| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (extended problem with normals, stationary points, or further geometry) |
| Difficulty | Moderate -0.3 This is a straightforward integration question requiring standard power rule integration (including fractional powers) and finding a constant using a boundary condition, followed by a routine normal line calculation. While it requires multiple steps and careful algebraic manipulation, all techniques are standard P1 material with no novel problem-solving required. |
| Spec | 1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| One of \(x^{-\frac{1}{2}} \to x^{\frac{1}{2}}\), \(-4 \to -4x\), \(x \to x^2\) | M1 | Integrates by raising power on one term; index does not need to be processed |
| Two terms correct of \(24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x\) | A1 | Or unsimplified equivalent appearing as a list; indices must be processed; allow \(x^1\) |
| \(24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x\) \((+c)\) | A1 | Or unsimplified equivalent; condone lack of \(+c\); allow \(x^1\) |
| Substitutes \(x=9\), \(y=8\) into \(f(x)\) to find \(c\) | dM1 | Dependent on previous M1; if no \(+c\) then cannot score |
| \(f(x) = 24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x - \frac{83}{2}\) | A1 | Withhold if coefficients made integers or rounded decimals used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f'(9) = \frac{12}{\sqrt{9}} + \frac{9}{3} - 4 = 3\) | M1 | Attempts to substitute \(x=9\) to find value for \(f'(9)\) |
| \(3 \to -\frac{1}{3}\) | dM1 | Negative reciprocal for gradient of normal; dependent on previous M1 |
| \(y - 8 = -\frac{1}{3}(0-9)\) | M1 | Attempt at normal line through \((9,8)\) using changed gradient; condone use of \(f''(9)\) instead of \(f'(9)\) |
| \((0, 11)\) | A1 | Or \(x=0\), \(y=11\) |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| One of $x^{-\frac{1}{2}} \to x^{\frac{1}{2}}$, $-4 \to -4x$, $x \to x^2$ | M1 | Integrates by raising power on one term; index does not need to be processed |
| Two terms correct of $24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x$ | A1 | Or unsimplified equivalent appearing as a list; indices must be processed; allow $x^1$ |
| $24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x$ $(+c)$ | A1 | Or unsimplified equivalent; condone lack of $+c$; allow $x^1$ |
| Substitutes $x=9$, $y=8$ into $f(x)$ to find $c$ | dM1 | Dependent on previous M1; if no $+c$ then cannot score |
| $f(x) = 24x^{\frac{1}{2}} + \frac{x^2}{6} - 4x - \frac{83}{2}$ | A1 | Withhold if coefficients made integers or rounded decimals used |
## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f'(9) = \frac{12}{\sqrt{9}} + \frac{9}{3} - 4 = 3$ | M1 | Attempts to substitute $x=9$ to find value for $f'(9)$ |
| $3 \to -\frac{1}{3}$ | dM1 | Negative reciprocal for gradient of normal; dependent on previous M1 |
| $y - 8 = -\frac{1}{3}(0-9)$ | M1 | Attempt at normal line through $(9,8)$ using changed gradient; condone use of $f''(9)$ instead of $f'(9)$ |
| $(0, 11)$ | A1 | Or $x=0$, $y=11$ |\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$\\
Given that
\begin{itemize}
\item $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4$
\item the point $P ( 9,8 )$ lies on $C$\\
(a) find, in simplest form, $\mathrm { f } ( x )$
\end{itemize}
The line $l$ is the normal to $C$ at $P$\\
(b) Find the coordinates of the point at which $l$ crosses the $y$-axis.
\hfill \mbox{\textit{Edexcel P1 2022 Q5 [9]}}