| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.3 Part (i) is a standard Simpson's rule application with straightforward function evaluation. Part (ii) requires recognizing that the integral can be split using linearity, with the first term integrating trivially to 2 and the second being 10 times the result from (i). This is routine manipulation once Simpson's rule is applied, requiring only basic integral properties rather than novel insight. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt calculation \(k(y + 4y + 2y + \ldots)\) | M1 | any constant \(k\); using \(y\) values with coefficients 1, 2, 4 each occurring at least once; brackets may be implied by subsequent calculation. Allow M1 for attempt using \(y\) values based on wrong \(x\) values such as 0, 1, 2, 3, 4; attempt based on \(k(y_0 + y_4) + 4y_1 + 2y_2 + 4y_3\) is M0 unless subsequent calculation shows missing brackets are 'present' |
| Obtain \(k(e^0 + 4e^{\sqrt{0.5}} + 2e + 4e^{\sqrt{1.5}} + e^{\sqrt{2}})\) | A1 | or equiv perhaps involving decimal values \(1, 2.02811\ldots, 2.71828\ldots, 3.40329\ldots, 4.11325\ldots\) |
| Use \(k = \frac{1}{2} \times \frac{1}{2}\) | A1 | |
| Obtain \(5.38\) | A1 | allow \(5.379\) but not, in final answer, greater 'accuracy'; answer \(5.38 + c\) is final A0. Answer only: 0/4 |
| Total | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt calculation of form \(10 \times (\text{answer to part i}) + k\) | M1 | implied by correct answer only or by answer following correctly from their incorrect part (i); any non-zero constant \(k\). Allow attempt involving second use of Simpson's rule: M1 for complete correct expression, A1 for answer |
| Obtain \(55.8\) or greater accuracy based on their part (i) – more than 3 s.f. acceptable | A1ft | following their answer to part (i) but A0 for \(55.8 + c\). Answer only \(54.8\) with no working earns M1A0 (as does \(10(\text{their ans}) + 1\)); otherwise incorrect answer with no working earns 0/2 |
| Total | [2] |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt calculation $k(y + 4y + 2y + \ldots)$ | M1 | any constant $k$; using $y$ values with coefficients 1, 2, 4 each occurring at least once; brackets may be implied by subsequent calculation. Allow M1 for attempt using $y$ values based on wrong $x$ values such as 0, 1, 2, 3, 4; attempt based on $k(y_0 + y_4) + 4y_1 + 2y_2 + 4y_3$ is M0 unless subsequent calculation shows missing brackets are 'present' |
| Obtain $k(e^0 + 4e^{\sqrt{0.5}} + 2e + 4e^{\sqrt{1.5}} + e^{\sqrt{2}})$ | A1 | or equiv perhaps involving decimal values $1, 2.02811\ldots, 2.71828\ldots, 3.40329\ldots, 4.11325\ldots$ |
| Use $k = \frac{1}{2} \times \frac{1}{2}$ | A1 | |
| Obtain $5.38$ | A1 | allow $5.379$ but not, in final answer, greater 'accuracy'; answer $5.38 + c$ is final A0. Answer only: 0/4 |
| **Total** | **[4]** | |
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## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt calculation of form $10 \times (\text{answer to part i}) + k$ | M1 | implied by correct answer only or by answer following correctly from their incorrect part (i); any non-zero constant $k$. Allow attempt involving second use of Simpson's rule: M1 for complete correct expression, A1 for answer |
| Obtain $55.8$ or greater accuracy based on their part (i) – more than 3 s.f. acceptable | A1ft | following their answer to part (i) but A0 for $55.8 + c$. Answer only $54.8$ with no working earns M1A0 (as does $10(\text{their ans}) + 1$); otherwise incorrect answer with no working earns 0/2 |
| **Total** | **[2]** | |
---3 (i) Use Simpson's rule with four strips to find an approximation to
$$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sqrt { x } } \mathrm {~d} x$$
giving your answer correct to 3 significant figures.\\
(ii) Deduce an approximation to $\int _ { 0 } ^ { 2 } \left( 1 + 10 \mathrm { e } ^ { \sqrt { x } } \right) \mathrm { d } x$.
\hfill \mbox{\textit{OCR C3 2014 Q3 [6]}}