| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Two Curves Intersection Area |
| Difficulty | Moderate -0.3 This is a standard P2/C2 area between curves question with straightforward steps: verify a point by substitution, solve a quartic that factors nicely to find intersection points, then integrate the difference of two polynomial functions. All techniques are routine for this level, though the algebra and integration require care. |
| 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 | Mark | Guidance |
| Substitutes \(x=1\) in \(C_1\): \(y = 10-1-8=1\) and in \(C_2\): \(y=1^3=1 \Rightarrow (1,1)\) lies on both curves | B1 | Substitutes \(x=1\) into both \(y=10x-x^2-8\) and \(y=x^3\) AND achieves \(y=1\) in both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(10x - x^2 - 8 = x^3 \Rightarrow x^3 + x^2 - 10x + 8 = 0\) | B1 | Sets equations equal, proceeds to \(x^3+x^2-10x+8=0\) |
| \((x-1)(x^2+2x-8)=0\) | M1 A1 | Divides by \((x-1)\) to form quadratic; correct quadratic factor \((x^2+2x-8)\) |
| \((x-1)(x+4)(x-2)=0 \Rightarrow x=2\) | M1 A1 | Factorising quadratic factor; achieves \(x=2\) |
| \((2, 8)\) | A1 | Coordinates of \(B=(2,8)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\{(10x-x^2-8)-x^3\}\,dx\) | M1 | Knowing area \(= \int\{(10x-x^2-8)-x^3\}\,dx\); may find separate areas and subtract |
| \(= 5x^2 - \dfrac{x^3}{3} - 8x - \dfrac{x^4}{4}\) | M1 A1 | Raising power of \(x\) in at least three terms; correct integration (allow \(\frac{10x^2}{2}\) for \(5x^2\)) |
| Using limits 2 and 1: \(\left(20-\dfrac{8}{3}-16-4\right)-\left(5-\dfrac{1}{3}-8-\dfrac{1}{4}\right)\) | M1 | Using limits "2" and 1 in integrated expression |
| \(= \dfrac{11}{12}\) | A1 | \(\dfrac{11}{12}\) or exact equivalent |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $x=1$ in $C_1$: $y = 10-1-8=1$ and in $C_2$: $y=1^3=1 \Rightarrow (1,1)$ lies on both curves | B1 | Substitutes $x=1$ into both $y=10x-x^2-8$ and $y=x^3$ **AND** achieves $y=1$ in both |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $10x - x^2 - 8 = x^3 \Rightarrow x^3 + x^2 - 10x + 8 = 0$ | B1 | Sets equations equal, proceeds to $x^3+x^2-10x+8=0$ |
| $(x-1)(x^2+2x-8)=0$ | M1 A1 | Divides by $(x-1)$ to form quadratic; correct quadratic factor $(x^2+2x-8)$ |
| $(x-1)(x+4)(x-2)=0 \Rightarrow x=2$ | M1 A1 | Factorising quadratic factor; achieves $x=2$ |
| $(2, 8)$ | A1 | Coordinates of $B=(2,8)$ |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\{(10x-x^2-8)-x^3\}\,dx$ | M1 | Knowing area $= \int\{(10x-x^2-8)-x^3\}\,dx$; may find separate areas and subtract |
| $= 5x^2 - \dfrac{x^3}{3} - 8x - \dfrac{x^4}{4}$ | M1 A1 | Raising power of $x$ in at least three terms; correct integration (allow $\frac{10x^2}{2}$ for $5x^2$) |
| Using limits 2 and 1: $\left(20-\dfrac{8}{3}-16-4\right)-\left(5-\dfrac{1}{3}-8-\dfrac{1}{4}\right)$ | M1 | Using limits "2" and 1 in integrated expression |
| $= \dfrac{11}{12}$ | A1 | $\dfrac{11}{12}$ or exact equivalent |
---8.
Figure 2
Figure 2 shows a sketch of part of the curves $C _ { 1 }$ and $C _ { 2 }$ with equations
$$\begin{array} { l l }
C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0 \\
C _ { 2 } : y = x ^ { 3 } & x > 0
\end{array}$$
The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Verify that the point $A$ has coordinates (1, 1)
\item Use algebra to find the coordinates of the point $B$
The finite region $R$ is bounded by $C _ { 1 }$ and $C _ { 2 }$
\item Use calculus to find the exact area of $R$\\
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582}
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2018 Q8 [12]}}