| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward trapezium rule question requiring only calculator work to complete a table, then mechanical application of the trapezium rule formula. Part (c) tests basic understanding that more strips improve accuracy. All steps are routine with no problem-solving or conceptual challenge beyond standard P2 numerical integration. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.25 | 0.5 | 0.75 | 1 |
| \(y\) | 1 | 1.251 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\): \(0, 0.25, 0.5, 0.75, 1\); \(y\): \(1, 1.251,\) 1.494, 1.741, \(2\) | B1 B1 | B1 for 1.494; B1 for 1.741 (1.740 is B0) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2} \times 0.25,\ \{(1+2) + 2(1.251 + 1.494 + 1.741)\}\) | B1 M1 A1ft | B1: need \(\frac{1}{2}\) of 0.25; M1: first bracket contains first plus last values and second bracket includes additional values; A1ft: follows their answers to part (a) |
| \(= 1.4965\) | A1 | Accept 1.4965, 1.497, or 1.50 only after correct work |
| Answer | Marks |
|---|---|
| Gives any valid reason: decrease the width of the strips / use more trapezia / increase the number of strips. Do not accept "use more decimal places" | B1 |
## Question 3:
### Part (a):
| $x$: $0, 0.25, 0.5, 0.75, 1$; $y$: $1, 1.251,$ **1.494**, **1.741**, $2$ | B1 B1 | B1 for 1.494; B1 for 1.741 (1.740 is B0) |
|---|---|---|
### Part (b):
| $\frac{1}{2} \times 0.25,\ \{(1+2) + 2(1.251 + 1.494 + 1.741)\}$ | B1 M1 A1ft | B1: need $\frac{1}{2}$ of 0.25; M1: first bracket contains first plus last values **and** second bracket includes additional values; A1ft: follows their answers to part (a) |
|---|---|---|
| $= 1.4965$ | A1 | Accept 1.4965, 1.497, or 1.50 only after correct work |
### Part (c):
| Gives any valid reason: decrease the width of the strips / use more trapezia / increase the number of strips. Do not accept "use more decimal places" | B1 | |
|---|---|---|
---3.
$$y = \sqrt { \left( 3 ^ { x } + x \right) }$$
\begin{enumerate}[label=(\alph*)]
\item Complete the table below, giving the values of $y$ to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.25 & 0.5 & 0.75 & 1 \\
\hline
$y$ & 1 & 1.251 & & & 2 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule with all the values of $y$ from your table to find an approximation for the value of
$$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } \mathrm { d } x$$
You must show clearly how you obtained your answer.
\item Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of
$$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$
\begin{center}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-10_2673_1948_107_118}
\end{center}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2018 Q3 [7]}}