| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with clearly provided data points. Students must recognize they need to find the area between two curves by applying the trapezium rule to the difference in y-values. While it requires careful arithmetic with the given coordinates, it's a standard C2 exercise with no conceptual challenges beyond routine application of the formula. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.09f Trapezium rule: numerical integration |
| Upper surface | Lower surface | ||
| \(x\) | \(y\) | \(x\) | \(y\) |
| 0 | 0 | 0 | 0 |
| 4 | 1.45 | 4 | - 0.85 |
| 8 | 1.56 | 8 | - 0.76 |
| 12 | 1.27 | 12 | - 0.55 |
| 16 | 1.04 | 16 | - 0.30 |
| 20 | 0 | 20 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([\cos A =] \frac{20^2 + 13^2 - 8^2}{2 \times 13 \times 20}\) | M1* | or \(8^2 = 20^2 + 13^2 - 2 \times 13 \times 20 \times \cos A\) |
| \([\cos A =] \frac{505}{520}\) oe soi | A1 | or 0.971 to 0.9712 |
| \(A = 13.79\) to \(13.8°\) or \(14°\) | A1 | or 0.24077 to 0.241 or 0.24 (radians); allow B3 if given to 3sf or more unsupported; or 15.32 (grad) |
| \([\text{Area} =] \frac{1}{2} \times 20 \times 13 \times \sin(\text{their } A)\) | M1dep* | or M1 for eg \(\frac{1}{2} \times 20 \times 8 \times \sin 22.8\), as long as angle calculated correctly from their \(A\); or \(\sqrt{\frac{41}{2}(\frac{41}{2}-8)(\frac{41}{2}-13)(\frac{41}{2}-20)}\); NB \(13\sin A = 3.099899192\) if \(\frac{1}{2} \times b \times h\) used |
| 30.99 to 31.01 isw or \(\frac{5\sqrt{615}}{4}\) oe isw | A1 | allow B2 for unsupported answer within range |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 4\) soi | B1 | |
| \(\frac{\text{their }4}{2} \times (0 + 0 + 2(1.45+1.56+1.27+1.04))\) or \(\frac{\text{their }4}{2} \times (0 + 0 + 2(\pm0.85 \pm 0.76 \pm 0.55 \pm 0.30))\) | M1* | shape of formula correct with 2, 3 or 4 \(y\)-values in inner bracket with their \(h\); allow recovery from bracket errors; M0 if any non-zero \(x\)-values used or if \(y\)-values used twice; eg \(\frac{\text{their }4}{2} \times \{1.45 + 1.04 + 2(1.56+1.27)\}\); signs must be consistent in 2nd alternative |
| B1 | all \(y\)-values correctly placed with their \(h\), condone omission of zeros and/or omission of outer brackets | |
| either 21.28 or \(\pm 9.84\) | A1 | or B1+B3* if area of 2 triangles and 3 trapezia calculated to give correct answer; NB \(2.9+6.02+5.66+4.62+2.08\) or \(\pm1.7 \pm 3.22 \pm 2.62 \pm 1.7 \pm 0.60\) with consistent signs throughout |
| their \(21.28 +\) their \(9.84\) | M1dep* | |
| 31.12 | A1 | ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 20 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 4\) soi | B1 | |
| attempt to find all \(y\)-values | M1 | \(y_{\text{upper}} - y_{\text{lower}}\); M0 if values are added to obtain 0.60, 0.80 etc |
| 2.3, 2.32, 1.82, 1.34 | A1 | all \(y\)-values correct |
| \(\frac{\text{their }4}{2} \times (0 + 0 + 2(2.3+2.32+1.82+1.34))\) | M1 | shape of formula correct with 2, 3 or 4 of their \(y\)-values in inner bracket with their \(h\); allow recovery from bracket errors; M0 if any non-zero \(x\)-values used or if \(y\)-values used twice; eg \(\frac{1}{2} \times 4 \times \{2.3 + 1.34 + 2(2.32+1.82)\}\) |
| B1FT | all their \(y\)-values correctly placed, condone omission of zeros and/or omission of outer brackets | |
| 31.12 | A1 | ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 20; or B1M1A1+B3 if area of 2 triangles and 3 trapezia calculated to give correct answer; NB \(4.6+9.24+8.28+6.32+2.68\) |
## Question 2(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[\cos A =] \frac{20^2 + 13^2 - 8^2}{2 \times 13 \times 20}$ | M1* | or $8^2 = 20^2 + 13^2 - 2 \times 13 \times 20 \times \cos A$ |
| $[\cos A =] \frac{505}{520}$ oe soi | A1 | or 0.971 to 0.9712 |
