| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard techniques: finding intercepts (trivial), differentiation to find maximum (routine quotient rule), applying trapezium rule with given intervals (mechanical calculation), and determining over/under-estimate from concavity (standard reasoning). All parts are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| State coordinates \((1, 0)\) | B1 |
| Answer | Marks |
|---|---|
| Use quotient or product rule | M1 |
| Obtain correct derivative, e.g. \(\frac{-\ln x}{x^2} + \frac{1}{x^2}\) | A1 |
| Equate derivative to zero and solve for \(x\) | M1 |
| Obtain \(x = e\) | A1 |
| Obtain \(y = \frac{1}{e}\) | A1 |
| Answer | Marks |
|---|---|
| Show or imply correct coordinates \(0, 0.34657..., 0.36620..., 0.34657...\) with \(h = 1\) and four ordinates | B1 |
| Use correct formula, or equivalent | A1 |
| Obtain answer \(0.89\) with no errors seen | A1 |
| Answer | Marks |
|---|---|
| Justify statement that the rule gives an under-estimate | B1 |
**(i)**
| State coordinates $(1, 0)$ | B1 |
**Total: 1 mark**
**(ii)**
| Use quotient or product rule | M1 |
| Obtain correct derivative, e.g. $\frac{-\ln x}{x^2} + \frac{1}{x^2}$ | A1 |
| Equate derivative to zero and solve for $x$ | M1 |
| Obtain $x = e$ | A1 |
| Obtain $y = \frac{1}{e}$ | A1 |
**Total: 5 marks**
**(iii)**
| Show or imply correct coordinates $0, 0.34657..., 0.36620..., 0.34657...$ with $h = 1$ and four ordinates | B1 |
| Use correct formula, or equivalent | A1 |
| Obtain answer $0.89$ with no errors seen | A1 |
**Total: 3 marks**
**(iv)**
| Justify statement that the rule gives an under-estimate | B1 |
**Total: 1 mark**
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{08210e25-0f0e-405b-b72d-1bf989689b0a-3_641_865_264_641}
The diagram shows the part of the curve $y = \frac { \ln x } { x }$ for $0 < x \leqslant 4$. The curve cuts the $x$-axis at $A$ and its maximum point is $M$.\\
(i) Write down the coordinates of $A$.\\
(ii) Show that the $x$-coordinate of $M$ is e, and write down the $y$-coordinate of $M$ in terms of e.\\
(iii) Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 1 } ^ { 4 } \frac { \ln x } { x } \mathrm {~d} x$$
correct to 2 decimal places.\\
(iv) 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 (iii).
\hfill \mbox{\textit{CAIE P2 2005 Q6 [10]}}