| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard P2 techniques: differentiation using product rule to find a maximum, verifying a tangent passes through a point, and applying the trapezium rule with clear instructions. All parts are routine applications of learned methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate using product or quotient rule | M1 | |
| Obtain derivative in any correct form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = 2\) correctly, with no other solution | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Find the gradient of the curve when \(x = 1\), must be simplified, allow 0.368 | B1 | |
| Form the equation of the tangent when \(x = 1\) | M1 | |
| Show that it passes through the origin | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply correct ordinates \(0.36787\ldots, 0.54134\ldots, 0.44808\ldots\) | B1 | |
| Use correct formula, or equivalent, correctly with \(h = 1\) and three ordinates | M1 | |
| Obtain answer 0.95 with no errors seen | A1 | [3] |
**(i)**
Differentiate using product or quotient rule | M1 |
Obtain derivative in any correct form | A1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain answer $x = 2$ correctly, with no other solution | A1 | [4]
**(ii)**
Find the gradient of the curve when $x = 1$, must be simplified, allow 0.368 | B1 |
Form the equation of the tangent when $x = 1$ | M1 |
Show that it passes through the origin | A1 | [3]
**(iii)**
State or imply correct ordinates $0.36787\ldots, 0.54134\ldots, 0.44808\ldots$ | B1 |
Use correct formula, or equivalent, correctly with $h = 1$ and three ordinates | M1 |
Obtain answer 0.95 with no errors seen | A1 | [3]8\\
\includegraphics[max width=\textwidth, alt={}, center]{8f815127-61b2-4a7f-8687-747950ea6597-3_693_1061_262_541}
The diagram shows the curve $y = x ^ { 2 } \mathrm { e } ^ { - x }$ and its maximum point $M$.\\
(i) Find the $x$-coordinate of $M$.\\
(ii) Show that the tangent to the curve at the point where $x = 1$ passes through the origin.\\
(iii) Use the trapezium rule, with two intervals, to estimate the value of
$$\int _ { 1 } ^ { 3 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P2 2007 Q8 [10]}}