| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.3 Part (i) is a routine trapezium rule application with straightforward function evaluation at given points. Part (ii) tests conceptual understanding of how trapezium rule accuracy relates to curve concavity, which is standard P3 content. The function evaluation requires calculator use but no algebraic manipulation or novel insight. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply 3 of the 4 ordinates \(1,\ 1.069389\ldots,\ 1.290994\ldots,\ 1.732050\ldots\) | B1 | SR: if only \(\sqrt{\frac{5}{3}}\) and/or \(\sqrt{3}\) are given and decimals not seen, B1 available |
| Use correct formula, or equivalent, with \(h = \frac{1}{12}\pi\) and four ordinates | M1 | Accept \(h = 0.26\) but not \(h = 15\) |
| Obtain answer \(0.98\) with no errors seen | A1 | SR: solutions with 2 or 4 intervals can score only M1 for correct expression |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Justify statement that the second estimate would be less than \(E\) | B1 |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply 3 of the 4 ordinates $1,\ 1.069389\ldots,\ 1.290994\ldots,\ 1.732050\ldots$ | B1 | SR: if only $\sqrt{\frac{5}{3}}$ and/or $\sqrt{3}$ are given and decimals not seen, B1 available |
| Use correct formula, or equivalent, with $h = \frac{1}{12}\pi$ and four ordinates | M1 | Accept $h = 0.26$ but not $h = 15$ |
| Obtain answer $0.98$ with no errors seen | A1 | SR: solutions with 2 or 4 intervals can score only M1 for correct expression |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Justify statement that the second estimate would be less than $E$ | B1 | |
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The diagram shows the curve $y = \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right)$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$.\\
(i) Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sqrt { } \left( 1 + 2 \tan ^ { 2 } x \right) \mathrm { d } x$$
giving your answer correct to 2 decimal places.\\
(ii) The estimate found in part (i) is denoted by $E$. Explain, without further calculation, whether another estimate found using the trapezium rule with six intervals would be greater than $E$ or less than $E$.
\hfill \mbox{\textit{CAIE P3 2009 Q2 [4]}}