| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a standard C2 trapezium rule and integration question with straightforward function evaluation and routine application of formulas. While it requires multiple steps (substitution, trapezium rule application, integration), all techniques are basic and commonly practiced. The integration is direct with no algebraic manipulation needed, making it slightly easier than average for A-level. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| \(y\) | 16.5 | 7.361 | 1.278 | 0.556 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 4\) (at \(x=2\)) | B1 | Accept 4.000 |
| \(y = 2.31\) (at \(x=2.5\)) | B1 | Accept 2.310 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h = 0.5\) or \(\frac{1}{2}\) | B1 | Need 0.25 or \(\frac{1}{2}\) of 0.5 |
| \(\frac{1}{2} \times 0.5 \times \{(16.5+0) + 2(7.361+4+2.31+1.278+0.556)\}\) | M1A1ft | First bracket: first \(y\) + last \(y\); second bracket: no additional \(y\) values |
| \(= 11.88\) | A1 | Accept 11.8775, 11.878, or 11.88 only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_1^4 \frac{16}{x^2} - \frac{x}{2} + 1\, dx\) → attempt to integrate | M1 | Power of \(x\) increased by 1 |
| \(\left[-\frac{16}{x} - \frac{x^2}{4} + x\right]_1^4\) | A1, A1 | Two correct terms (A1), all three correct unsimplified (A1) |
| \(= [-4-4+4] - [-16 - \frac{1}{4} + 1]\) | dM1 | Uses limits 4 and 1, subtracts |
| \(= 11\frac{1}{4}\) or equivalent | A1 | Accept 11.25 or \(\frac{45}{4}\); penalise negative final answer |
## Question 6:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 4$ (at $x=2$) | B1 | Accept 4.000 |
| $y = 2.31$ (at $x=2.5$) | B1 | Accept 2.310 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h = 0.5$ or $\frac{1}{2}$ | B1 | Need 0.25 or $\frac{1}{2}$ of 0.5 |
| $\frac{1}{2} \times 0.5 \times \{(16.5+0) + 2(7.361+4+2.31+1.278+0.556)\}$ | M1A1ft | First bracket: first $y$ + last $y$; second bracket: no additional $y$ values |
| $= 11.88$ | A1 | Accept 11.8775, 11.878, or 11.88 only |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_1^4 \frac{16}{x^2} - \frac{x}{2} + 1\, dx$ → attempt to integrate | M1 | Power of $x$ increased by 1 |
| $\left[-\frac{16}{x} - \frac{x^2}{4} + x\right]_1^4$ | A1, A1 | Two correct terms (A1), all three correct unsimplified (A1) |
| $= [-4-4+4] - [-16 - \frac{1}{4} + 1]$ | dM1 | Uses limits 4 and 1, subtracts |
| $= 11\frac{1}{4}$ or equivalent | A1 | Accept 11.25 or $\frac{45}{4}$; penalise negative final answer |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-07_611_1326_280_310}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the graph of the curve with equation
$$y = \frac { 16 } { x ^ { 2 } } - \frac { x } { 2 } + 1 , \quad x > 0$$
The finite region $R$, bounded by the lines $x = 1$, the $x$-axis and the curve, is shown shaded in Figure 1. The curve crosses the $x$-axis at the point $( 4,0 )$.
\begin{enumerate}[label=(\alph*)]
\item Complete the table with the values of $y$ corresponding to $x = 2$ and 2.5
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\
\hline
$y$ & 16.5 & 7.361 & & & 1.278 & 0.556 & 0 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule with all the values in the completed table to find an approximate value for the area of $R$, giving your answer to 2 decimal places.
\item Use integration to find the exact value for the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2012 Q6 [11]}}