| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Simpson's rule estimation |
| Difficulty | Standard +0.8 This question requires multiple techniques (Simpson's rule, exponential conversion, exact integration) and culminates in a non-obvious deduction connecting numerical and exact results to approximate ln 2. The final part requires algebraic manipulation and insight beyond routine integration, elevating it above standard C3 questions. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(h = 1\) | B1 | |
| Attempt calculation involving attempts at \(y\) values | M1 | addition with each of coefficients 1, 2, 4 occurring at least once; involving at least 5 \(y\) values any constant \(a\) |
| Obtain \((1 + 4x2 + 2x4 + 4x8 + 2x16 + 4x32 + 64)A1\) | ||
| Obtain 91 | A1 4 | |
| (ii) State \(e^{\ln 2}\) or \(k = \ln 2\) | B1 | allow decimal equiv such as \(e^{0.69x}\) |
| Integrate \(e^{kx}\) to obtain \(\frac{1}{e}e^{kx}\) | M1 | any constant \(k\) or in terms of general \(k\) |
| Obtain \(\frac{1}{\ln 2}(e^{6\ln 2} - e^0)\) | A1 | or exact equiv |
| Simplify to obtain \(\frac{63}{\ln 2}\) | A1 4 | allow if simplification in part (iii) |
| (iii) Equate answers to (i) and (ii) | M1 | provided \(\ln 2\) involved other than in power of \(e\) |
| Obtain \(\frac{63}{91}\) and hence \(\frac{9}{13}\) | A1 2 | AG; necessary correct detail required |
**(i)** State or imply $h = 1$ | B1 |
Attempt calculation involving attempts at $y$ values | M1 | addition with each of coefficients 1, 2, 4 occurring at least once; involving at least 5 $y$ values any constant $a$
Obtain $(1 + 4x2 + 2x4 + 4x8 + 2x16 + 4x32 + 64)A1$ |
Obtain 91 | A1 4 |
**(ii)** State $e^{\ln 2}$ or $k = \ln 2$ | B1 | allow decimal equiv such as $e^{0.69x}$
Integrate $e^{kx}$ to obtain $\frac{1}{e}e^{kx}$ | M1 | any constant $k$ or in terms of general $k$
Obtain $\frac{1}{\ln 2}(e^{6\ln 2} - e^0)$ | A1 | or exact equiv
Simplify to obtain $\frac{63}{\ln 2}$ | A1 4 | allow if simplification in part (iii)
**(iii)** Equate answers to (i) and (ii) | M1 | provided $\ln 2$ involved other than in power of $e$
Obtain $\frac{63}{91}$ and hence $\frac{9}{13}$ | A1 2 | AG; necessary correct detail required
---8 The definite integral $I$ is defined by
$$I = \int _ { 0 } ^ { 6 } 2 ^ { x } \mathrm {~d} x$$
(i) Use Simpson's rule with 6 strips to find an approximate value of $I$.\\
(ii) By first writing $2 ^ { x }$ in the form $\mathrm { e } ^ { k x }$, where the constant $k$ is to be determined, find the exact value of $I$.\\
(iii) Use the answers to parts (i) and (ii) to deduce that $\ln 2 \approx \frac { 9 } { 13 }$.
\hfill \mbox{\textit{OCR C3 2008 Q8 [10]}}