| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the trapezium rule with clear intervals. Part (b) requires volume of revolution using π∫y² dx, which simplifies nicely since y² = 1 + e^{0.5x} integrates directly. Both parts are standard textbook exercises requiring routine application of formulas with no problem-solving insight needed, making this slightly easier than average. |
| Spec | 1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | Use y-values 2, 1+e, 1+e2, 1+e3 | B1 |
| Use correct formula, or equivalent, with h=2 | M1 | |
| Obtain 15.7 | A1 | Or greater accuracy. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(b) | State or imply that volume is π(1+e0.5x )dx | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 2 | M1 | Where k k 0 with or without π . |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain π(x+2e0.5x ) or x+2e0.5x | A1 | |
| Obtain 4π+2πe3 | A1 | Or exact equivalent. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(a) ---
4(a) | Use y-values 2, 1+e, 1+e2, 1+e3 | B1 | Or decimal equivalents
Use correct formula, or equivalent, with h=2 | M1
Obtain 15.7 | A1 | Or greater accuracy.
3
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | State or imply that volume is π(1+e0.5x )dx | B1 | Implied if π appears only at the end.
Integrate to obtain form k x+k e0.5x
1 2 | M1 | Where k k 0 with or without π .
1 2
Obtain π(x+2e0.5x ) or x+2e0.5x | A1
Obtain 4π+2πe3 | A1 | Or exact equivalent.
4
Question | Answer | Marks | Guidance\includegraphics{figure_4}
The diagram shows the curve with equation $y = \sqrt{1 + e^{0.5x}}$. The shaded region is bounded by the curve and the straight lines $x = 0$, $x = 6$ and $y = 0$.
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures. [3]
\item The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid produced. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q4 [7]}}