| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.8 Part (a) is a straightforward integration using the reverse chain rule with a linear function inside cosine—pure recall. Part (b) is a standard trapezium rule application with three intervals requiring only substitution into the formula and calculator work. Both parts are routine textbook exercises with no problem-solving or insight required, making this easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| (a) Integrate to obtain form \(k\sin(\frac{1}{3}x + 2)\) where \(k \neq 4\) | M1 |
| Obtain \(12\sin(\frac{1}{3}x + 2)\) \((+ c)\) | A1 [2] |
| (b) State or imply correct \(y\)-values \(2, \sqrt{20}, \sqrt{68}, \sqrt{148}\) | B1 |
| Use correct formula, or equivalent, with \(h = 4\) and four \(y\)-values | M1 |
| Obtain \(79.2\) | A1 [3] |
(a) Integrate to obtain form $k\sin(\frac{1}{3}x + 2)$ where $k \neq 4$ | M1 |
Obtain $12\sin(\frac{1}{3}x + 2)$ $(+ c)$ | A1 [2] |
(b) State or imply correct $y$-values $2, \sqrt{20}, \sqrt{68}, \sqrt{148}$ | B1 |
Use correct formula, or equivalent, with $h = 4$ and four $y$-values | M1 |
Obtain $79.2$ | A1 [3] |3
\begin{enumerate}[label=(\alph*)]
\item Find $\int 4 \cos \left( \frac { 1 } { 3 } x + 2 \right) \mathrm { d } x$.
\item Use the trapezium rule with three intervals to find an approximation to
$$\int _ { 0 } ^ { 12 } \sqrt { } \left( 4 + x ^ { 2 } \right) \mathrm { d } x$$
giving your answer correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2014 Q3 [5]}}