| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Two Curves Intersection Area |
| Difficulty | Standard +0.3 This is a straightforward integration question requiring finding intersection points by solving a cubic equation (which factorises nicely given one root), then computing area between curves. The algebraic manipulation is routine and the integration involves only polynomial terms. Slightly easier than average due to the guided structure and standard techniques. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Substitutes \(x = \frac{1}{2}\) into \(y = 2x^3 + 10\) and \(y = 42x - 15x^2 - 7\) and finds the \(y\) values for both | M1 | 1.1b |
| Achieves \(\frac{41}{4}\) o.e. for both and makes a valid conclusion | A1* | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Sets \(42x - 15x^2 - 7 = 2x^3 + 10 \Rightarrow 2x^3 + 15x^2 - 42x + 17 = 0\) | M1 | 1.1b |
| Deduces that \((2x-1)\) is a factor and attempts to divide | dM1 | 2.1 |
| \(2x^3 + 15x^2 - 42x + 17 = (2x-1)(x^2 + 8x - 17)\) | A1 | 1.1b |
| Solves their \(x^2 + 8x - 17 = 0\) using suitable method | M1 | 1.1b |
| Deduces \(x = -4 + \sqrt{33}\) only (must not include \(x = -4 - \sqrt{33}\)) | A1 | 2.2a |
# Question 11:
## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Substitutes $x = \frac{1}{2}$ into $y = 2x^3 + 10$ **and** $y = 42x - 15x^2 - 7$ and finds the $y$ values for both | M1 | 1.1b |
| Achieves $\frac{41}{4}$ o.e. for both and makes a valid conclusion | A1* | 2.4 |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Sets $42x - 15x^2 - 7 = 2x^3 + 10 \Rightarrow 2x^3 + 15x^2 - 42x + 17 = 0$ | M1 | 1.1b |
| Deduces that $(2x-1)$ is a factor and attempts to divide | dM1 | 2.1 |
| $2x^3 + 15x^2 - 42x + 17 = (2x-1)(x^2 + 8x - 17)$ | A1 | 1.1b |
| Solves their $x^2 + 8x - 17 = 0$ using suitable method | M1 | 1.1b |
| Deduces $x = -4 + \sqrt{33}$ only (must not include $x = -4 - \sqrt{33}$) | A1 | 2.2a |
---11.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-28_647_855_244_605}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a sketch of part of the curve $C _ { 1 }$ with equation
$$y = 2 x ^ { 3 } + 10 \quad x > 0$$
and part of the curve $C _ { 2 }$ with equation
$$y = 42 x - 15 x ^ { 2 } - 7 \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Verify that the curves intersect at $x = \frac { 1 } { 2 }$
The curves intersect again at the point $P$
\item Using algebra and showing all stages of working, find the exact $x$ coordinate of $P$
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q11 [7]}}