| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.3 This is a straightforward trapezium rule application with standard follow-up questions about over/under-estimation and accuracy improvement. Part (a) requires routine calculation with 4 intervals (h=0.5), part (b) tests understanding that the trapezium rule over-estimates for concave functions, and part (c) asks for the standard response about using more intervals. While it requires careful arithmetic, it involves no problem-solving or novel insight—slightly easier than average due to its predictable structure. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State the 4 correct non-zero \(y\)-values and no others | B1 | Exact values (including unsimplified) or decimal equivs \((0, 1.12, 1.73, 2.29, 2.83)\) – 3sf or better. B0 if other ordinates seen unless clearly not intended to be used |
| \(0.5 \times 0.5\left\{0 + 2\sqrt{2} + 2\left(\frac{\sqrt{5}}{2} + \sqrt{3} + \frac{\sqrt{21}}{2}\right)\right\}\) — Attempt to find area between \(x=1\) and \(x=3\), using \(k\{y_0 + y_n + 2(y_1 + \ldots + y_{n-1})\}\) | M1\* | Big brackets need to be seen or implied. \(y\)-values must be correctly placed. Must be using attempts for at least 4 \(y\)-values (but no need to see \(y=0\) explicitly). Condone using other than 4 intervals as long as values equally spaced between \(x=1\) and \(x=3\) |
| Use \(k = 0.5 \times 0.5\) soi | M1d\* | Dep on previous M1. Or using \(k = 0.5h\), \(h\) consistent with their different number of intervals |
| \(= 3.28\) | A1 | Allow answers to \(> 3\)sf, as long as they round to 3.28 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Under-estimate, as the tops of the trapezia are below the curve | B1 | Condone just 'trapezia under curve'. Or curve is concave / decreasing gradient (not decreasing function). Accept explanation on diagrams. Allow comparing to true value \((3.36)\). B0 if any additional incorrect or contradictory statements |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use more trapezia, of a lesser width, between the same limits | B1 | Convincing reason. Condone just 'more trapezia' or 'narrower trapezia'. Could refer to strips or intervals |
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State the 4 correct non-zero $y$-values and no others | **B1** | Exact values (including unsimplified) or decimal equivs $(0, 1.12, 1.73, 2.29, 2.83)$ – 3sf or better. B0 if other ordinates seen unless clearly not intended to be used |
| $0.5 \times 0.5\left\{0 + 2\sqrt{2} + 2\left(\frac{\sqrt{5}}{2} + \sqrt{3} + \frac{\sqrt{21}}{2}\right)\right\}$ — Attempt to find area between $x=1$ and $x=3$, using $k\{y_0 + y_n + 2(y_1 + \ldots + y_{n-1})\}$ | **M1\*** | Big brackets need to be seen or implied. $y$-values must be correctly placed. Must be using attempts for at least 4 $y$-values (but no need to see $y=0$ explicitly). Condone using other than 4 intervals as long as values equally spaced between $x=1$ and $x=3$ |
| Use $k = 0.5 \times 0.5$ soi | **M1d\*** | Dep on previous M1. Or using $k = 0.5h$, $h$ consistent with their different number of intervals |
| $= 3.28$ | **A1** | Allow answers to $> 3$sf, as long as they round to 3.28 |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Under-estimate, as the tops of the trapezia are below the curve | **B1** | Condone just 'trapezia under curve'. Or curve is concave / decreasing gradient (not decreasing function). Accept explanation on diagrams. Allow comparing to true value $(3.36)$. B0 if any additional incorrect or contradictory statements |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use more trapezia, of a lesser width, between the same limits | **B1** | Convincing reason. Condone just 'more trapezia' or 'narrower trapezia'. Could refer to strips or intervals |
---1\\
\includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-4_303_451_358_242}
The diagram shows part of the curve $y = \sqrt { x ^ { 2 } - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with 4 intervals to find an estimate for $\int _ { 1 } ^ { 3 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x$.
Give your answer correct to $\mathbf { 3 }$ significant figures.
\item State whether the value from part (a) is an under-estimate or an over-estimate, giving a reason for your answer.
\item Explain how the trapezium rule could be used to obtain a more accurate estimate.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2022 Q1 [6]}}