| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 7 |
| 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 differentiate f'(x) to find a using the condition f''(27)=0, then integrate f'(x) and use the point (1,-8) to find the constant. All steps are routine applications of standard techniques with no conceptual challenges beyond basic calculus mechanics. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x)=ax-12x^{\frac{1}{3}}\Rightarrow f''(x)=a-4x^{-\frac{2}{3}}\) | B1 | States or uses \(f''(x)=a-4x^{-\frac{2}{3}}\); may be unsimplified |
| Sets \(f''(27)=0\Rightarrow 0=a-4\times\dfrac{1}{9}\) | M1 | Sets their \(f''(27)=0\) and proceeds to a value for \(a\); dependent on one correct term in \(f''(x)\) |
| \(a=\dfrac{4}{9}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(x)=ax-12x^{\frac{1}{3}}\Rightarrow f(x)=\dfrac{1}{2}ax^2-9x^{\frac{4}{3}}+c\) | M1 A1ft | Integrates with one term correct e.g. \(\dfrac{1}{2}ax^2\) or \(-\dfrac{12x^{\frac{4}{3}}}{\frac{4}{3}}\); follow through on \(a\) or numerical \(a\); must include \(+c\) |
| Substitutes \(x=1\), \(f(x)=-8\Rightarrow c=\ldots\) | dM1 | Must have numerical \(a\) now; depends on first M mark |
| \(f(x)=\dfrac{2}{9}x^2-9x^{\frac{4}{3}}+\dfrac{7}{9}\) | A1 | Allow equivalent correct fractions for \(\dfrac{2}{9}\), \(\dfrac{7}{9}\) or recurring decimals e.g. \(0.\dot{2}\), \(0.\dot{7}\) |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x)=ax-12x^{\frac{1}{3}}\Rightarrow f''(x)=a-4x^{-\frac{2}{3}}$ | B1 | States or uses $f''(x)=a-4x^{-\frac{2}{3}}$; may be unsimplified |
| Sets $f''(27)=0\Rightarrow 0=a-4\times\dfrac{1}{9}$ | M1 | Sets their $f''(27)=0$ and proceeds to a value for $a$; dependent on one correct term in $f''(x)$ |
| $a=\dfrac{4}{9}$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(x)=ax-12x^{\frac{1}{3}}\Rightarrow f(x)=\dfrac{1}{2}ax^2-9x^{\frac{4}{3}}+c$ | M1 A1ft | Integrates with one term correct e.g. $\dfrac{1}{2}ax^2$ or $-\dfrac{12x^{\frac{4}{3}}}{\frac{4}{3}}$; follow through on $a$ or numerical $a$; must include $+c$ |
| Substitutes $x=1$, $f(x)=-8\Rightarrow c=\ldots$ | dM1 | Must have numerical $a$ now; depends on first M mark |
| $f(x)=\dfrac{2}{9}x^2-9x^{\frac{4}{3}}+\dfrac{7}{9}$ | A1 | Allow equivalent correct fractions for $\dfrac{2}{9}$, $\dfrac{7}{9}$ or recurring decimals e.g. $0.\dot{2}$, $0.\dot{7}$ |
The image appears to be essentially blank/empty - it only shows "PMT" in the top right corner and a Pearson Education Limited copyright notice at the bottom. There is no mark scheme content visible on this page to extract.
Could you please share the actual mark scheme pages that contain the questions, answers, and mark allocations? This appears to be either a blank page or the back cover of a mark scheme document.10. A curve has equation $y = \mathrm { f } ( x ) , x > 0$
Given that
\begin{itemize}
\item $\mathrm { f } ^ { \prime } ( x ) = a x - 12 x ^ { \frac { 1 } { 3 } }$, where $a$ is a constant
\item $\mathrm { f } ^ { \prime \prime } ( x ) = 0$ when $x = 27$
\item the curve passes through the point $( 1 , - 8 )$
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$.
\item Hence find $\mathrm { f } ( x )$.\\
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q10 [7]}}