| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.3 Part (i) is a straightforward trapezium rule application with 4 strips requiring routine calculation with a calculator-friendly function. Part (ii) tests conceptual understanding of concavity and trapezium rule behavior, which is standard C4 material. The question is slightly easier than average due to its mechanical nature and clear structure, though the function involving √(sin x) adds minor computational complexity. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks |
|---|---|
| B2, 1, 0 | For values 0.4493, 0.6792, 0.9498 (4 d.p. or better soi) [accept truncated to 4 figs after dec point] |
| M1 | Use of the trapezium rule. Trapezium rule formula for 4 strips must be seen, with or without substitution seen. Correct \(h\) must be soi. [accept separate trapezia added] |
| A1 | 0.538 www 3 d.p. only (NB using 1.325 is ww) |
| SC B0: 0.538 without any working as no indication of strips or method used | |
| SC B1: 0.538 with some indication of 4 strips but no values seen | |
| Correct values followed by 0.538 scores B2 B0 | |
| Correct values followed by correct formula for 4 strips, with or without substitution seen, then \(A = 0.538\) scores 4/4. | |
| Correct formula for 4 strips and values of form \(\left(\left(\frac{\pi}{16}\right)^3 + \sqrt{\sin\frac{\pi}{16}}\ldots\right)\) followed by correct answer scores 4/4 (or \(\frac{3}{4}\) with wrong d.p.) | |
| NB Values given in the table to only 3 d.p. give apparently the correct answer, but scores B0, M1 A0 ww |
| Answer | Marks |
|---|---|
| B1 | Not possible to say, e.g. some trapezia are above and some below curve o.e. |
| Need a reason. Must be without further calculation. |
## (i)
B2, 1, 0 | For values 0.4493, 0.6792, 0.9498 (4 d.p. or better soi) [accept truncated to 4 figs after dec point]
M1 | Use of the trapezium rule. Trapezium rule formula for 4 strips must be seen, with or without substitution seen. Correct $h$ must be soi. [accept separate trapezia added]
A1 | 0.538 www 3 d.p. only (NB using 1.325 is ww)
| SC B0: 0.538 without any working as no indication of strips or method used
| SC B1: 0.538 with some indication of 4 strips but no values seen
| Correct values followed by 0.538 scores B2 B0
| Correct values followed by correct formula for 4 strips, with or without substitution seen, then $A = 0.538$ scores 4/4.
| Correct formula for 4 strips and values of form $\left(\left(\frac{\pi}{16}\right)^3 + \sqrt{\sin\frac{\pi}{16}}\ldots\right)$ followed by correct answer scores 4/4 (or $\frac{3}{4}$ with wrong d.p.)
| NB Values given in the table to only 3 d.p. give apparently the correct answer, but scores B0, M1 A0 ww
## (ii)
B1 | Not possible to say, e.g. some trapezia are above and some below curve o.e.
| Need a reason. Must be without further calculation.
---1 Fig. 3 shows the curve $y = x ^ { 3 } + \sqrt { ( \sin x ) }$ for $0 \leqslant x \leqslant \frac { \pi } { 4 }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-1_587_540_393_768}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
(i) Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the $x$-axis and the line $x = \frac { \pi } { 4 }$, giving your answer to 3 decimal places.\\
(ii) Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say.
\hfill \mbox{\textit{OCR MEI C4 Q1 [5]}}