| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial with rational/modulus curves |
| Difficulty | Moderate -0.3 This is a multi-part question requiring standard P1 skills: identifying asymptotes, factorising a cubic, sketching a polynomial curve, and interpreting intersections graphically. Part (d) requires recognizing that the equation can be rewritten as the intersection of the two curves, but this is a routine technique. The factorisation is straightforward (common factor then perfect square), and the curve sketching follows standard procedures. Slightly easier than average due to the guided structure and routine nature of each component. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Scheme | Marks | Guidance |
| \(x = -2\) | B1 | \(x = -2\) only. Accept labelled on the diagram but do not accept just \(-2\). If other equations are given (other than \(y = 0\)) then B0. Beware that \(x = -2\) is one of the solutions to the cubic so the equation of the asymptote must be seen in (a), on the diagram or by the question. |
| (a) Total: 1 mark | ||
| \(x^3 + 4x^2 + 4x = x(x+2)^2\) | M1A1 | M1: Attempts to take out a linear factor e.g \(x(x^2+4x±4)\) or \((x+2)(x^2±2x)\). May be implied by the correct answer. A1: \(x(x+2)^2\) or \(x(x+2)(x+2)\). Accept \((x+2)(x+2)x\) and condone a missing trailing bracket on the final linear factor e.g. \(x(x+2)(x+2\) but \(x(x+2)^2\) is A0. Condone \((x+0)(x+2)^2\). Isw once a correct expression is seen and ignore spurious \(=0\). |
| (b) Total: 2 marks | ||
| Sketch a positive cubic anywhere on a set of axes. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. The cubic does not need to have two turning points for this mark. Examples of acceptable cubic shapes (including where poor curvature would be condoned): [Examples provided showing cubic curves] | B1B1B1 | B1: Sketches a positive cubic anywhere on a set of axes. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. The cubic does not need to have two turning points for this mark. A cubic which: has a turning point where the x-axis and vertical asymptote intersect, passes through the origin (from quadrants 3 to 1 or from quadrants 2 to 4 – it cannot start or stop at the origin) It also cannot be a turning point. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. B1: \((-2, 0)\) indicated on the graph where their graph crosses or turns on the negative x-axis. Do not be concerned regarding any other points where the graph crosses or turns on the x-axis including the origin. Condone -2 labelled on the x-axis or the coordinates the wrong way round as \((0, -2)\) or with missing brackets. The asymptote labelled \(x = -2\) does not score this mark and do not be concerned with the point of intersection relative to the asymptote. |
| (c) Total: 3 marks | ||
| 2 as the graphs intersect (each other) twice (since \((x+2)(x^3+4x^2+4x) = 1\) is the same as \(x^3+4x^2+4x = \frac{1}{x+2}\)) | B1 | 2 (real) roots and a valid reason e.g. (graphs/curves/functions/equations/they) intersect/meet/cross/touch (each other) twice. e.g. "intersect twice" or "2 intersections" can score this mark. May also refer to an intersection in the first and third quadrants. Only withhold the mark if it is clear that they are referring to intersections between the cubic graph and the x-axis. Do not accept responses relating to the roots of the equation, use of the discriminant or other algebraic methods which do not use the graphs. |
| (d) Total: 1 mark | ||
| Total for Question 4: 7 marks |
| Answer/Scheme | Marks | Guidance |
|---|---|---|
| $x = -2$ | B1 | $x = -2$ only. Accept labelled on the diagram but do not accept just $-2$. If other equations are given (other than $y = 0$) then B0. Beware that $x = -2$ is one of the solutions to the cubic so the equation of the asymptote must be seen in (a), on the diagram or by the question. |
| **(a) Total: 1 mark** | | |
| $x^3 + 4x^2 + 4x = x(x+2)^2$ | M1A1 | M1: Attempts to take out a linear factor e.g $x(x^2+4x±4)$ or $(x+2)(x^2±2x)$. May be implied by the correct answer. A1: $x(x+2)^2$ or $x(x+2)(x+2)$. Accept $(x+2)(x+2)x$ and condone a missing trailing bracket on the final linear factor e.g. $x(x+2)(x+2$ but $x(x+2)^2$ is A0. Condone $(x+0)(x+2)^2$. Isw once a correct expression is seen and ignore spurious $=0$. |
| **(b) Total: 2 marks** | | |
| Sketch a positive cubic anywhere on a set of axes. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. The cubic does not need to have two turning points for this mark. Examples of acceptable cubic shapes (including where poor curvature would be condoned): [Examples provided showing cubic curves] | B1B1B1 | B1: Sketches a positive cubic anywhere on a set of axes. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. The cubic does not need to have two turning points for this mark. A cubic which: has a turning point where the x-axis and vertical asymptote intersect, passes through the origin (from quadrants 3 to 1 or from quadrants 2 to 4 – it cannot start or stop at the origin) It also cannot be a turning point. Do not penalise poor curvature provided the intention is clear and ignore if the curvature looks asymptotic. B1: $(-2, 0)$ indicated on the graph where their graph crosses or turns on the negative x-axis. Do not be concerned regarding any other points where the graph crosses or turns on the x-axis including the origin. Condone -2 labelled on the x-axis or the coordinates the wrong way round as $(0, -2)$ or with missing brackets. The asymptote labelled $x = -2$ does not score this mark and do not be concerned with the point of intersection relative to the asymptote. |
| **(c) Total: 3 marks** | | |
| 2 as the graphs intersect (each other) twice (since $(x+2)(x^3+4x^2+4x) = 1$ is the same as $x^3+4x^2+4x = \frac{1}{x+2}$) | B1 | 2 (real) roots and a valid reason e.g. (graphs/curves/functions/equations/they) intersect/meet/cross/touch (each other) twice. e.g. "intersect twice" or "2 intersections" can score this mark. May also refer to an intersection in the first and third quadrants. Only withhold the mark if it is clear that they are referring to intersections between the cubic graph and the x-axis. Do not accept responses relating to the roots of the equation, use of the discriminant or other algebraic methods which do not use the graphs. |
| **(d) Total: 1 mark** | | |
| **Total for Question 4: 7 marks** | | |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-08_687_775_248_646}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve $C$ with equation $y = \frac { 1 } { x + 2 }$
\begin{enumerate}[label=(\alph*)]
\item State the equation of the asymptote of $C$ that is parallel to the $y$-axis.
\item Factorise fully $x ^ { 3 } + 4 x ^ { 2 } + 4 x$
A copy of Figure 1, labelled Diagram 1, is shown on the next page.
\item On Diagram 1, add a sketch of the curve with equation
$$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$
On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
\item Hence state the number of real solutions of the equation
$$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$
giving a reason for your answer.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
\end{center}
Only use the copy of Diagram 1 if you need to redraw your answer to part (c).
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2023 Q4 [7]}}