| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find curve equation from derivative |
| Difficulty | Moderate -0.3 This is a straightforward integration question requiring students to find a tangent line and then integrate a polynomial/power function. Part (a) involves simple substitution and linear equation work. Part (b) requires integrating terms with fractional powers (including x^{-2} and x^{1/3}), which is standard P1/C1 material with no conceptual challenges beyond applying the reverse power rule and using the given point to find the constant of integration. Slightly easier than average due to routine techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(8) = \frac{32}{3 \times 8^2} + 3 - 2\sqrt[3]{8} \left(= -\frac{5}{6}\right)\) | M1 | Substitutes \(x = 8\); condone slips; \(-\frac{5}{6}\) seen implies mark |
| \(y - 2 = -\frac{5}{6}(x - 8)\) | dM1 | Line through \((8,2)\) using their \(f'(8)\); dependent on previous M1; if \(y = mx + c\) used must reach \(c = \ldots\) |
| \(y = -\frac{5}{6}x + \frac{26}{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = \frac{32}{3x^2} + 3 - 2\sqrt[3]{x} = \ldots x^{-2} + 3 + \ldots x^{\frac{1}{3}}\) | ||
| \(x^{-2} \to x^{-1},\quad 3 \to 3x,\quad x^{\frac{1}{3}} \to x^{\frac{4}{3}}\) | M1 | |
| \(f(x) = \int \frac{32}{3}x^{-2} + 3 - 2x^{\frac{1}{3}}\,dx = -\frac{32}{3}x^{-1} + 3x - \frac{3}{2}x^{\frac{4}{3}} + c\) | A1A1 | |
| \(2 = -\frac{32}{3} \times 8^{-1} + 3 \times 8 - \frac{3}{2} \times 8^{\frac{4}{3}} + c \Rightarrow c = \ldots\) | dM1 | Substitutes \((8, 2)\) to find \(c\); dependent on previous M1 |
| \(f(x) = -\frac{32}{3}x^{-1} + 3x - \frac{3}{2}x^{\frac{4}{3}} + \frac{10}{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Raises power on one term, e.g. \(x^{-2} \to x^{-1}\), \(3 \to 3x\), \(x^{\frac{1}{3}} \to x^{\frac{4}{3}}\) | M1 | At least one term correctly integrated |
| Two of: \(-\frac{32}{3}x^{-1}\), \(+3x\), \(-\frac{3}{2}x^{\frac{4}{3}}\) correct | A1 | Indices must be processed; unsimplified equivalents accepted |
| \(-\frac{32}{3}x^{-1}+3x-\frac{3}{2}x^{\frac{4}{3}}(+c)\) | A1 | Condone lack of \(+c\); \(-10.7x^{-1}\) not correct but allow \(-10.\dot{6}x^{-1}\) |
| Substitutes \(x=8\), \(y=2\) into \(f(x)\) to find \(c\) | dM1 | Dependent on previous M1; condone slips in rearrangement |
| \(f(x) = -\frac{32}{3}x^{-1}+3x-\frac{3}{2}x^{\frac{4}{3}}+\frac{10}{3}\) | A1 | Accept \(-\frac{32}{3x}+3x-\frac{3}{2}x^{\frac{4}{3}}+\frac{10}{3}\); do not accept rounded decimals for coefficients |
# Question 6(a):
$f'(8) = \frac{32}{3 \times 8^2} + 3 - 2\sqrt[3]{8} \left(= -\frac{5}{6}\right)$ | M1 | Substitutes $x = 8$; condone slips; $-\frac{5}{6}$ seen implies mark
$y - 2 = -\frac{5}{6}(x - 8)$ | dM1 | Line through $(8,2)$ using their $f'(8)$; dependent on previous M1; if $y = mx + c$ used must reach $c = \ldots$
$y = -\frac{5}{6}x + \frac{26}{3}$ | A1 |
---
# Question 6(b):
$f'(x) = \frac{32}{3x^2} + 3 - 2\sqrt[3]{x} = \ldots x^{-2} + 3 + \ldots x^{\frac{1}{3}}$ | |
$x^{-2} \to x^{-1},\quad 3 \to 3x,\quad x^{\frac{1}{3}} \to x^{\frac{4}{3}}$ | M1 |
$f(x) = \int \frac{32}{3}x^{-2} + 3 - 2x^{\frac{1}{3}}\,dx = -\frac{32}{3}x^{-1} + 3x - \frac{3}{2}x^{\frac{4}{3}} + c$ | A1A1 |
$2 = -\frac{32}{3} \times 8^{-1} + 3 \times 8 - \frac{3}{2} \times 8^{\frac{4}{3}} + c \Rightarrow c = \ldots$ | dM1 | Substitutes $(8, 2)$ to find $c$; dependent on previous M1
$f(x) = -\frac{32}{3}x^{-1} + 3x - \frac{3}{2}x^{\frac{4}{3}} + \frac{10}{3}$ | A1 |
# Integration Question (Question 6 based on context):
| Answer | Mark | Guidance |
|--------|------|----------|
| Raises power on one term, e.g. $x^{-2} \to x^{-1}$, $3 \to 3x$, $x^{\frac{1}{3}} \to x^{\frac{4}{3}}$ | M1 | At least one term correctly integrated |
| Two of: $-\frac{32}{3}x^{-1}$, $+3x$, $-\frac{3}{2}x^{\frac{4}{3}}$ correct | A1 | Indices must be processed; unsimplified equivalents accepted |
| $-\frac{32}{3}x^{-1}+3x-\frac{3}{2}x^{\frac{4}{3}}(+c)$ | A1 | Condone lack of $+c$; $-10.7x^{-1}$ not correct but allow $-10.\dot{6}x^{-1}$ |
| Substitutes $x=8$, $y=2$ into $f(x)$ to find $c$ | dM1 | Dependent on previous M1; condone slips in rearrangement |
| $f(x) = -\frac{32}{3}x^{-1}+3x-\frac{3}{2}x^{\frac{4}{3}}+\frac{10}{3}$ | A1 | Accept $-\frac{32}{3x}+3x-\frac{3}{2}x^{\frac{4}{3}}+\frac{10}{3}$; do not accept rounded decimals for coefficients |
---6. The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$
Given that
\begin{itemize}
\item $C$ passes through the point $P ( 8,2 )$
\item $\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )$
\begin{enumerate}[label=(\alph*)]
\item find the equation of the tangent to $C$ at $P$. Write your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found.\\
(3)
\item Find, in simplest form, $\mathrm { f } ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2021 Q6 [8]}}