| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Deduce inequality solutions from sketch |
| Difficulty | Moderate -0.8 This is a straightforward P1 question testing basic curve sketching concepts. Part (a) requires reading roots from a given sketch, (b) is routine algebraic expansion, (c) involves simple substitution and differentiation, and (d) applies a standard horizontal translation. All parts are textbook exercises requiring recall and direct application of techniques with no problem-solving insight needed. |
| 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 sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) \leqslant 0 \Rightarrow x \leqslant -\frac{5}{2}\), \(x = 3\) | M1 A1 | M1 for either \(x \leqslant -\frac{5}{2}\) or \(x = 3\); condone \(x < -\frac{5}{2}\) for M1. A1 for both; accept "and"/"or" between them. Answers like \(-\frac{5}{2} \leqslant x \leqslant 3\) are M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) = (2x+5)(x-3)^2 = (2x+5)(x^2-6x+9)\) | M1 | Attempts to multiply two brackets achieving \(x^2\), \(x\) and constant terms |
| \(= 2x^3 - 12x^2 + 18x + 5x^2 - 30x + 45\) | M1 | Multiplies result by third bracket to reach four-term cubic |
| \(= 2x^3 - 7x^2 - 12x + 45\) | A1 | Ignore any reference to "\(=0\)" or "\(<0\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(P(0, 45)\) | B1ft | Following through on their \(d\); do not accept \((45, 0)\) or just \(45\) |
| (ii) Gradient \(= -12\) | B1ft | Following through on their \(c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(g(x) = (2(x-2)+5)(x-2-3)^2 = (2x+1)(x-5)^2\) | M1 A1 | M1 attempts to replace \(x\) with \((x-2)\); allow M1 for one correct bracket if no incorrect working seen. A1 correct factorised form; if substituting into expanded form must factorise correctly |
| (ii) \(25\) | B1 | Accept \((0, 25)\) |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) \leqslant 0 \Rightarrow x \leqslant -\frac{5}{2}$, $x = 3$ | M1 A1 | M1 for either $x \leqslant -\frac{5}{2}$ or $x = 3$; condone $x < -\frac{5}{2}$ for M1. A1 for both; accept "and"/"or" between them. Answers like $-\frac{5}{2} \leqslant x \leqslant 3$ are M0A0 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = (2x+5)(x-3)^2 = (2x+5)(x^2-6x+9)$ | M1 | Attempts to multiply two brackets achieving $x^2$, $x$ and constant terms |
| $= 2x^3 - 12x^2 + 18x + 5x^2 - 30x + 45$ | M1 | Multiplies result by third bracket to reach four-term cubic |
| $= 2x^3 - 7x^2 - 12x + 45$ | A1 | Ignore any reference to "$=0$" or "$<0$" |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) $P(0, 45)$ | B1ft | Following through on their $d$; do not accept $(45, 0)$ or just $45$ |
| (ii) Gradient $= -12$ | B1ft | Following through on their $c$ |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) $g(x) = (2(x-2)+5)(x-2-3)^2 = (2x+1)(x-5)^2$ | M1 A1 | M1 attempts to replace $x$ with $(x-2)$; allow M1 for one correct bracket if no incorrect working seen. A1 correct factorised form; if substituting into expanded form must factorise correctly |
| (ii) $25$ | B1 | Accept $(0, 25)$ |10.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-22_592_665_251_676}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{center}
\end{figure}
Figure 6 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where
$$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Deduce the values of $x$ for which $\mathrm { f } ( x ) \leqslant 0$
The curve crosses the $y$-axis at the point $P$, as shown.
\item Expand $\mathrm { f } ( x )$ to the form
$$a x ^ { 3 } + b x ^ { 2 } + c x + d$$
where $a$, $b$, $c$ and $d$ are integers to be found.
\item Hence, or otherwise, find
\begin{enumerate}[label=(\roman*)]
\item the coordinates of $P$,
\item the gradient of the curve at $P$.
The curve with equation $y = \mathrm { f } ( x )$ is translated two units in the positive $x$ direction to a curve with equation $y = \mathrm { g } ( x )$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { g } ( x )$, giving your answer in a simplified factorised form.
\item Hence state the $y$ intercept of the curve with equation $y = \mathrm { g } ( x )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2019 Q10 [10]}}