| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Trapezium Rule Approximation with Area |
| Difficulty | Moderate -0.3 This is a standard P2 trapezium rule question with straightforward integration and area subtraction. Part (a) is routine application of the trapezium rule formula with given values, part (b) is basic integration of power functions, and part (c) requires simple subtraction of areas. The multi-part structure and combination of numerical/analytical methods is typical for P2, but no step requires novel insight or complex problem-solving. |
| 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 |
| \(x\) | 2.5 | 2.75 | 3 | 3.25 | 3.5 | 3.75 | 4 |
| \(y\) | 5.453 | 7.764 | 9.375 | 9.964 | 9.367 | 7.626 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = \frac{1}{4}\) | B1 | \(h=0.25\) or equivalent, e.g. \(\frac{1}{2}\times\frac{1}{4}\) outside brackets |
| \(A \approx \left(\frac{1}{2}\times\frac{1}{4}\right)\{5.453 + 5 + 2(7.764 + \ldots + 7.626)\}\) | M1 | Correct bracket structure for trapezium rule: \(\frac{h}{2}\)(first + last + 2(sum of remaining)) |
| \(A \approx \left(\frac{1}{2}\times\frac{1}{4}\right)\{5.453 + 5 + 2(7.764 + 9.375 + 9.964 + 9.367 + 7.626)\}\) | A1 | All 7 \(y\) values used correctly |
| \(= \text{awrt } 12.33\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int x^{\frac{3}{2}} - 3x + 9\, dx = kx^{\frac{5}{2}} - lx^2 + mx\) | M1 | Attempts to integrate; power increases by 1 in at least two terms |
| \(\int x^{\frac{3}{2}} - 3x + 9\, dx = \frac{2}{5}x^{\frac{5}{2}} - \frac{3}{2}x^2 + 9x\,(+c)\) | A1A1 | A1: any two terms correct; A1: fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left[\frac{2}{5}x^{\frac{5}{2}} - \frac{3}{2}x^2 + 9x\right]_{\frac{5}{2}}^{4} = \left(\frac{2}{5}\times 4^{\frac{5}{2}} - \frac{3}{2}\times 4^2 + 9\times 4\right) - \left(\frac{2}{5}\left(\frac{5}{2}\right)^{\frac{5}{2}} - \frac{3}{2}\left(\frac{5}{2}\right)^2 + 9\left(\frac{5}{2}\right)\right)\) | M1 | Applies limits 2.5 and 4; limits may be either way round |
| \((= 24.8 - 17.077\ldots = 7.722\ldots)\) | ||
| Area \(R =\) "12.33" \(-\) "7.722" \(= \ldots\) | M1 | Subtracts area under \(C_2\) from answer to (a); must subtract a positive number |
| \(= \text{awrt } 4.6\) | A1 | Following correct answer to (a) |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = \frac{1}{4}$ | B1 | $h=0.25$ or equivalent, e.g. $\frac{1}{2}\times\frac{1}{4}$ outside brackets |
| $A \approx \left(\frac{1}{2}\times\frac{1}{4}\right)\{5.453 + 5 + 2(7.764 + \ldots + 7.626)\}$ | M1 | Correct bracket structure for trapezium rule: $\frac{h}{2}$(first + last + 2(sum of remaining)) |
| $A \approx \left(\frac{1}{2}\times\frac{1}{4}\right)\{5.453 + 5 + 2(7.764 + 9.375 + 9.964 + 9.367 + 7.626)\}$ | A1 | All 7 $y$ values used correctly |
| $= \text{awrt } 12.33$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int x^{\frac{3}{2}} - 3x + 9\, dx = kx^{\frac{5}{2}} - lx^2 + mx$ | M1 | Attempts to integrate; power increases by 1 in at least two terms |
| $\int x^{\frac{3}{2}} - 3x + 9\, dx = \frac{2}{5}x^{\frac{5}{2}} - \frac{3}{2}x^2 + 9x\,(+c)$ | A1A1 | A1: any two terms correct; A1: fully correct |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left[\frac{2}{5}x^{\frac{5}{2}} - \frac{3}{2}x^2 + 9x\right]_{\frac{5}{2}}^{4} = \left(\frac{2}{5}\times 4^{\frac{5}{2}} - \frac{3}{2}\times 4^2 + 9\times 4\right) - \left(\frac{2}{5}\left(\frac{5}{2}\right)^{\frac{5}{2}} - \frac{3}{2}\left(\frac{5}{2}\right)^2 + 9\left(\frac{5}{2}\right)\right)$ | M1 | Applies limits 2.5 and 4; limits may be either way round |
| $(= 24.8 - 17.077\ldots = 7.722\ldots)$ | | |
| Area $R =$ "12.33" $-$ "7.722" $= \ldots$ | M1 | Subtracts area under $C_2$ from answer to (a); must subtract a positive number |
| $= \text{awrt } 4.6$ | A1 | Following correct answer to (a) |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-14_547_1084_269_420}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the graph of the curves $C _ { 1 }$ and $C _ { 2 }$\\
The curves intersect when $x = 2.5$ and when $x = 4$
A table of values for some points on the curve $C _ { 1 }$ is shown below, with $y$ values given to 3 decimal places as appropriate.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 2.5 & 2.75 & 3 & 3.25 & 3.5 & 3.75 & 4 \\
\hline
$y$ & 5.453 & 7.764 & 9.375 & 9.964 & 9.367 & 7.626 & 5 \\
\hline
\end{tabular}
\end{center}
Using the trapezium rule with all the values of $y$ in the table,
\begin{enumerate}[label=(\alph*)]
\item find, to 2 decimal places, an estimate for the area bounded by the curve $C _ { 1 }$, the line with equation $x = 2.5$, the $x$-axis and the line with equation $x = 4$
The curve $C _ { 2 }$ has equation
$$y = x ^ { \frac { 3 } { 2 } } - 3 x + 9 \quad x > 0$$
\item Find $\int \left( x ^ { \frac { 3 } { 2 } } - 3 x + 9 \right) \mathrm { d } x$
The region $R$, shown shaded in Figure 2, is bounded by the curves $C _ { 1 }$ and $C _ { 2 }$
\item Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region $R$.\\
(3)
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q5 [10]}}