| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find curve equation from derivative |
| Difficulty | Standard +0.3 This is a straightforward integration problem requiring students to: (1) find the second derivative and use a given condition to find k, (2) integrate to find f(x), and (3) use a point to find the constant of integration. While it involves multiple steps and requires careful algebraic manipulation with surds and fractional powers, each individual step uses standard A-level techniques (differentiation, integration of powers) with no novel problem-solving insight required. It's slightly easier than average due to the structured guidance provided by parts (a) and (b). |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07p Points of inflection: using second derivative1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x) = 4\sqrt{x^3} + \dfrac{k}{x^2} = 4x^{\frac{3}{2}} + kx^{-2}\) | — | Starting expression |
| \(f''(x) = 6x^{\frac{1}{2}} - 2kx^{-3}\) | M1, A1 | M1: Differentiates and achieves at least one index correct. Look for \(px^{\frac{1}{2}} - qx^{"}\) or \(px^{"} - qx^{-3}\). A1: Correct differentiation, may be left unsimplified e.g. \(4 \times \dfrac{3}{2} \times x^{\frac{1}{2}} - 2k \times x^{-3}\) or \(6x^{\frac{1}{2}} - 2kx^{-3}\) |
| \(f''(2) = 6\sqrt{2} - 2k \times \dfrac{1}{8} = 0 \Rightarrow k = 24\sqrt{2}\) | dM1, A1 | dM1: Sets \(f''(2) = 0\) and proceeds to a value for \(k\). Dependent on previous M mark. Allow even if called something else. A1: \(k = 24\sqrt{2}\) or exact equivalent. ISW after correct exact value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x) = 4x^{\frac{3}{2}} + kx^{-2} \Rightarrow f(x) = 4 \times \dfrac{2}{5}x^{\frac{5}{2}} - kx^{-1}\ (+c)\) | M1, A1 ft | M1: Integrates and achieves at least one index correct. Look for \(px^{\frac{5}{2}} - qx^{"}\) or \(px^{"} - qx^{-1}\). A1ft: Correct integration to \(4 \times \dfrac{2}{5}x^{\frac{5}{2}} - kx^{-1}\). No requirement to simplify or have \(+c\). Follow through on their \(k\), also allow with "\(k\)" |
| Uses \(P\!\left(2, 8\sqrt{2}\right) \Rightarrow 8\sqrt{2} = 4 \times \dfrac{2}{5} \times 2^{\frac{5}{2}} - \dfrac{k}{2} + c \Rightarrow c = p\sqrt{2}\) | dM1 | dM1: Uses \(P(2, 8\sqrt{2})\) and their \(k = ...\sqrt{2}\) to find \(c\) as a multiple of \(\sqrt{2}\). Dependent on previous M mark. Both indices must now be correct. The \(...x^{\frac{5}{2}}\) term must produce a term in \(\sqrt{2}\) |
| \(f(x) = \dfrac{8}{5}x^{\frac{5}{2}} - \dfrac{24\sqrt{2}}{x} + \dfrac{68}{5}\sqrt{2}\) | A1 | Achieves \(\dfrac{8}{5}x^{\frac{5}{2}} - \dfrac{24\sqrt{2}}{x} + \dfrac{68}{5}\sqrt{2}\) for \(f(x)\). Accept any other simplified equivalent. No requirement for \(f(x) =\) |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x) = 4\sqrt{x^3} + \dfrac{k}{x^2} = 4x^{\frac{3}{2}} + kx^{-2}$ | — | Starting expression |
| $f''(x) = 6x^{\frac{1}{2}} - 2kx^{-3}$ | M1, A1 | M1: Differentiates and achieves at least one index correct. Look for $px^{\frac{1}{2}} - qx^{"}$ or $px^{"} - qx^{-3}$. A1: Correct differentiation, may be left unsimplified e.g. $4 \times \dfrac{3}{2} \times x^{\frac{1}{2}} - 2k \times x^{-3}$ or $6x^{\frac{1}{2}} - 2kx^{-3}$ |
| $f''(2) = 6\sqrt{2} - 2k \times \dfrac{1}{8} = 0 \Rightarrow k = 24\sqrt{2}$ | dM1, A1 | dM1: Sets $f''(2) = 0$ and proceeds to a value for $k$. Dependent on previous M mark. Allow even if called something else. A1: $k = 24\sqrt{2}$ or exact equivalent. ISW after correct exact value |
**(4 marks)**
---
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x) = 4x^{\frac{3}{2}} + kx^{-2} \Rightarrow f(x) = 4 \times \dfrac{2}{5}x^{\frac{5}{2}} - kx^{-1}\ (+c)$ | M1, A1 ft | M1: Integrates and achieves at least one index correct. Look for $px^{\frac{5}{2}} - qx^{"}$ or $px^{"} - qx^{-1}$. A1ft: Correct integration to $4 \times \dfrac{2}{5}x^{\frac{5}{2}} - kx^{-1}$. No requirement to simplify or have $+c$. Follow through on their $k$, also allow with "$k$" |
| Uses $P\!\left(2, 8\sqrt{2}\right) \Rightarrow 8\sqrt{2} = 4 \times \dfrac{2}{5} \times 2^{\frac{5}{2}} - \dfrac{k}{2} + c \Rightarrow c = p\sqrt{2}$ | dM1 | dM1: Uses $P(2, 8\sqrt{2})$ and their $k = ...\sqrt{2}$ to find $c$ as a multiple of $\sqrt{2}$. Dependent on previous M mark. Both indices must now be correct. The $...x^{\frac{5}{2}}$ term must produce a term in $\sqrt{2}$ |
| $f(x) = \dfrac{8}{5}x^{\frac{5}{2}} - \dfrac{24\sqrt{2}}{x} + \dfrac{68}{5}\sqrt{2}$ | A1 | Achieves $\dfrac{8}{5}x^{\frac{5}{2}} - \dfrac{24\sqrt{2}}{x} + \dfrac{68}{5}\sqrt{2}$ for $f(x)$. Accept any other simplified equivalent. No requirement for $f(x) =$ |
**(4 marks) — (8 marks total)**\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$\\
Given that
\begin{itemize}
\item the point $P ( 2,8 \sqrt { 2 } )$ lies on $C$
\item $\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }$ where $k$ is a constant
\item $\mathrm { f } ^ { \prime \prime } ( x ) = 0$ at $P$\\
(a) find the exact value of $k$,\\
(b) find $\mathrm { f } ( x )$, giving your answer in simplest form.
\end{itemize}
\hfill \mbox{\textit{Edexcel P1 2024 Q10 [8]}}