| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| 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 trapezium rule application requiring straightforward substitution into the formula. Part (ii) requires recognizing that log₁₀√(2+x) = ½log₁₀(2+x), making the answer half of part (i). This is slightly easier than average as it's a routine two-part question with the second part being a simple deduction rather than requiring independent calculation. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(\int_1^5 \log_{10}(2+x)dx = \frac{1}{x} \times (\log 5 + 2\log 5.5 + 2\log 6 + 2\log 6.5 + \log 7)\) ≈ 1.55 | M1, M1, M1, A1 |
| (ii) | \(\int_1^5 \log_{10}(2+x)^2 dx = \frac{1}{2}\int_1^5 \log_{10}(2+x)dx = \frac{1}{2} \times 1.55 = 0.78\) | B1√, B1 |
(i) | $\int_1^5 \log_{10}(2+x)dx = \frac{1}{x} \times (\log 5 + 2\log 5.5 + 2\log 6 + 2\log 6.5 + \log 7)$ ≈ 1.55 | M1, M1, M1, A1 | Attempt y-coords for at least 4 of the correct 5 x-coords only; Use correct trapezium rule, any h, to find area between $x = 3$ and $x = 5$; Correct h (soi) for their y-values; Obtain 1.55 |
(ii) | $\int_1^5 \log_{10}(2+x)^2 dx = \frac{1}{2}\int_1^5 \log_{10}(2+x)dx = \frac{1}{2} \times 1.55 = 0.78$ | B1√, B1 | Divide by 2, or equiv, at any stage to obtain 0.78 or 0.77, following their answer to (i); Explicitly use $\log \sqrt{a} = \frac{1}{2}\log a$ on a single term |4 (i) Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for
$$\int _ { 3 } ^ { 5 } \log _ { 10 } ( 2 + x ) d x$$
giving your answer correct to 3 significant figures.\\
(ii) Use your answer to part (i) to deduce an approximate value for $\int _ { 3 } ^ { 5 } \log _ { 10 } \sqrt { 2 + x } \mathrm {~d} x$, showing your method clearly.
\hfill \mbox{\textit{OCR C2 2010 Q4 [6]}}