| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard techniques: (i) differentiate using quotient rule, set derivative to zero, and solve ln x = 2 to find the stationary point; (ii) apply trapezium rule with given intervals. Both parts are routine applications of A-level methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Use product or quotient rule | M1* | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1*(dep) | |
| Obtain \(x = e^{0.5}\) or \(\sqrt{e}\) | A1 | |
| Obtain \(\frac{1}{2e}\), or equivalent | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply correct ordinates 0, 0.17328..., 0.12206..., 0.08664... | B1 | |
| Use correct formula, or equivalent, correctly with \(h = 1\) and four ordinates | M1 | |
| Obtain answer 0.34 with no errors seen | A1 | [3] |
**(i)**
| Use product or quotient rule | M1* |
| Obtain correct derivative in any form | A1 |
| Equate derivative to zero and solve for $x$ | M1*(dep) |
| Obtain $x = e^{0.5}$ or $\sqrt{e}$ | A1 |
| Obtain $\frac{1}{2e}$, or equivalent | A1 | [5] |
**(ii)**
| State or imply correct ordinates 0, 0.17328..., 0.12206..., 0.08664... | B1 |
| Use correct formula, or equivalent, correctly with $h = 1$ and four ordinates | M1 |
| Obtain answer 0.34 with no errors seen | A1 | [3] |7\\
\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-3_611_1084_648_532}
The diagram shows the curve $y = \frac { \ln x } { x ^ { 2 } }$ and its maximum point $M$.\\
(i) Find the exact coordinates of $M$.\\
(ii) Use the trapezium rule with three intervals to estimate the value of
$$\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P2 2010 Q7 [8]}}