| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2004 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring standard techniques: differentiation using the product rule to find a maximum (routine), applying the trapezium rule formula with given intervals (direct application), and recognizing concavity to determine over/under-estimation (standard reasoning). All parts are textbook exercises with no novel problem-solving required, making it easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State derivative of the form \((e^{-x} \pm xe^{-x})\). Allow \(xe^x \pm e^x\) via quotient rule | M1 | |
| Obtain correct derivative of \(e^{\pm x} - xe^{-x}\) | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = 1\) | A1 | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show or imply correct ordinates 0, 0.367879…, 0.27067… | B1 | |
| Use correct formula, or equivalent, with \(h = 1\) and three ordinates | M1 | |
| Obtain answer 0.50 with no errors seen | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Justify statement that the rule gives an under-estimate | B1 | Total: 1 |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State derivative of the form $(e^{-x} \pm xe^{-x})$. Allow $xe^x \pm e^x$ via quotient rule | M1 | |
| Obtain correct derivative of $e^{\pm x} - xe^{-x}$ | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain answer $x = 1$ | A1 | **Total: 4** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show or imply correct ordinates 0, 0.367879…, 0.27067… | B1 | |
| Use correct formula, or equivalent, with $h = 1$ and three ordinates | M1 | |
| Obtain answer 0.50 with no errors seen | A1 | **Total: 3** |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Justify statement that the rule gives an under-estimate | B1 | **Total: 1** |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{34177829-f05d-449e-8881-5ab4d852c4ce-3_458_643_285_751}
The diagram shows the part of the curve $y = x \mathrm { e } ^ { - x }$ for $0 \leqslant x \leqslant 2$, and its maximum point $M$.\\
(i) Find the $x$-coordinate of $M$.\\
(ii) Use the trapezium rule with two intervals to estimate the value of
$$\int _ { 0 } ^ { 2 } x \mathrm { e } ^ { - x } \mathrm {~d} x$$
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 2004 Q5 [8]}}