| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: (i) differentiation using quotient rule to find stationary point, (ii) routine trapezium rule application with 2 intervals, (iii) conceptual understanding of trapezium rule accuracy based on concavity. All parts are textbook-standard 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct quotient rule or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = 2\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply ordinates \(1.6487\ldots,\ 1.3591\ldots,\ 1.4938\ldots\) | B1 | |
| Use correct formula, or equivalent, with \(h = 1\) and three ordinates | M1 | |
| Obtain answer 2.93 only | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Explain why the estimate would be less than \(E\) | B1 | |
| Total: 1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct quotient rule or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain $x = 2$ | A1 | |
| **Total: 4** | | |
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply ordinates $1.6487\ldots,\ 1.3591\ldots,\ 1.4938\ldots$ | B1 | |
| Use correct formula, or equivalent, with $h = 1$ and three ordinates | M1 | |
| Obtain answer 2.93 only | A1 | |
| **Total: 3** | | |
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Explain why the estimate would be less than $E$ | B1 | |
| **Total: 1** | | |
---7\\
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The diagram shows a sketch of the curve $y = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x }$ for $x > 0$, and its minimum point $M$.\\
(i) Find the $x$-coordinate of $M$.\\
(ii) Use the trapezium rule with two intervals to estimate the value of
$$\int _ { 1 } ^ { 3 } \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.\\
(iii) The estimate found in part (ii) is denoted by $E$. Explain, without further calculation, whether another estimate found using the trapezium rule with four intervals would be greater than $E$ or less than $E$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q7 [8]}}