| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Show trapezium rule gives specific value |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the trapezium rule formula with given intervals—pure procedural calculation. Part (b) requires understanding that ln x is concave (curves below chords), making trapezium rule an underestimate, but this is a standard conceptual point taught explicitly in P2. The question is slightly easier than average due to its routine nature and the 'show that' format providing a target answer. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(y\)-values \([\ln 1]\), \(\ln 2\), \(\ln 3\), \(\ln 4\) | B1 | but not *their* decimal equivalents |
| Use correct formula, or equivalent, with \(h=1\) | M1 | allow with decimal equivalents |
| Use both relevant logarithm properties correctly | M1 | |
| Obtain \(\frac{1}{2}[\ln 1 + 2\ln 2 + 2\ln 3 + \ln 4]\) and hence \(\ln 12\) | A1 | AG – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch correct graph of \(y = \ln x\) for at least \(y \geqslant 0\) | *B1 | |
| Indicate that top of each trapezium is below curve or clear equivalent | DB1 | but B0 if only one chord shown from \((1, 0)\) to \((4, \ln 4)\); allow 'concave down' |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $y$-values $[\ln 1]$, $\ln 2$, $\ln 3$, $\ln 4$ | B1 | but not *their* decimal equivalents |
| Use correct formula, or equivalent, with $h=1$ | M1 | allow with decimal equivalents |
| Use both relevant logarithm properties correctly | M1 | |
| Obtain $\frac{1}{2}[\ln 1 + 2\ln 2 + 2\ln 3 + \ln 4]$ and hence $\ln 12$ | A1 | AG – necessary detail needed |
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## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch correct graph of $y = \ln x$ for at least $y \geqslant 0$ | *B1 | |
| Indicate that top of each trapezium is below curve or clear equivalent | DB1 | but B0 if only one chord shown from $(1, 0)$ to $(4, \ln 4)$; allow 'concave down' |
---4
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with three intervals to show that the value of $\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x$ is approximately $\ln 12$.
\item Use a graph of $y = \ln x$ to show that $\ln 12$ is an under-estimate of the true value of $\int _ { 1 } ^ { 4 } \ln x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q4 [6]}}