| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| 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 basic calculus skills: finding coordinates (trivial), differentiation with product rule to find maximum, applying trapezium rule formula (standard procedure), and interpreting concavity. All parts are routine A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.06a Exponential function: a^x and e^x graphs and properties1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| State coordinates (0, 1) for \(A\) | B1 | [1 mark] |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate using the product rule | M1* | |
| Obtain derivative in any correct form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1* | |
| Obtain \(x = \frac{1}{4}\pi\) or 0.785 (allow 45°) | A1 | [4 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Show or imply correct ordinates 1, 1.4619…, 1.4248…, 0 | B1 | |
| Use correct formula or equivalent with \(h = \frac{1}{6}\pi\) and four ordinates | M1 | |
| Obtain correct answer 1.77 with no errors seen | A1 | [3 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Justify statement that the trapezium rule gives and underestimate | B1 | [1 mark] |
**(i)**
State coordinates (0, 1) for $A$ | B1 | [1 mark] |
**(ii)**
Differentiate using the product rule | M1* | |
Obtain derivative in any correct form | A1 | |
Equate derivative to zero and solve for $x$ | M1* | |
Obtain $x = \frac{1}{4}\pi$ or 0.785 (allow 45°) | A1 | [4 marks] |
**(iii)**
Show or imply correct ordinates 1, 1.4619…, 1.4248…, 0 | B1 | |
Use correct formula or equivalent with $h = \frac{1}{6}\pi$ and four ordinates | M1 | |
Obtain correct answer 1.77 with no errors seen | A1 | [3 marks] |
**(iv)**
Justify statement that the trapezium rule gives and underestimate | B1 | [1 mark] |7\\
\includegraphics[max width=\textwidth, alt={}, center]{9d93ad8c-0a22-4de7-8342-387606e4e510-3_584_675_945_735}
The diagram shows the part of the curve $y = \mathrm { e } ^ { x } \cos x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. The curve meets the $y$-axis at the point $A$. The point $M$ is a maximum point.\\
(i) Write down the coordinates of $A$.\\
(ii) Find the $x$-coordinate of $M$.\\
(iii) Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } e ^ { x } \cos x d x$$
giving your answer 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 2007 Q7 [9]}}