| $A = 13.79$ to $13.8°$ or $14°$ | A1 | or 0.24077 to 0.241 or 0.24 (radians); allow B3 if given to 3sf or more unsupported; or 15.32 (grad) |
| $[\text{Area} =] \frac{1}{2} \times 20 \times 13 \times \sin(\text{their } A)$ | M1dep* | or M1 for eg $\frac{1}{2} \times 20 \times 8 \times \sin 22.8$, as long as angle calculated correctly from their $A$; or $\sqrt{\frac{41}{2}(\frac{41}{2}-8)(\frac{41}{2}-13)(\frac{41}{2}-20)}$; NB $13\sin A = 3.099899192$ if $\frac{1}{2} \times b \times h$ used |
| 30.99 to 31.01 isw or $\frac{5\sqrt{615}}{4}$ oe isw | A1 | allow B2 for unsupported answer within range |
---
## Question 2(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 4$ soi | B1 | |
| $\frac{\text{their }4}{2} \times (0 + 0 + 2(1.45+1.56+1.27+1.04))$ or $\frac{\text{their }4}{2} \times (0 + 0 + 2(\pm0.85 \pm 0.76 \pm 0.55 \pm 0.30))$ | M1* | shape of formula correct with 2, 3 or 4 $y$-values in inner bracket with their $h$; allow recovery from bracket errors; M0 if any non-zero $x$-values used or if $y$-values used twice; eg $\frac{\text{their }4}{2} \times \{1.45 + 1.04 + 2(1.56+1.27)\}$; signs must be consistent in 2nd alternative |
| | B1 | all $y$-values correctly placed with their $h$, condone omission of zeros and/or omission of outer brackets |
| either 21.28 or $\pm 9.84$ | A1 | or B1+B3* if area of 2 triangles and 3 trapezia calculated to give correct answer; NB $2.9+6.02+5.66+4.62+2.08$ or $\pm1.7 \pm 3.22 \pm 2.62 \pm 1.7 \pm 0.60$ with consistent signs throughout |
| their $21.28 +$ their $9.84$ | M1dep* | |
| 31.12 | A1 | ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 20 |
**Alternatively:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 4$ soi | B1 | |
| attempt to find all $y$-values | M1 | $y_{\text{upper}} - y_{\text{lower}}$; M0 if values are added to obtain 0.60, 0.80 etc |
| 2.3, 2.32, 1.82, 1.34 | A1 | all $y$-values correct |
| $\frac{\text{their }4}{2} \times (0 + 0 + 2(2.3+2.32+1.82+1.34))$ | M1 | shape of formula correct with 2, 3 or 4 of their $y$-values in inner bracket with their $h$; allow recovery from bracket errors; M0 if any non-zero $x$-values used or if $y$-values used twice; eg $\frac{1}{2} \times 4 \times \{2.3 + 1.34 + 2(2.32+1.82)\}$ |
| | B1FT | all their $y$-values correctly placed, condone omission of zeros and/or omission of outer brackets |
| 31.12 | A1 | ignore subsequent rounding, but A0 if answer spoiled by eg multiplication by 20; or B1M1A1+B3 if area of 2 triangles and 3 trapezia calculated to give correct answer; NB $4.6+9.24+8.28+6.32+2.68$ |
---2
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4dcf71fc-2585-4247-a21d-8b14f11ce0d0-1_239_1478_439_335}
\captionsetup{labelformat=empty}
\caption{Fig. 9.1}
\end{center}
\end{figure}
(i) Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.\\
(ii) Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of $x$ and $y$ representing one centimetre.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4dcf71fc-2585-4247-a21d-8b14f11ce0d0-1_415_1662_1081_240}
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
\end{center}
\end{figure}
Here are some of the coordinates that Jean used to draw the new cross-section.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{Upper surface} & \multicolumn{2}{c|}{Lower surface} \\
\hline
$x$ & $y$ & $x$ & $y$ \\
\hline
0 & 0 & 0 & 0 \\
\hline
4 & 1.45 & 4 & - 0.85 \\
\hline
8 & 1.56 & 8 & - 0.76 \\
\hline
12 & 1.27 & 12 & - 0.55 \\
\hline
16 & 1.04 & 16 & - 0.30 \\
\hline
20 & 0 & 20 & 0 \\
\hline
\end{tabular}
\end{center}
Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
\hfill \mbox{\textit{OCR MEI C2 Q2 [11]}}