| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | October |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line-Axis Bounded Region |
| Difficulty | Standard +0.3 This is a standard C1/C2 integration question requiring finding intersection points (solving a quadratic after substitution), verifying a root, and calculating area between curves. All techniques are routine: solving equations, substitution, and integration of powers including fractional indices. The multi-part structure and 'exact area' requirement add slight complexity, but this remains a textbook exercise with no novel problem-solving required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\frac{1}{3}x+5=4x^{\frac{1}{2}}-x+5 \Rightarrow x=3x^{\frac{1}{2}}\) | M1 | Sets line = curve and obtains an equation of the form \(\alpha x = \beta x^{\frac{1}{2}}\) or equivalent e.g. \(\alpha x - \beta x^{\frac{1}{2}}=0\) |
| \(x=9\) | A1 | Obtains \(x=9\) from a correct equation |
| \((0,5)\) | B1 | Correct point. Coordinates not necessary and may be seen on the diagram. |
| \((9,8)\) | A1 | Correct point. Coordinates not necessary and may be seen as values and/or on the diagram. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(x=25 \Rightarrow 4(25)^{\frac{1}{2}}-25+5=20-25+5=0\), so \(x\)-coordinate of \(F\) is 25 | B1 | Shows \(F\)'s \(x\) coordinate is 25. Need to see \(4(25)^{\frac{1}{2}}\) evaluated as \(4\times5\) or 20 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\int(4x^{\frac{1}{2}}-x+5)\,dx = \frac{8}{3}x^{\frac{3}{2}}-\frac{x^2}{2}+5x\) or \(\int\left(4x^{\frac{1}{2}}-\frac{4}{3}x\right)dx=\frac{8}{3}x^{\frac{3}{2}}-\frac{2}{3}x^2\) | M1A1 | M1: \(x^n \to x^{n+1}\) seen at least once. A1: Correct integration, simplified or unsimplified. Score as soon as the correct integration is seen. Can be awarded for the curve or their \(\pm(\text{curve}-\text{line})\). Award even if mistakes made in 'simplifying' their \(\pm(\text{curve}-\text{line})\) as long as subsequent integration is correct. |
| \(\left[\frac{8}{3}x^{\frac{3}{2}}-\frac{x^2}{2}+5x\right]_{"9"}^{25} = \frac{875}{6}-\frac{153}{2}\) | M1 | Uses the limits 25 and "9" in their integrated (changed) curve and subtracts either way round. |
| Area of trapezium \(=\frac{(\text{"8"}+5)}{2}\times\text{"9"}=58.5\) or Triangle + Rectangle \(=\text{"5"}\times\text{"9"}+\frac{\text{"5"}\times\text{"9"}}{2}=58.5\) | M1 (process 1 or 2 used) | Correct trapezium area method or may be done as triangle + rectangle or as \(\int_0^{"9"}\left(\frac{1}{3}x+5\right)dx=\left[\frac{1}{6}x^2+5x\right]_0^{"9"}=58.5\). Must be correct integration and correct use of limits. |
| Uses process 1 and process 2 | dM1 | Dependent on the previous M |
| \(R=\frac{208}{3}+58.5=\ldots\) | dM1 | Adds their areas. Dependent on all the previous M marks. |
| \(=\frac{767}{6}\) | A1 | cao |
## Question 10(a):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\frac{1}{3}x+5=4x^{\frac{1}{2}}-x+5 \Rightarrow x=3x^{\frac{1}{2}}$ | M1 | Sets line = curve and obtains an equation of the form $\alpha x = \beta x^{\frac{1}{2}}$ or equivalent e.g. $\alpha x - \beta x^{\frac{1}{2}}=0$ |
| $x=9$ | A1 | Obtains $x=9$ from a correct equation |
| $(0,5)$ | B1 | Correct point. Coordinates not necessary and may be seen on the diagram. |
| $(9,8)$ | A1 | Correct point. Coordinates not necessary and may be seen as values and/or on the diagram. |
---
## Question 10(b):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $x=25 \Rightarrow 4(25)^{\frac{1}{2}}-25+5=20-25+5=0$, so $x$-coordinate of $F$ is 25 | B1 | Shows $F$'s $x$ coordinate is 25. Need to see $4(25)^{\frac{1}{2}}$ evaluated as $4\times5$ or 20 |
---
## Question 10(c):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\int(4x^{\frac{1}{2}}-x+5)\,dx = \frac{8}{3}x^{\frac{3}{2}}-\frac{x^2}{2}+5x$ or $\int\left(4x^{\frac{1}{2}}-\frac{4}{3}x\right)dx=\frac{8}{3}x^{\frac{3}{2}}-\frac{2}{3}x^2$ | M1A1 | M1: $x^n \to x^{n+1}$ seen at least once. A1: Correct integration, simplified or unsimplified. Score as soon as the correct integration is seen. Can be awarded for the curve or their $\pm(\text{curve}-\text{line})$. Award even if mistakes made in 'simplifying' their $\pm(\text{curve}-\text{line})$ as long as subsequent integration is correct. |
| $\left[\frac{8}{3}x^{\frac{3}{2}}-\frac{x^2}{2}+5x\right]_{"9"}^{25} = \frac{875}{6}-\frac{153}{2}$ | M1 | Uses the limits 25 and "9" in their integrated (changed) curve and subtracts either way round. |
| Area of trapezium $=\frac{(\text{"8"}+5)}{2}\times\text{"9"}=58.5$ or Triangle + Rectangle $=\text{"5"}\times\text{"9"}+\frac{\text{"5"}\times\text{"9"}}{2}=58.5$ | M1 (process 1 or 2 used) | Correct trapezium area method or may be done as triangle + rectangle or as $\int_0^{"9"}\left(\frac{1}{3}x+5\right)dx=\left[\frac{1}{6}x^2+5x\right]_0^{"9"}=58.5$. Must be correct integration and correct use of limits. |
| Uses process 1 **and** process 2 | dM1 | Dependent on the previous M |
| $R=\frac{208}{3}+58.5=\ldots$ | dM1 | Adds their areas. Dependent on all the previous M marks. |
| $=\frac{767}{6}$ | A1 | cao |10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-28_826_1632_264_153}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The finite region $R$, which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line $l$ with equation $y = \frac { 1 } { 3 } x + 5$ and the curve $C$ with equation $y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0$
The line $l$ meets the curve $C$ at the point $D$ on the $y$-axis and at the point $E$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Use algebra to find the coordinates of the points $D$ and $E$.
The curve $C$ crosses the $x$-axis at the point $F$.
\item Verify that the $x$ coordinate of $F$ is 25
\item Use algebraic integration to find the exact area of the shaded region $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2018 Q10 [11]}}