| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Apply trapezium rule to given table |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule formula with given y-values, followed by a simple area calculation. Part (a) requires only substitution into the standard formula, and part (b) involves integrating a simple polynomial and basic percentage calculation. No problem-solving insight needed—purely procedural execution of standard P2 techniques. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 3 | 2.6833 | 2.4 | 2.1466 | 1.92 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 0.5\) | B1 | Seen or implied |
| \(\frac{1}{2} \times 0.5 \times [3 + 1.92 + 2(2.6833 + 2.4 + 2.1466)]\) | M1 | Full attempt at trapezium rule; condone copying slips; missing brackets score M0 unless recovered |
| \(4.845\) | A1 | awrt \(4.845\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_0^2 2 - \frac{1}{4}x^2\, dx = \left[2x - \frac{x^3}{12}\right]_0^2 = \frac{10}{3}\) | M1A1 | M1 for integrating \(2 - \frac{1}{4}x^2\); A1 for \(\frac{10}{3}\) seen or implied |
| \(\% = \frac{\text{"4.845"} - \frac{10}{3}}{6}\) | dM1 | Difference between part (a) and area under \(C_2\) (must be positive), divided by 6; must follow attempt to integrate \(C_2\) |
| \(= 25.2\%\) | A1 | awrt \(25.2(\%)\); do not allow \(-25.2\%\) or \(0.252\); allow \(\frac{907}{36}(\%)\) |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.5$ | B1 | Seen or implied |
| $\frac{1}{2} \times 0.5 \times [3 + 1.92 + 2(2.6833 + 2.4 + 2.1466)]$ | M1 | Full attempt at trapezium rule; condone copying slips; missing brackets score M0 unless recovered |
| $4.845$ | A1 | awrt $4.845$ |
## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_0^2 2 - \frac{1}{4}x^2\, dx = \left[2x - \frac{x^3}{12}\right]_0^2 = \frac{10}{3}$ | M1A1 | M1 for integrating $2 - \frac{1}{4}x^2$; A1 for $\frac{10}{3}$ seen or implied |
| $\% = \frac{\text{"4.845"} - \frac{10}{3}}{6}$ | dM1 | Difference between part (a) and area under $C_2$ (must be positive), divided by 6; must follow attempt to integrate $C_2$ |
| $= 25.2\%$ | A1 | awrt $25.2(\%)$; do not allow $-25.2\%$ or $0.252$; allow $\frac{907}{36}(\%)$ |
---\begin{enumerate}
\item The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$.
\end{enumerate}
A table of values of $x$ and $y$ for $y = \mathrm { f } ( x )$ is shown below, with the $y$ values rounded to 4 decimal places where appropriate.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & 3 & 2.6833 & 2.4 & 2.1466 & 1.92 \\
\hline
\end{tabular}
\end{center}
(a) Use the trapezium rule with all the values of $y$ in the table to find an approximation for
$$\int _ { 0 } ^ { 2 } f ( x ) d x$$
giving your answer to 3 decimal places.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_629_592_1105_402}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_540_456_1194_1192}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
The region $R$, shown shaded in Figure 1, is bounded by
\begin{itemize}
\item the curve $C _ { 1 }$
\item the curve $C _ { 2 }$ with equation $y = 2 - \frac { 1 } { 4 } x ^ { 2 }$
\item the line with equation $x = 2$
\item the $y$-axis
\end{itemize}
The region $R$ forms part of the design for a logo shown in Figure 2.\\
The design consists of the shaded region $R$ inside a rectangle of width 2 and height 3 Using calculus and the answer to part (a),\\
(b) calculate an estimate for the percentage of the logo which is shaded.
\hfill \mbox{\textit{Edexcel P2 2022 Q6 [7]}}