| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.8 Part (a) requires standard differentiation and solving dy/dx=0 (routine calculus). Part (b) applies the trapezium rule formula with given ordinates—a straightforward numerical method requiring only substitution and arithmetic. Both parts are below-average difficulty for A-level, being direct applications of standard C2 techniques with no problem-solving insight required. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dy}{dx} = 4 - x^{-2}\) | M1 A1 | |
| For minimum, \(4 - x^{-2} = 0\) | M1 | |
| \(x^2 = \frac{1}{4}\) | M1 | |
| \(x > 0 \quad \therefore x = \frac{1}{2} \quad \therefore \left(\frac{1}{2}, 4\right)\) | A2 | |
| (b) | \(x\) | \(1\) |
| \(4x + x^{-1}\) | \(5\) | \(8\frac{1}{2}\) |
| Area | \(= \frac{1}{2} \times 1 \times [5 + 16\frac{1}{4} + 2(8\frac{1}{2} + 12\frac{1}{3})]\) | B1 M1 A1 |
| \(= 31.5\) (3sf) | A1 | (10 marks) |
**(a)** $\frac{dy}{dx} = 4 - x^{-2}$ | M1 A1 |
For minimum, $4 - x^{-2} = 0$ | M1 |
$x^2 = \frac{1}{4}$ | M1 |
$x > 0 \quad \therefore x = \frac{1}{2} \quad \therefore \left(\frac{1}{2}, 4\right)$ | A2 |
**(b)** | $x$ | $1$ | $2$ | $3$ | $4$ |
| | $4x + x^{-1}$ | $5$ | $8\frac{1}{2}$ | $12\frac{1}{3}$ | $16\frac{1}{4}$ | B1 |
| Area | $= \frac{1}{2} \times 1 \times [5 + 16\frac{1}{4} + 2(8\frac{1}{2} + 12\frac{1}{3})]$ | B1 M1 A1 |
| | $= 31.5$ (3sf) | A1 | **(10 marks)**6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3e44996a-4635-46f6-bd45-7799a8c49463-3_589_894_248_397}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with equation $y = 4 x + \frac { 1 } { x } , x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the minimum point of the curve.
The shaded region $R$ is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 4$.
\item Use the trapezium rule with three intervals of equal width to estimate the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [10]}}