| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.3 Part (a) requires quotient rule differentiation and solving a quadratic equation for stationary points—standard calculus techniques. Part (b) is a routine trapezium rule application with three intervals. Both parts are straightforward applications of well-practiced methods with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differentiate using quotient rule (or product rule) | \*M1 | |
| Obtain \(\dfrac{(x^2+8) - 2x(x-2)}{(x^2+8)^2}\) | A1 | OE |
| Equate first derivative to zero and attempt solution to get \(x = \ldots\) | DM1 | |
| Obtain \(2 \pm \sqrt{12}\) or exact equivalents | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(y\) values \((0),\ \dfrac{4}{44},\ \dfrac{8}{108},\ \dfrac{12}{204}\) or decimal equivalents | B1 | Decimal equivalents need to be to at least 2 decimal places |
| Use correct formula, or equivalent, with \(h = 4\) | M1 | |
| Obtain \(2\!\left(0 + 2\times\dfrac{4}{44} + 2\times\dfrac{8}{108} + \dfrac{12}{204}\right)\) or equivalent and hence \(0.78\) | A1 | |
| Total | 3 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate using quotient rule (or product rule) | \*M1 | |
| Obtain $\dfrac{(x^2+8) - 2x(x-2)}{(x^2+8)^2}$ | A1 | OE |
| Equate first derivative to zero and attempt solution to get $x = \ldots$ | DM1 | |
| Obtain $2 \pm \sqrt{12}$ or exact equivalents | A1 | |
| **Total** | **4** | |
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## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $y$ values $(0),\ \dfrac{4}{44},\ \dfrac{8}{108},\ \dfrac{12}{204}$ or decimal equivalents | B1 | Decimal equivalents need to be to at least 2 decimal places |
| Use correct formula, or equivalent, with $h = 4$ | M1 | |
| Obtain $2\!\left(0 + 2\times\dfrac{4}{44} + 2\times\dfrac{8}{108} + \dfrac{12}{204}\right)$ or equivalent and hence $0.78$ | A1 | |
| **Total** | **3** | |
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\includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-06_460_1445_260_349}
The diagram shows the curve with equation $y = \frac { x - 2 } { x ^ { 2 } + 8 }$. The shaded region is bounded by the curve and the lines $x = 14$ and $y = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence determine the exact $x$-coordinates of the stationary points.
\item Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give the answer correct to 2 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q4 [7]}}