| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Standard +0.2 This is a straightforward multi-part question on standard differentiation techniques. Part (a) uses the condition that dy/dx = 0 at a stationary point to find k (simple algebra). Part (b) requires solving a quadratic inequality. Part (c) involves routine definite integration. All parts are textbook exercises requiring no novel insight, making this easier than average for A-level. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \frac{4}{3}x^3 - 11x^2 + kx \Rightarrow \frac{dy}{dx} = 4x^2 - 22x + k\) | M1 | At least one index correct; must be seen in part (a) |
| Uses \(x=2\), \(\frac{dy}{dx}=0\): \(0 = 16 - 44 + k \Rightarrow k = 28\) | dM1 A1* | Substitutes \(x=2\) into \(\frac{dy}{dx}\) of form \(ax^2+bx+k\) and sets \(=0\); achieves \(k=28\) via correct intermediate line, no missing "=0" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 4x^2 - 22x + 28 = 0 \Rightarrow (2x-4)(2x-7) = 0 \Rightarrow x = \ldots\) | M1 | Attempts to find critical values |
| \(x < 2,\ x > \frac{7}{2}\) | A1 | Do not accept \(\frac{7}{2} < x < 2\); accept alternative set notations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\left(\frac{4}{3}x^3 - 11x^2 + 28x\right)dx \Rightarrow \frac{1}{3}x^4 - \frac{11}{3}x^3 + 14x^2\) | M1 A1 | At least one index correct; need not be simplified |
| Correct \(y\)-coordinate of \(M = \frac{68}{3}\) | B1 | Accept awrt 22.7; may be seen anywhere e.g. on sketch |
| Complete method: \(R = 2 \times \frac{68}{3} - \int_0^2\left(\frac{4}{3}x^3 - 11x^2 + 28x\right)dx\) | M1 | Lower limit may be implied; integral must be a changed function |
| \(= 2 \times \frac{68}{3} - \left(\frac{1}{3}\times 2^4 - \frac{11}{3}\times 2^3 + 14\times 2^2\right) = \frac{40}{3}\) | A1 | Exact equivalent accepted |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \frac{4}{3}x^3 - 11x^2 + kx \Rightarrow \frac{dy}{dx} = 4x^2 - 22x + k$ | M1 | At least one index correct; must be seen in part (a) |
| Uses $x=2$, $\frac{dy}{dx}=0$: $0 = 16 - 44 + k \Rightarrow k = 28$ | dM1 A1* | Substitutes $x=2$ into $\frac{dy}{dx}$ of form $ax^2+bx+k$ and sets $=0$; achieves $k=28$ via correct intermediate line, no missing "=0" |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 4x^2 - 22x + 28 = 0 \Rightarrow (2x-4)(2x-7) = 0 \Rightarrow x = \ldots$ | M1 | Attempts to find critical values |
| $x < 2,\ x > \frac{7}{2}$ | A1 | Do not accept $\frac{7}{2} < x < 2$; accept alternative set notations |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\left(\frac{4}{3}x^3 - 11x^2 + 28x\right)dx \Rightarrow \frac{1}{3}x^4 - \frac{11}{3}x^3 + 14x^2$ | M1 A1 | At least one index correct; need not be simplified |
| Correct $y$-coordinate of $M = \frac{68}{3}$ | B1 | Accept awrt 22.7; may be seen anywhere e.g. on sketch |
| Complete method: $R = 2 \times \frac{68}{3} - \int_0^2\left(\frac{4}{3}x^3 - 11x^2 + 28x\right)dx$ | M1 | Lower limit may be implied; integral must be a changed function |
| $= 2 \times \frac{68}{3} - \left(\frac{1}{3}\times 2^4 - \frac{11}{3}\times 2^3 + 14\times 2^2\right) = \frac{40}{3}$ | A1 | Exact equivalent accepted |
---8. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-24_739_736_411_605}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve $C$ with equation
$$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$
The point $M$ is the maximum turning point of $C$ and is shown in Figure 2.\\
Given that the $x$ coordinate of $M$ is 2
\begin{enumerate}[label=(\alph*)]
\item show that $k = 28$
\item Determine the range of values of $x$ for which $y$ is increasing.
The line $l$ passes through $M$ and is parallel to the $x$-axis.\\
The region $R$, shown shaded in Figure 2, is bounded by the curve $C$, the line $l$ and the $y$-axis.
\item Find, by algebraic integration, the exact area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q8 [10]}}