Question 10 - A-Level Maths

Edexcel P2 2024 January — Question 10 9 marks

Exam Board	Edexcel
Module	P2 (Pure Mathematics 2)
Year	2024
Session	January
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Chain Rule
Type	Find stationary points and nature
Difficulty	Standard +0.3 This is a straightforward two-part question requiring standard differentiation using the chain rule to find a stationary point (with the x-coordinate given), followed by routine integration. The algebraic manipulation is slightly more involved than basic examples due to the fractional power, but both parts follow standard procedures with no novel problem-solving required. This is slightly easier than average for A-level.
Spec	1.07i Differentiate x^n: for rational n and sums 1.07n Stationary points: find maxima, minima using derivatives 1.08e Area between curve and x-axis: using definite integrals

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-30_646_741_376_662} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point $P$ is the only stationary point on the curve.

Use calculus to show that the $x$ coordinate of $P$ is 9 The line $l$ passes through the point $P$ and is parallel to the $x$-axis.
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line $l$ and the line with equation $x = 4$
Use algebraic integration to find the exact area of $R$.

Show mark scheme Show mark scheme source

Question 10(a):

Answer	Marks	Guidance
$\frac{dy}{dx} = x - 2187x^{-\frac{5}{2}}$	M1, A1	M1: Attempts to differentiate with one index correct e.g. $\ldots x^2 \to \ldots x$ or $x^{-\frac{3}{2}} \to x^{-\frac{3}{2}-1}$ but not for $74 \to 0$. A1: Correct differentiation, indices must be processed.
Sets $x - 2187x^{-\frac{5}{2}} = 0 \Rightarrow x^{\frac{7}{2}} = 2187 \Rightarrow x = 9$	dM1A1\*	dM1: Solves $\frac{dy}{dx} = 0$ of form $x - \ldots x^m = 0$ where $m$ is a fraction via $x^n = A$ or $x = A^{\frac{1}{n}}$. A1\*: Correct calculations and working proving $x$-coordinate of $P$ is $9$.

Question 10(b):

Answer	Marks	Guidance
$\int\left\{\frac{1}{2}x^2 + 1458x^{-\frac{3}{2}} - 74\right\}dx = \frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x$	M1A1ft	M1: Attempts to integrate with one index correct. A1ft: Correct integration $\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x$ or follow through if curve–line is integrated.
$y$ value at $P$ is $20.5$	B1	May be seen as part of wider calculation or on diagram. $y = 20.5$ or $\frac{41}{2}$.
Area $R = \left[\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x\right]_4^9 - (9-4)\times 20.5$ $= \left(\frac{1}{6}\times 9^3 - 2916\times 9^{-\frac{1}{2}} - 74\times 9\right) - \left(\frac{1}{6}\times 4^3 - 2916\times 4^{-\frac{1}{2}} - 74\times 4\right) - (9-4)\times 20.5$	dM1	Full method to find area of $R$. Values embedded sufficient. Method for finding $y$ at $P$ must be correct.
$\left(-1516\frac{1}{2} + 1743\frac{1}{3} - 102\frac{1}{2}\right) = 124\frac{1}{3}$	A1	Correct answer $124\frac{1}{3}$ o.e. $\frac{373}{3}$, including $124.\dot{3}$ or $124.\overline{3}$ or $124.33\ldots$ but not rounded $124.3$. Can only be scored provided all previous marks scored.

## Question 10(a):

$\frac{dy}{dx} = x - 2187x^{-\frac{5}{2}}$ | **M1, A1** | M1: Attempts to differentiate with one index correct e.g. $\ldots x^2 \to \ldots x$ or $x^{-\frac{3}{2}} \to x^{-\frac{3}{2}-1}$ but not for $74 \to 0$. A1: Correct differentiation, indices must be processed.

Sets $x - 2187x^{-\frac{5}{2}} = 0 \Rightarrow x^{\frac{7}{2}} = 2187 \Rightarrow x = 9$ | **dM1A1\*** | dM1: Solves $\frac{dy}{dx} = 0$ of form $x - \ldots x^m = 0$ where $m$ is a fraction via $x^n = A$ or $x = A^{\frac{1}{n}}$. A1\*: Correct calculations and working proving $x$-coordinate of $P$ is $9$.

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## Question 10(b):

$\int\left\{\frac{1}{2}x^2 + 1458x^{-\frac{3}{2}} - 74\right\}dx = \frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x$ | **M1A1ft** | M1: Attempts to integrate with one index correct. A1ft: Correct integration $\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x$ or follow through if curve–line is integrated.

