| (b)(ii) | \(I \approx \frac{\pi}{24}(1.3229 + 2 \times (1.2973 + 1.2247 + 1.1180) + 1) \approx 1.257\) | B1 M1 |
(a) | $y\left(\frac{\pi}{6}\right) \approx 1.2247$, $y\left(\frac{\pi}{4}\right) = 1.1180$ | B1 B1 | accept awrt 4 d.p. |
| $I \approx \frac{\pi}{12}(1.3229 + 2 \times 1.2247 + 1) \approx 1.249$ | B1 M1 | B1 for $\frac{\pi}{12}$, cao |
(b)(i) | $I \approx \frac{\pi}{12}(1.3229 + 2 \times 1.2247 + 1) \approx 1.249$ | B1 M1 | B1 for $\frac{\pi}{12}$, cao |
(b)(ii) | $I \approx \frac{\pi}{24}(1.3229 + 2 \times (1.2973 + 1.2247 + 1.1180) + 1) \approx 1.257$ | B1 M1 | B1 for $\frac{\pi}{24}$, cao |
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{280ae2a5-7344-4ba3-907f-235fba3fd5b3-02_684_767_274_589}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows part of the curve with equation $y = \sqrt { } \left( 0.75 + \cos ^ { 2 } x \right)$. The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis, the $x$-axis and the line with equation $x = \frac { \pi } { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Complete the table with values of $y$ corresponding to $x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 4 }$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & $\frac { \pi } { 12 }$ & $\frac { \pi } { 6 }$ & $\frac { \pi } { 4 }$ & $\frac { \pi } { 3 }$ \\
\hline
$y$ & 1.3229 & 1.2973 & & & 1 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule
\begin{enumerate}[label=(\roman*)]
\item with the values of $y$ at $x = 0 , x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 3 }$ to find an estimate of the area of $R$. Give your answer to 3 decimal places.
\item with the values of $y$ at $x = 0 , x = \frac { \pi } { 12 } , x = \frac { \pi } { 6 } , x = \frac { \pi } { 4 }$ and $x = \frac { \pi } { 3 }$ to find a further estimate of the area of $R$. Give your answer to 3 decimal places.\\
(6)
\section*{LU}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2010 Q1 [8]}}