| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line Intersection Area |
| Difficulty | Standard +0.3 This is a standard curve-line intersection problem requiring finding intersection points by solving a polynomial equation (which factors nicely), then computing area between curves using definite integration. The fractional power (3/2) adds minor complexity but integration is straightforward. Slightly easier than average as it's a routine textbook exercise with clear steps and no novel insight required. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Set \(4x+3 = 2x^{3/2} - 2x + 3\), collect terms to form \(Ax = Bx^{3/2}\), \(A,B \neq 0\); so \(6x = 2x^{3/2}\) | M1 | Must be simplified before M1A1 can be scored |
| \(x = 9\) | A1 | May ignore any reference to \(x=0\) |
| One of \((0,3)\), \((9,39)\) | B1 | |
| Both \((0,3)\) and \((9,39)\) | B1 | Just the answers with no equation scores 0,0,1,1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int 2x^{3/2} - 2x + 3\,dx = \frac{2}{5/2}x^{5/2} - x^2 + 3x\) | M1 A1 | Attempted integration; accept \(x^n \to x^{n+1}\) on any term. Correct integration, may be unsimplified |
| Uses upper limit and subtracts lower limit (often 0) to obtain an area | M1 | If lower limit is 0, subtraction need not be shown |
| Area of trapezium \(= \frac{1}{2}(3+39)\times 9 = 189\) OR \(2\times 9^2 + 3\times 9\) | B1 | May be implied by correct final answer of 48.6 |
| Area = Area of trapezium \(-\) Area beneath curve | M1 | |
| \(= 189 - 140.4 = 48.6\) | A1 cso | Note: \(-48.6\) is A0 even if candidate loses the sign |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\pm\!\left(\int 6x - 2x^{3/2}\,dx = 3x^2 - \frac{2}{5/2}x^{5/2}\right)\) | M1 A1 | |
| Uses upper limit and subtracts lower limit (often 0) | M1 | |
| Implied by final answer of 48.6 or \(-48.6\) | B1 | |
| Implied by subtraction in the integration | M1 | |
| \(= 48.6\) | A1 cso |
## Question 14:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Set $4x+3 = 2x^{3/2} - 2x + 3$, collect terms to form $Ax = Bx^{3/2}$, $A,B \neq 0$; so $6x = 2x^{3/2}$ | M1 | Must be simplified before M1A1 can be scored |
| $x = 9$ | A1 | May ignore any reference to $x=0$ |
| One of $(0,3)$, $(9,39)$ | B1 | |
| Both $(0,3)$ and $(9,39)$ | B1 | Just the answers with no equation scores 0,0,1,1 |
### Part (b):
**Method 1:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int 2x^{3/2} - 2x + 3\,dx = \frac{2}{5/2}x^{5/2} - x^2 + 3x$ | M1 A1 | Attempted integration; accept $x^n \to x^{n+1}$ on any term. Correct integration, may be unsimplified |
| Uses upper limit and subtracts lower limit (often 0) to obtain an area | M1 | If lower limit is 0, subtraction need not be shown |
| Area of trapezium $= \frac{1}{2}(3+39)\times 9 = 189$ OR $2\times 9^2 + 3\times 9$ | B1 | May be implied by correct final answer of 48.6 |
| Area = Area of trapezium $-$ Area beneath curve | M1 | |
| $= 189 - 140.4 = 48.6$ | A1 cso | Note: $-48.6$ is A0 even if candidate loses the sign |
**Method 2:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\pm\!\left(\int 6x - 2x^{3/2}\,dx = 3x^2 - \frac{2}{5/2}x^{5/2}\right)$ | M1 A1 | |
| Uses upper limit and subtracts lower limit (often 0) | M1 | |
| Implied by final answer of 48.6 or $-48.6$ | B1 | |
| Implied by subtraction in the integration | M1 | |
| $= 48.6$ | A1 cso | |
---14.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-21_641_920_260_568}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The finite region $R$, which is shown shaded in Figure 3, is bounded by the straight line $l$ with equation $y = 4 x + 3$ and the curve $C$ with equation $y = 2 x ^ { \frac { 3 } { 2 } } - 2 x + 3 , x \geqslant 0$
The line $l$ meets the curve $C$ at the point $A$ on the $y$-axis and $l$ meets $C$ again at the point $B$, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Use algebra to find the coordinates of $A$ and $B$.
\item Use integration to find the area of the shaded region $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2015 Q14 [10]}}