| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the trapezium rule with clear intervals. Part (ii) requires recognizing that region B can be found by subtracting region A from a rectangle (2×3), then explaining the geometric reason for underestimation—this involves some insight but is a standard technique for relating areas. The question tests understanding beyond mechanical calculation but remains accessible for P2 level. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show or imply correct ordinates \(1, \sqrt{2}\) or \(1.414, 3\) | B1 | |
| Use correct formula, or equivalent, with \(h = 1\) | M1 | |
| Obtain \(3.41\) | A1 | [3] |
| (ii) Obtain \(6 - 3.41\) and hence \(2.59\), following their answer to (i) provided less than \(6\) | B1√ | |
| Refer, in some form, to two line segments replacing curve and conclude with clear justification of given result that answer is an under-estimate | B1 | [2] |
**(i)** Show or imply correct ordinates $1, \sqrt{2}$ or $1.414, 3$ | B1 |
Use correct formula, or equivalent, with $h = 1$ | M1 |
Obtain $3.41$ | A1 | [3]
**(ii)** Obtain $6 - 3.41$ and hence $2.59$, following their answer to (i) provided less than $6$ | B1√ |
Refer, in some form, to two line segments replacing curve and conclude with clear justification of given result that answer is an under-estimate | B1 | [2]2\\
\includegraphics[max width=\textwidth, alt={}, center]{2c27f384-5289-4c1b-9199-6b4c6ac81e38-2_645_750_429_699}
The diagram shows the curve $y = \sqrt { } \left( 1 + x ^ { 3 } \right)$. Region $A$ is bounded by the curve and the lines $x = 0$, $x = 2$ and $y = 0$. Region $B$ is bounded by the curve and the lines $x = 0$ and $y = 3$.\\
(i) Use the trapezium rule with two intervals to find an approximation to the area of region $A$. Give your answer correct to 2 decimal places.\\
(ii) Deduce an approximation to the area of region $B$ and explain why this approximation underestimates the true area of region $B$.
\hfill \mbox{\textit{CAIE P2 2011 Q2 [5]}}