$y$ value at $P$ is $20.5$ | **B1** | May be seen as part of wider calculation or on diagram. $y = 20.5$ or $\frac{41}{2}$.

Area $R = \left[\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x\right]_4^9 - (9-4)\times 20.5$ $= \left(\frac{1}{6}\times 9^3 - 2916\times 9^{-\frac{1}{2}} - 74\times 9\right) - \left(\frac{1}{6}\times 4^3 - 2916\times 4^{-\frac{1}{2}} - 74\times 4\right) - (9-4)\times 20.5$ | **dM1** | Full method to find area of $R$. Values embedded sufficient. Method for finding $y$ at $P$ must be correct.

$\left(-1516\frac{1}{2} + 1743\frac{1}{3} - 102\frac{1}{2}\right) = 124\frac{1}{3}$ | **A1** | Correct answer $124\frac{1}{3}$ o.e. $\frac{373}{3}$, including $124.\dot{3}$ or $124.\overline{3}$ or $124.33\ldots$ but **not** rounded $124.3$. **Can only be scored provided all previous marks scored.**

Show LaTeX source

\begin{enumerate}
  \item In this question you must show detailed reasoning.
\end{enumerate}

Solutions relying entirely on calculator technology are not acceptable.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-30_646_741_376_662}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Figure 2 shows a sketch of the curve with equation

$$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$

The point $P$ is the only stationary point on the curve.\\
(a) Use calculus to show that the $x$ coordinate of $P$ is 9

The line $l$ passes through the point $P$ and is parallel to the $x$-axis.\\
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line $l$ and the line with equation $x = 4$\\
(b) Use algebraic integration to find the exact area of $R$.

\hfill \mbox{\textit{Edexcel P2 2024 Q10 [9]}}

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Q1 3 Q2 3 Q3 6 Q4 9 Q5 8 Q6 8 Q7 9 Q8 6 Q9 14 Q10 9

Answer	Marks	Guidance
\(\int\left\{\frac{1}{2}x^2 + 1458x^{-\frac{3}{2}} - 74\right\}dx = \frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x\)	M1A1ft	M1: Attempts to integrate with one index correct. A1ft: Correct integration \(\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x\) or follow through if curve–line is integrated.
\(y\) value at \(P\) is \(20.5\)	B1	May be seen as part of wider calculation or on diagram. \(y = 20.5\) or \(\frac{41}{2}\).
Area \(R = \left[\frac{1}{6}x^3 - 2916x^{-\frac{1}{2}} - 74x\right]_4^9 - (9-4)\times 20.5\) \(= \left(\frac{1}{6}\times 9^3 - 2916\times 9^{-\frac{1}{2}} - 74\times 9\right) - \left(\frac{1}{6}\times 4^3 - 2916\times 4^{-\frac{1}{2}} - 74\times 4\right) - (9-4)\times 20.5\)	dM1	Full method to find area of \(R\). Values embedded sufficient. Method for finding \(y\) at \(P\) must be correct.
\(\left(-1516\frac{1}{2} + 1743\frac{1}{3} - 102\frac{1}{2}\right) = 124\frac{1}{3}\)	A1	Correct answer \(124\frac{1}{3}\) o.e. \(\frac{373}{3}\), including \(124.\dot{3}\) or \(124.\overline{3}\) or \(124.33\ldots\) but not rounded \(124.3\). Can only be scored provided all previous marks scored.

Answer	Marks	Guidance
\(\frac{dy}{dx} = x - 2187x^{-\frac{5}{2}}\)	M1, A1	M1: Attempts to differentiate with one index correct e.g. \(\ldots x^2 \to \ldots x\) or \(x^{-\frac{3}{2}} \to x^{-\frac{3}{2}-1}\) but not for \(74 \to 0\). A1: Correct differentiation, indices must be processed.
Sets \(x - 2187x^{-\frac{5}{2}} = 0 \Rightarrow x^{\frac{7}{2}} = 2187 \Rightarrow x = 9\)	dM1A1\*	dM1: Solves \(\frac{dy}{dx} = 0\) of form \(x - \ldots x^m = 0\) where \(m\) is a fraction via \(x^n = A\) or \(x = A^{\frac{1}{n}}\). A1\*: Correct calculations and working proving \(x\)-coordinate of \(P\) is \(9\).