| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward C2 trapezium rule question requiring routine calculation (filling in two table values, applying the standard formula) and basic reasoning about concavity. The conceptual demand is minimal—students simply need to recognize that trapeziums overestimate for concave curves—making this easier than average A-level questions. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| \(\sqrt { } \left( 2 ^ { x } + 1 \right)\) | 1.414 | 1.554 | 1.732 | 1.957 | 3 |
| Answer | Marks |
|---|---|
| \(x = 2\) gives \(2.236\) (allow AWRT) Accept \(\sqrt{5}\) | B1 |
| \(x = 2.5\) gives \(2.580\) (allow AWRT) Accept \(2.58\) | B1 |
| Answer | Marks |
|---|---|
| \(\left(\frac{1}{2} \times \frac{1}{2}\right) \quad [(1.414 + 3) + 2(1.554 + 1.732 + 1.957 + 2.236 + 2.580)]\) | B1, [M1A1ft] |
| \(= 6.133\) (AWRT 6.13, even following minor slips) | A1 |
| Answer | Marks |
|---|---|
| Overestimate | B1 |
| 'Since the trapezia lie above the curve', or an equivalent explanation, or sketch of (one or more) trapezium above the curve on a diagram (or on the given diagram, in which case there should be reference to this). (Note that there must be some reference to a trapezium or trapezium in the explanation or diagram). | dB1 |
**Part (a)**
$x = 2$ gives $2.236$ (allow AWRT) Accept $\sqrt{5}$ | B1 |
$x = 2.5$ gives $2.580$ (allow AWRT) Accept $2.58$ | B1 |
**Part (b)**
$\left(\frac{1}{2} \times \frac{1}{2}\right) \quad [(1.414 + 3) + 2(1.554 + 1.732 + 1.957 + 2.236 + 2.580)]$ | B1, [M1A1ft] |
$= 6.133$ (AWRT 6.13, even following minor slips) | A1 |
**Part (c)**
Overestimate | B1 |
'Since the trapezia lie above the curve', or an equivalent explanation, or sketch of (one or more) trapezium above the curve on a diagram (or on the given diagram, in which case there should be reference to this). (Note that there must be some reference to a trapezium or trapezium in the explanation or diagram). | dB1 |
**Guidance:**
**Part (b):**
B1 for $\frac{1}{2} \times \frac{1}{2}$ or equivalent.
For the M mark, the first bracket must contain the 'first and last' values, and the second bracket (which must be multiplied by 2) must have no additional values. If the only mistake is to omit one of the values from the second bracket, this can be considered as a slip and the M mark can be allowed.
Bracketing mistake: i.e. $\left(\frac{1}{2} \times \frac{1}{2}\right)(1.414 + 3) + 2(1.554 + 1.732 + 1.957 + 2.236 + 2.580)$ scores B1 M1 A0 A0 unless the final answer implies that the calculation has been done correctly (then full marks can be given).
**Alternative:** Separate trapezia may be used, and this can be marked equivalently.
$$\left[\frac{1}{4}(1.414 + 1.554) + \frac{1}{4}(1.554 + 1.732) + \ldots\ldots\ldots + \frac{1}{4}(2.580 + 3)\right]$$
$1^{\text{st}}$ A1ft for correct expression, ft their 2.236 and their 2.580
**Part (c):**
$1^{\text{st}}$ B1 for 'overestimate', ignoring earlier mistakes and ignoring any reasons given.
$2^{\text{nd}}$ B1 is dependent upon the $1^{\text{st}}$ B1 (overestimate).
---4. (a) Complete the table below, giving values of $\sqrt { } \left( 2 ^ { x } + 1 \right)$ to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
$\sqrt { } \left( 2 ^ { x } + 1 \right)$ & 1.414 & 1.554 & 1.732 & 1.957 & & & 3 \\
\hline
\end{tabular}
\end{center}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-05_653_595_616_676}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the region $R$ which is bounded by the curve with equation $y = \sqrt { } \left( 2 ^ { x } + 1 \right)$, the $x$-axis and the lines $x = 0$ and $x = 3$\\
(b) Use the trapezium rule, with all the values from your table, to find an approximation for the area of $R$.\\
(c) By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of $R$.\\
\hfill \mbox{\textit{Edexcel C2 2009 Q4 [8]}}