| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | October |
| Marks | 4 |
| 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 two-part question requiring (a) calculator substitution into a given formula and (b) direct application of the trapezium rule formula with provided strip width. Both parts are routine procedural tasks with no problem-solving or conceptual challenge, making it easier than the average A-level question. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | - 0.25 | 0 | 0.25 | 0.5 | 0.75 |
| \(y\) | 0.462 | 0.653 | 0.698 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x\): \(-0.25\), \(0\), \(0.25\), \(0.5\), \(0.75\); \(y\): \(0.462\), 0.577, \(0.653\), 0.686, \(0.698\) | B1 | Allow awrt these values. Also allow exact value for 0.577 e.g. \(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 0.25\) | B1 | Correct strip width. May be implied by \(\frac{1}{8}\) or \(\frac{1}{2}\times0.25\) |
| \(A \approx \frac{1}{2}\times\text{"0.25"}\{0.462+0.698+2(\text{"0.577"}+0.653+\text{"0.686"})\}\) | M1 | Correct application of trapezium rule with their \(h\). Must use all \(y\)-values. Missing brackets score M0 unless implied by subsequent work |
| \(\approx\) awrt \(0.624\) or \(\frac{78}{125}\) e.g. \(\frac{312}{500}\) | A1 | Accept awrt 0.624 or exact fraction. Calculator answer: \(0.6265569683...\) |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x$: $-0.25$, $0$, $0.25$, $0.5$, $0.75$; $y$: $0.462$, **0.577**, $0.653$, **0.686**, $0.698$ | B1 | Allow awrt these values. Also allow exact value for 0.577 e.g. $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$ |
**(1 mark)**
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.25$ | B1 | Correct strip width. May be implied by $\frac{1}{8}$ or $\frac{1}{2}\times0.25$ |
| $A \approx \frac{1}{2}\times\text{"0.25"}\{0.462+0.698+2(\text{"0.577"}+0.653+\text{"0.686"})\}$ | M1 | Correct application of trapezium rule with their $h$. Must use all $y$-values. Missing brackets score M0 unless implied by subsequent work |
| $\approx$ awrt $0.624$ or $\frac{78}{125}$ e.g. $\frac{312}{500}$ | A1 | Accept awrt 0.624 or exact fraction. Calculator answer: $0.6265569683...$ |
**(3 marks) — Total 4**
---2.
$$y = \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } }$$
\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
$x$ & - 0.25 & 0 & 0.25 & 0.5 & 0.75 \\
\hline
$y$ & 0.462 & & 0.653 & & 0.698 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule,with all the values of $y$ from the completed table,to find an approximate value for\\
.
$$\int _ { - 0.25 } ^ { 0.75 } \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } } \mathrm { d } x$$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2020 Q2 [4]}}