| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Easy -1.2 This is a straightforward C2 question requiring only calculator work to evaluate a function at given points and then apply the trapezium rule formula. No conceptual understanding of integration is needed, just mechanical application of a standard numerical method with all values provided. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 2.646 | 3.630 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1.732,\ 2.058,\ 5.196\) awrt | B1 B1 | One or two correct B1 B0, all correct B1 B1. Accept awrt (less accuracy loses marks). Accept exact answers e.g. \(\sqrt{3}\) at \(x=0\), \(\sqrt{27}\) or \(3\sqrt{3}\) at \(x=2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2} \times 0.5 \times \ldots\) | B1 | |
| \(\ldots\{(1.732 + 5.196) + 2(2.058 + 2.646 + 3.630)\}\) | M1 A1ft | First bracket must contain first and last values only; second bracket must have no additional values. Omitting one value from second bracket is a slip and M mark can be allowed |
| \(= 5.899\) (awrt 5.9, allowed even after minor slips in values) | A1 | Bracketing mistake: \(\frac{1}{2}\times 0.5(1.732+5.196)+2(2.058+2.646+3.630)\) scores B1 M1 A0 A0 unless final answer implies correct calculation. \(x\) values used in brackets: M0 |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1.732,\ 2.058,\ 5.196$ awrt | B1 B1 | One or two correct B1 B0, all correct B1 B1. Accept awrt (less accuracy loses marks). Accept exact answers e.g. $\sqrt{3}$ at $x=0$, $\sqrt{27}$ or $3\sqrt{3}$ at $x=2$ |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2} \times 0.5 \times \ldots$ | B1 | |
| $\ldots\{(1.732 + 5.196) + 2(2.058 + 2.646 + 3.630)\}$ | M1 A1ft | First bracket must contain first and last values only; second bracket must have no additional values. Omitting one value from second bracket is a slip and M mark can be allowed |
| $= 5.899$ (awrt 5.9, allowed even after minor slips in values) | A1 | Bracketing mistake: $\frac{1}{2}\times 0.5(1.732+5.196)+2(2.058+2.646+3.630)$ scores B1 M1 A0 A0 unless final answer implies correct calculation. $x$ values used in brackets: M0 |
---2.
$$y = \sqrt { } \left( 5 ^ { x } + 2 \right)$$
\begin{enumerate}[label=(\alph*)]
\item Complete the table below, giving the values of $y$ to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & & & 2.646 & 3.630 & \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule, with all the values of $y$ from your table, to find an approximation for the value of $\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2008 Q2 [6]}}