| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule accuracy improvement explanation |
| Difficulty | Moderate -0.8 This is a straightforward trapezium rule question with standard follow-up parts. Part (a) requires direct application of the formula with given values, part (b) tests understanding of improving accuracy (use more strips), and part (c) involves simple scaling/translation of integrals using linearity properties. All techniques are routine A-level procedures with no problem-solving or novel insight required. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 0.5 | 0.6742 | 0.8284 | 0.9686 | 1.0981 |
| Answer | Marks |
|---|---|
| B1 | Outside brackets \(0.5\) or \(\frac{0.5}{2}\) or \(0.25\) or \(\frac{1}{4}\) |
| M1 | For structure of trapezium rule. No errors are allowed, e.g. an omission of a \(y\)-ordinate or an extra \(y\)-ordinate or a repeated \(y\)-ordinate. |
| Answer | Marks |
|---|---|
| A1 | Correct method leading to a correct answer only of \(1.635\) |
| Answer | Marks |
|---|---|
| B1 | Any valid reason, for example: |
| Answer | Marks |
|---|---|
| B1ft | \(\int_1^3 \frac{5x}{1+x} \, dx = 5(\text{``}1.635\text{''}) = 8.175\) or a value which is \(5 \times\) their answer to part (a) |
| Answer | Marks |
|---|---|
| B1ft | \(\int_1^3 \frac{x}{1+x} \, dx = 6(2) + (\text{``}1.635\text{''}) = 13.635\) or a value which is \(12 +\) their answer to part (a) |
# Question 1
## 1(a)
B1 | Outside brackets $0.5$ or $\frac{0.5}{2}$ or $0.25$ or $\frac{1}{4}$
M1 | For structure of trapezium rule. No errors are allowed, e.g. an omission of a $y$-ordinate or an extra $y$-ordinate or a repeated $y$-ordinate.
Area$(R) = 0.5[0.5 + 2(0.6742 + 0.8284 + 0.9686) + 1.0981] = 6.5405 = 1.635125 = 1.635$ (3 dp)
A1 | Correct method leading to a correct answer only of $1.635$
(3 marks)
## 1(b)
B1 | Any valid reason, for example:
- Increase the number of strips
- Decrease the width of the strips
- Use more trapezia between $x = 1$ and $x = 3$
(1 mark)
## 1(c)(i)
B1ft | $\int_1^3 \frac{5x}{1+x} \, dx = 5(\text{``}1.635\text{''}) = 8.175$ or a value which is $5 \times$ their answer to part (a)
Note: Allow B1ft for $8.176$ (to 3 dp) which is found from $5(1.631 25) = 8.175625$
Note: Do not allow an answer of $8.1886\ldots$ which is found directly from integration
(1 mark)
## 1(c)(ii)
B1ft | $\int_1^3 \frac{x}{1+x} \, dx = 6(2) + (\text{``}1.635\text{''}) = 13.635$ or a value which is $12 +$ their answer to part (a)
Note: Do not allow an answer of $13.6377\ldots$ which is found directly from integration
(1 mark)
(6 marks total)1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-02_659_853_349_607}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0$\\
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the line with equation $x = 1$, the $x$-axis and the line with equation $x = 3$
The table below shows corresponding values of $x$ and $y$ for $y = \frac { x } { 1 + \sqrt { } x }$
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
$y$ & 0.5 & 0.6742 & 0.8284 & 0.9686 & 1.0981 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for the area of $R$, giving your answer to 3 decimal places.
\item Explain how the trapezium rule can be used to give a better approximation for the area of $R$.
\item Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
\begin{enumerate}[label=(\roman*)]
\item $\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x$
\item $\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 Q1 [6]}}