| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the trapezium rule with just two intervals—routine calculation requiring substitution of three x-values. Part (ii) requires recognizing that the volume of revolution formula simplifies nicely when squaring the given function (the square root disappears), then integrating 1 + 3cos²(x/2) using the double angle identity cos²θ = (1+cos2θ)/2. While this requires multiple steps and careful algebraic manipulation, it follows standard A-level techniques without requiring novel insight. The 8 total marks and mix of routine numerical work with algebraic integration place this slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| 6(i) | Use y values 2, 2.5, 1 or equivalents | B1 |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks |
|---|---|
| 2 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 6(ii) | State or imply volume is ∫π(1+3cos2 1x)dx | |
| 2 | B1 | Allow if πappears later; condone omission of dx |
| Answer | Marks |
|---|---|
| 1 2 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 2 2 | A1 | Condone omission of dx |
| Answer | Marks |
|---|---|
| 2 2 2 2 | A1 |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
--- 6(i) ---
6(i) | Use y values 2, 2.5, 1 or equivalents | B1
Use correct formula, or equivalent, with h= 1π and three y values
2 | M1
Obtain 1×1π(2+2 2.5+1) or equivalent and hence 4.84
2 2 | A1
3
Question | Answer | Marks | Guidance
--- 6(ii) ---
6(ii) | State or imply volume is ∫π(1+3cos2 1x)dx
2 | B1 | Allow if πappears later; condone omission of dx
Use appropriate identity to express integrand in form k +k cosx
1 2 | M1
Obtain ∫π(5+ 3cosx)dx or ∫(5 + 3cosx)dx
2 2 2 2 | A1 | Condone omission of dx
Integrate to obtain π(5x+ 3sinx) or 5x+ 3sinx
2 2 2 2 | A1
Obtain 5π2 with no errors seen
2 | A1
5
Question | Answer | Marks | Guidance\includegraphics{figure_6}
The diagram shows the curve with equation $y = \sqrt{1 + 3\cos^2(\frac{1}{2}x)}$ for $0 \leqslant x \leqslant \pi$. The region $R$ is bounded by the curve, the axes and the line $x = \pi$.
\begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with two intervals to find an approximation to the area of $R$, giving your answer correct to 3 significant figures. [3]
\item The region $R$ is rotated completely about the $x$-axis. Without using a calculator, find the exact volume of the solid produced. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2018 Q6 [8]}}