| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve transformed function equations |
| Difficulty | Standard +0.3 This is a multi-part question testing standard transformations and differentiation. Parts (a) and (b) involve straightforward substitution into transformations, (c) requires expanding and differentiating a cubic (routine but algebraically involved), and (d) applies the derivative to solve an inequality. While there are several steps and some algebra, all techniques are standard P1 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x)1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4 \times -2 \times -9 = 72\), \(p = "72" - 50\) | M1 | Correct method for \(p\). Condone sign slips in product \(4\times-2\times-9\). Alt: multiplies to cubic \((\pm72)\) and subtracts 50. Note: \(50-72=-22\) is M0 |
| \(p = 22\) | A1 | cao; 22 with no working is M1A1. Accept \(y=f(x)-22\) but not \(f(x)-22 \Rightarrow p=-22\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(q = -4,\ 2,\ 4.5\) | B1B1 | B1: any 1 correct value. B1: all three correct (cao). If they change the sign then B0. If any value excluded withhold second mark. Check next to question, ignore references to \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x)=(x+4)(x-2)(2x-9)\), \(f(x)=(x^2+2x-8)(2x-9) = ...x^3 \pm ...x^2 \pm ...x\ (\pm...)\) | M1 | Correct method to expand to cubic \(ax^3+bx^2+cx\ (+d)\) where \(a,b,c\) all non-zero; terms need not be collected. Expansion may appear in (a) or (b) |
| \(= 2x^3 - 5x^2 - 34x\ (+72)\) | A1 | Simplified or unsimplified; not concerned with value of \(d\) |
| \(f'(x) = 6x^2 - 10x - 34\) | M1A1 | M1: \(x^n \to x^{n-1}\) on at least one term of cubic. A1: \(6x^2-10x-34\) following correct expansion. If constant incorrect (not just omitted) then A0; isw e.g. \(6x^2-10x-34=0\) can score A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \("6x^2-10x-34" = -18\), \("6x^2-10x-16" = 0 \Rightarrow x = ...\left(-1,\ \frac{8}{3}\right)\) | M1 | Sets \(f'(x)=\pm18\), collects terms and solves 3TQ by factorising, quadratic formula or completing the square. Condone slips in rearrangement |
| \(-1 < x < \frac{8}{3}\) | dM1A1 | dM1: attempts "inside" region for their values; not concerned whether \(<\) or \(\leqslant\); dependent on previous M. A1: \(-1 |
# Question 7:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4 \times -2 \times -9 = 72$, $p = "72" - 50$ | M1 | Correct method for $p$. Condone sign slips in product $4\times-2\times-9$. Alt: multiplies to cubic $(\pm72)$ and subtracts 50. Note: $50-72=-22$ is M0 |
| $p = 22$ | A1 | cao; 22 with no working is M1A1. Accept $y=f(x)-22$ but not $f(x)-22 \Rightarrow p=-22$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $q = -4,\ 2,\ 4.5$ | B1B1 | B1: any 1 correct value. B1: all three correct (cao). If they change the sign then B0. If any value excluded withhold second mark. Check next to question, ignore references to $x$ |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x)=(x+4)(x-2)(2x-9)$, $f(x)=(x^2+2x-8)(2x-9) = ...x^3 \pm ...x^2 \pm ...x\ (\pm...)$ | M1 | Correct method to expand to cubic $ax^3+bx^2+cx\ (+d)$ where $a,b,c$ all non-zero; terms need not be collected. Expansion may appear in (a) or (b) |
| $= 2x^3 - 5x^2 - 34x\ (+72)$ | A1 | Simplified or unsimplified; not concerned with value of $d$ |
| $f'(x) = 6x^2 - 10x - 34$ | M1A1 | M1: $x^n \to x^{n-1}$ on at least one term of cubic. A1: $6x^2-10x-34$ following correct expansion. If constant incorrect (not just omitted) then A0; isw e.g. $6x^2-10x-34=0$ can score A1 |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $"6x^2-10x-34" = -18$, $"6x^2-10x-16" = 0 \Rightarrow x = ...\left(-1,\ \frac{8}{3}\right)$ | M1 | Sets $f'(x)=\pm18$, collects terms and solves 3TQ by factorising, quadratic formula or completing the square. Condone slips in rearrangement |
| $-1 < x < \frac{8}{3}$ | dM1A1 | dM1: attempts "inside" region for their values; not concerned whether $<$ or $\leqslant$; dependent on previous M. A1: $-1<x<\frac{8}{3}$ or equivalent e.g. $\{x\in\mathbb{R}: -1<x<\frac{8}{3}\}$, $x>-1 \cap x<\frac{8}{3}$. Do not accept "$x>-1$ or $x<\frac{8}{3}$" or "$x>-1 \cup x<\frac{8}{3}$" |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6c320b71-8793-461a-a078-e4f64c144a3a-20_618_841_267_555}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where
$$f ( x ) = ( x + 4 ) ( x - 2 ) ( 2 x - 9 )$$
Given that the curve with equation $y = \mathrm { f } ( x ) - p$ passes through the point with coordinates $( 0,50 )$
\begin{enumerate}[label=(\alph*)]
\item find the value of the constant $p$.
Given that the curve with equation $y = \mathrm { f } ( x + q )$ passes through the origin,
\item write down the possible values of the constant $q$.
\item Find $\mathrm { f } ^ { \prime } ( x )$.
\item Hence find the range of values of $x$ for which the gradient of the curve with equation $y = \mathrm { f } ( x )$ is less than - 18\\
\includegraphics[max width=\textwidth, alt={}, center]{6c320b71-8793-461a-a078-e4f64c144a3a-23_68_37_2617_1914}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2022 Q7 [11]}}