| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch with inequalities or regions |
| Difficulty | Standard +0.8 This question requires factorisation, curve sketching with stationary points, and determining the range of k for three intersections by finding turning points using calculus. While each component is standard A-level technique, the multi-step nature and the need to synthesize calculus (finding turning points) with graphical reasoning (determining when y=k intersects three times) elevates it above routine exercises. The set notation requirement and 'no calculator' constraint add modest additional challenge. |
| Spec | 1.02h Express solutions: using 'and', 'or', set and interval notation1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(9x-x^3=x(9-x^2)\) | M1 | Takes out factor of \(x\) or \(-x\); scored for \(\pm x(\pm9\pm x^2)\); may be implied by correct answer or \(\pm x(\pm x\pm3)(\pm x\pm3)\) |
| \(9x-x^3=x(3-x)(3+x)\) | A1 | Allow eg \(-x(x-3)(x+3)\), \(x(x-3)(-x-3)\); condone \(=0\) on end |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Cubic with correct orientation (negative cubic, minimum left, maximum right) | B1 | Be tolerant of pen slips; judge intent |
| Passes through origin, \((3,0)\) and \((-3,0)\) | B1 | Points may be indicated as just \(3\) and \(-3\) on axes; condone \(x\) and \(y\) wrong way round eg \((0,-3)\) for \((-3,0)\) if on correct axis; do not allow \((-3,0)\) labelled as \((3,0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y=9x-x^3\Rightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=9-3x^2=0\Rightarrow x=(\pm)\sqrt{3}\Rightarrow y=...\) | M1 | Differentiates to quadratic, solves \(\frac{\mathrm{d}y}{\mathrm{d}x}=0\), uses \(x\) to find \(y\); cannot be scored for answer without working |
| \(y=(\pm)6\sqrt{3}\) | A1 | Either or both values |
\(\left\{k\in\mathbb{R}:-6\sqrt{3}| A1ft |
Correct set notation following their \(6\sqrt{3}\); condone \(\left\{"-6\sqrt{3}" | |
## Question 7:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $9x-x^3=x(9-x^2)$ | M1 | Takes out factor of $x$ or $-x$; scored for $\pm x(\pm9\pm x^2)$; may be implied by correct answer or $\pm x(\pm x\pm3)(\pm x\pm3)$ |
| $9x-x^3=x(3-x)(3+x)$ | A1 | Allow eg $-x(x-3)(x+3)$, $x(x-3)(-x-3)$; condone $=0$ on end |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Cubic with correct orientation (negative cubic, minimum left, maximum right) | B1 | Be tolerant of pen slips; judge intent |
| Passes through origin, $(3,0)$ and $(-3,0)$ | B1 | Points may be indicated as just $3$ and $-3$ on axes; condone $x$ and $y$ wrong way round eg $(0,-3)$ for $(-3,0)$ if on correct axis; do not allow $(-3,0)$ labelled as $(3,0)$ |
**Part (c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $y=9x-x^3\Rightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=9-3x^2=0\Rightarrow x=(\pm)\sqrt{3}\Rightarrow y=...$ | M1 | Differentiates to quadratic, solves $\frac{\mathrm{d}y}{\mathrm{d}x}=0$, uses $x$ to find $y$; **cannot be scored for answer without working** |
| $y=(\pm)6\sqrt{3}$ | A1 | Either or both values |
| $\left\{k\in\mathbb{R}:-6\sqrt{3}<k<6\sqrt{3}\right\}$ | A1ft | Correct set notation following their $6\sqrt{3}$; condone $\left\{"-6\sqrt{3}"<k<"6\sqrt{3}"\right\}$; must be in terms of $k$; do not accept union notation |
---\begin{enumerate}
\item (a) Factorise completely $9 x - x ^ { 3 }$
\end{enumerate}
The curve $C$ has equation
$$y = 9 x - x ^ { 3 }$$
(b) Sketch $C$ showing the coordinates of the points at which the curve cuts the $x$-axis.
The line $l$ has equation $y = k$ where $k$ is a constant.\\
Given that $C$ and $l$ intersect at 3 distinct points,\\
(c) find the range of values for $k$, writing your answer in set notation.
Solutions relying on calculator technology are not acceptable.
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q7 [7]}}