| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.3 Part (a) is a straightforward integration of a rational function requiring a simple substitution or recognition of ln form, followed by basic logarithm manipulation. Part (b) is a standard trapezium rule application with clearly specified intervals. Both parts are routine A-level techniques with no problem-solving insight required, making this slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Integrate to obtain form \(k\ln\left(2x - 7\right)\) | M1 | |
| Obtain correct \(3\ln\left(2x - 7\right)\) | A1 | |
| Substitute limits correctly (dependent on first M1) | DM1 | |
| Use law for logarithm of a quotient or power (dependent on first M1) | DM1 | |
| Confirm \(\ln 125\) following correct work and sufficient detail (AG) | A1 | [5] |
| (b) Evaluate \(y\) at \((1), 5, 9, 13, 17\) | M1 | |
| Use correct formula, or equivalent, with \(h = 4\) and five \(y\)-values | M1 | |
| Obtain \(13.5\) | A1 | [3] |
**(a)** Integrate to obtain form $k\ln\left(2x - 7\right)$ | M1 |
Obtain correct $3\ln\left(2x - 7\right)$ | A1 |
Substitute limits correctly (dependent on first M1) | DM1 |
Use law for logarithm of a quotient or power (dependent on first M1) | DM1 |
Confirm $\ln 125$ following correct work and sufficient detail (AG) | A1 | [5]
**(b)** Evaluate $y$ at $(1), 5, 9, 13, 17$ | M1 |
Use correct formula, or equivalent, with $h = 4$ and five $y$-values | M1 |
Obtain $13.5$ | A1 | [3]6
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 6 } ^ { 16 } \frac { 6 } { 2 x - 7 } \mathrm {~d} x = \ln 125$.
\item Use the trapezium rule with four intervals to find an approximation to
$$\int _ { 1 } ^ { 17 } \log _ { 10 } x d x$$
giving your answer correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2014 Q6 [8]}}