| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining basic calculus techniques. Part (i) requires simple differentiation and solving dy/dx=0. Part (ii) is a standard trapezium rule application with clear intervals. Part (iii) tests understanding of concavity, requiring the second derivative test. All techniques are routine for P2/C3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain correct derivative | B1 | |
| Obtain \(x = 2\) only | B1 | [2] |
| (ii) State or imply correct ordinates \(0.61370..., 0.80277..., 1.22741..., 1.78112...\) | B1 | |
| Use correct formula, or equivalent, correctly with \(h = 1\) and four ordinates | M1 | |
| Obtain answer \(3.23\) with no errors seen | A1 | [3] |
| (iii) Justify statement that the trapezium rule gives an over-estimate | B1 | [1] |
**(i)** Obtain correct derivative | B1 |
Obtain $x = 2$ only | B1 | [2] |
**(ii)** State or imply correct ordinates $0.61370..., 0.80277..., 1.22741..., 1.78112...$ | B1 |
Use correct formula, or equivalent, correctly with $h = 1$ and four ordinates | M1 |
Obtain answer $3.23$ with no errors seen | A1 | [3] |
**(iii)** Justify statement that the trapezium rule gives an over-estimate | B1 | [1] |3\\
\includegraphics[max width=\textwidth, alt={}, center]{322eb555-d40a-460c-8c71-5780f5772bcd-2_535_1041_573_552}
The diagram shows the curve $y = x - 2 \ln x$ and its minimum point $M$.\\
(i) Find the $x$-coordinate of $M$.\\
(ii) Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 2 } ^ { 5 } ( x - 2 \ln x ) \mathrm { d } x$$
giving your answer correct to 2 decimal places.\\
(iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
\hfill \mbox{\textit{CAIE P2 2011 Q3 [6]}}