| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with clearly specified parameters (4 strips, width 1.5), requiring only substitution into the standard formula. Part (ii) tests understanding of concavity but is a standard conceptual question. Slightly easier than average due to the routine nature and clear structure, though the logarithm function and reasoning component prevent it from being trivial. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State, or use, \(y\)-values of \(\lg 9\), \(\lg 12\), \(\lg 15\), \(\lg 18\) and \(\lg 21\) | B1 | B0 if other \(y\)-values also found (unless not used in trap rule). Allow decimal equivs (0.95, 1.08, 1.18, 1.26, 1.32 or better) |
| Attempt correct trapezium rule, any \(h\), to find area between \(x=4\) and \(x=10\) | M1 | Correct structure required, including correct placing of \(y\)-values. The 'big brackets' must be seen, or implied by later working. Could be implied by stating general rule in terms of \(y_0\) etc. Could use other than 4 strips as long as of equal width. Using \(x\)-values is M0. Can give M1 even if error in \(y\)-values eg using 9, 12, 15, 18, 21 or using incorrect function eg \(\log(2x)+1\). Allow BoD if first or last \(y\)-value incorrect, unless clearly from an incorrect \(x\)-value |
| Use correct \(h\) in recognisable attempt at trap rule | M1 | Must be in attempt at trap rule, not Simpson's rule. Allow if muddle over placing \(y\)-values (but M0 for \(x\)-values). Allow if \(\frac{1}{2}\) missing. Allow other than 4 strips as long as \(h\) is consistent. Allow slips which result in \(x\)-values not equally spaced |
| \(0.5 \times 1.5 \times \{\lg 9 + 2(\lg 12 + \lg 15 + \lg 18) + \lg 21\} = 6.97\); Obtain 6.97, or better | A1 | Allow answers in range [6.970, 6.975] if \(>3\)sf. Answer only is 0/4. Using trap rule on result of integration attempt is 0/4. Using 4 separate trapezia can get full marks — if other than 4 trapezia then mark as above. However, using only one trapezium is 0/4 |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| tops of trapezia are below curve | B1 | B0 for 'the trapezium is below the curve' (ie 'top' not used). Sketch with explanation is fine, even if just arrow and 'gap'. Sketching rectangles/triangles is B0, as is a trapezium that doesn't have both top vertices intended to be on curve. Concave/convex is B0, as is comparing to exact area. B1 for reference to decreasing gradient |
| [1] |
## Question 2:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State, or use, $y$-values of $\lg 9$, $\lg 12$, $\lg 15$, $\lg 18$ and $\lg 21$ | B1 | B0 if other $y$-values also found (unless not used in trap rule). Allow decimal equivs (0.95, 1.08, 1.18, 1.26, 1.32 or better) |
| Attempt correct trapezium rule, any $h$, to find area between $x=4$ and $x=10$ | M1 | Correct structure required, including correct placing of $y$-values. The 'big brackets' must be seen, or implied by later working. Could be implied by stating general rule in terms of $y_0$ etc. Could use other than 4 strips as long as of equal width. Using $x$-values is M0. Can give M1 even if error in $y$-values eg using 9, 12, 15, 18, 21 or using incorrect function eg $\log(2x)+1$. Allow BoD if first or last $y$-value incorrect, unless clearly from an incorrect $x$-value |
| Use correct $h$ in recognisable attempt at trap rule | M1 | Must be in attempt at trap rule, not Simpson's rule. Allow if muddle over placing $y$-values (but M0 for $x$-values). Allow if $\frac{1}{2}$ missing. Allow other than 4 strips as long as $h$ is consistent. Allow slips which result in $x$-values not equally spaced |
| $0.5 \times 1.5 \times \{\lg 9 + 2(\lg 12 + \lg 15 + \lg 18) + \lg 21\} = 6.97$; Obtain 6.97, or better | A1 | Allow answers in range [6.970, 6.975] if $>3$sf. Answer only is 0/4. Using trap rule on result of integration attempt is 0/4. Using 4 separate trapezia can get full marks — if other than 4 trapezia then mark as above. However, using only one trapezium is 0/4 |
| **[4]** | | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| tops of trapezia are below curve | B1 | B0 for 'the trapezium is below the curve' (ie 'top' not used). Sketch with explanation is fine, even if just arrow and 'gap'. Sketching rectangles/triangles is B0, as is a trapezium that doesn't have both top vertices intended to be on curve. Concave/convex is B0, as is comparing to exact area. B1 for reference to decreasing gradient |
| **[1]** | | |
---2\\
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_536_917_1016_577}
The diagram shows the curve $y = \log _ { 10 } ( 2 x + 1 )$.\\
(i) Use the trapezium rule with 4 strips each of width 1.5 to find an approximation to the area of the region bounded by the curve, the $x$-axis and the lines $x = 4$ and $x = 10$. Give your answer correct to 3 significant figures.\\
(ii) Explain why this approximation is an under-estimate.
\hfill \mbox{\textit{OCR C2 2012 Q2 [5]}}