| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with only two intervals (minimal computation), followed by a standard concavity argument using a sketch. The sec x function is familiar, and determining over/under-estimate from convexity is a routine skill tested at this level. Slightly easier than average due to the small number of intervals and predictable reasoning required. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| Show or imply correct coordinates \(1, 1.15470..., 2\) | B1 |
| Use correct formula, or equivalent, with \(h = \frac{1}{6}\pi\) and three ordinates | M1 |
| Obtain answer 1.39 with no errors seen | A1 |
| Answer | Marks |
|---|---|
| Make recognisable sketch of \(y = \sec x\) for \(0 \le x \le \frac{1}{4}\pi\) | B1 |
| Using a correct graph, explain that the rule gives an over-estimate | B1 |
**(i)**
| Show or imply correct coordinates $1, 1.15470..., 2$ | B1 |
| Use correct formula, or equivalent, with $h = \frac{1}{6}\pi$ and three ordinates | M1 |
| Obtain answer 1.39 with no errors seen | A1 |
**Total: [3]**
**(ii)**
| Make recognisable sketch of $y = \sec x$ for $0 \le x \le \frac{1}{4}\pi$ | B1 |
| Using a correct graph, explain that the rule gives an over-estimate | B1 |
**Total: [2]**
---3 (i) Use the trapezium rule with two intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sec x \mathrm {~d} x$$
giving your answer correct to 2 decimal places.\\
(ii) Using a sketch of the graph of $y = \sec x$ for $0 \leqslant x \leqslant \frac { 1 } { 3 } \pi$, explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).
\hfill \mbox{\textit{CAIE P2 2009 Q3 [5]}}