| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule symmetry argument |
| Difficulty | Moderate -0.3 This is a straightforward trapezium rule application with only 2 intervals requiring basic substitution into the formula, followed by a simple observation about the curve's shape. The reasoning part (ii) is accessible—students need only note the curve is approximately linear over the interval. Below average difficulty due to minimal computation and standard bookwork nature. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| State or imply ordinates \(0.915929\ldots, 1, 1.112485\ldots\) | B1 | |
| Use correct formula, or equivalent, with \(h = 1.2\) and three ordinates | M1 | |
| Obtain answer \(2.42\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| Justify the given statement | B1 |
## Question 1(i):
| Answer | Mark | Notes |
|--------|------|-------|
| State or imply ordinates $0.915929\ldots, 1, 1.112485\ldots$ | B1 | |
| Use correct formula, or equivalent, with $h = 1.2$ and three ordinates | M1 | |
| Obtain answer $2.42$ only | A1 | |
## Question 1(ii):
| Answer | Mark | Notes |
|--------|------|-------|
| Justify the given statement | B1 | |1\\
\includegraphics[max width=\textwidth, alt={}, center]{746d2c39-7d78-4478-bc36-15ea5e3ba72a-02_460_807_258_667}
The diagram shows a sketch of the curve $y = \frac { 3 } { \sqrt { } \left( 9 - x ^ { 3 } \right) }$ for values of $x$ from - 1.2 to 1.2 .\\
(i) Use the trapezium rule, with two intervals, to estimate the value of
$$\int _ { - 1.2 } ^ { 1.2 } \frac { 3 } { \sqrt { \left( 9 - x ^ { 3 } \right) } } \mathrm { d } x$$
giving your answer correct to 2 decimal places.\\
(ii) Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.\\
\hfill \mbox{\textit{CAIE P3 2017 Q1 [4]}}