| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring direct application of standard A-level techniques: (i) trapezium rule with just 2 intervals (minimal calculation), and (ii) iterative formula already provided (no rearrangement needed), requiring only repeated substitution until convergence. Both parts are routine procedural exercises with no problem-solving or conceptual challenge beyond basic execution. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| (i) State or imply correct coordinates \(1.4142..., 1.1370..., 1\) | B1 |
| Use correct formula, or equivalent, correctly with \(h = \frac{\pi}{4}\) and three ordinates | M1 |
| Obtain answer 1.84 with no errors seen | A1 |
| [3] | |
| (ii) Use the iterative formula correctly at least once | M1 |
| Obtain final answer 1.06 | A1 |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval \((1.055, 1.065)\) | B1 |
| [3] |
**(i)** State or imply correct coordinates $1.4142..., 1.1370..., 1$ | B1 |
Use correct formula, or equivalent, correctly with $h = \frac{\pi}{4}$ and three ordinates | M1 |
Obtain answer 1.84 with no errors seen | A1 |
| | [3] |
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.06 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval $(1.055, 1.065)$ | B1 |
| | [3] |4\\
\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-2_499_787_922_678}
The diagram shows the part of the curve $y = \sqrt { } ( 2 - \sin x )$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Use the trapezium rule with 2 intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 2 - \sin x ) \mathrm { d } x$$
giving your answer correct to 2 decimal places.\\
(ii) The line $y = x$ intersects the curve $y = \sqrt { } ( 2 - \sin x )$ at the point $P$. Use the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( 2 - \sin x _ { n } \right)$$
to determine the $x$-coordinate of $P$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2012 Q4 [6]}}