| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule accuracy improvement explanation |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with clearly specified parameters (4 strips, width 1.5). Part (i) requires only substitution into the standard formula with a simple function evaluation. Part (ii) tests basic understanding that more strips improve accuracy—a standard bookwork response. Below average difficulty as it's purely procedural with no problem-solving or conceptual challenge. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State the 5 correct \(y\)-values, and no others | B1 | B0 if other \(y\)-values also found (unless not used); allow unsimplified; allow decimal equivs |
| \(0.5 \times 1.5 \times (\sqrt{7} + 2(\sqrt{10} + \sqrt{13} + \sqrt{16}) + \sqrt{19})\) | M1* | Correct placing of \(y\)-values required; \(y\)-values may not be correct but must be from attempt at correct \(x\)-values (allow 7, 10 etc, ie no \(\sqrt{}\)); 'big brackets' must be seen or implied; M0 for just one strip |
| M1d* | Use \(k = 0.5 \times 1.5\); or \(k = 0.5 \times h\) where \(h\) is consistent with number of strips used | |
| \(= 21.4\) | A1 | Obtain \(21.4\) or better; allow answers in range \([21.40, 21.41]\) if \(>3\)sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use more strips / narrower strips | B1 | No need to explicitly state it is over the same interval; ignore any reference to under/over-estimate; penalise contradictory statements e.g. "use more strips, which are wider" |
## Question 2:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State the 5 correct $y$-values, and no others | B1 | B0 if other $y$-values also found (unless not used); allow unsimplified; allow decimal equivs |
| $0.5 \times 1.5 \times (\sqrt{7} + 2(\sqrt{10} + \sqrt{13} + \sqrt{16}) + \sqrt{19})$ | M1* | Correct placing of $y$-values required; $y$-values may not be correct but must be from attempt at correct $x$-values (allow 7, 10 etc, ie no $\sqrt{}$); 'big brackets' must be seen or implied; M0 for just one strip |
| | M1d* | Use $k = 0.5 \times 1.5$; or $k = 0.5 \times h$ where $h$ is consistent with number of strips used |
| $= 21.4$ | A1 | Obtain $21.4$ or better; allow answers in range $[21.40, 21.41]$ if $>3$sf |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use more strips / narrower strips | B1 | No need to explicitly state it is over the same interval; ignore any reference to under/over-estimate; penalise contradictory statements e.g. "use more strips, which are wider" |
---2 (i) Use the trapezium rule, with 4 strips each of width 1.5, to estimate the value of
$$\int _ { 4 } ^ { 10 } \sqrt { 2 x - 1 } \mathrm {~d} x ,$$
giving your answer correct to 3 significant figures.\\
(ii) Explain how the trapezium rule could be used to obtain a more accurate estimate.
\hfill \mbox{\textit{OCR C2 2015 Q2 [5]}}