| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Solve Polynomial Equation |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring routine techniques: sketching a simple rational function with asymptotes, equating two functions to form a cubic, and using factor theorem with polynomial division. While it has multiple parts (7 marks typical), each step is standard AS-level material with no novel problem-solving required. The 'show that' in part (b) is algebraic verification, and part (c) is textbook polynomial division given one root. Slightly easier than average due to the scaffolded structure and explicit hint. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct positive reciprocal graph shape with both branches | M1 | Both branches anywhere on axes; condone poor curvature if no clear maximum/minimum intended |
| Fully correct graph with intersection labelled at \(x = -2\) on graph | A1 | Condone labelled as \((0, -2)\); if two horizontal/vertical asymptotes drawn then mark cannot be scored |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Asymptotes: \(y = 3\), \(x = 0\) | B1 | cao (not \(x \neq 0\)); must be correct for their graph; cannot be awarded for just stating 0 under \(y\)-axis and 3 on dashed horizontal line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{6}{x} + 3 = 3x^2 - 4x - 10 \Rightarrow 6 + 3x = 3x^3 - 4x^2 - 10x\) | M1 | Sets \(\frac{6}{x} + 3 = 3x^2 - 4x - 10\) and rearranges to form a cubic; terms need not be collected |
| \(3x^3 - 4x^2 - 13x - 6 = 0\) | A1* | No errors seen; must have at least one stage of intermediate working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - 2x \pm \ldots\) e.g. \(3x + 2 \overline{)3x^3 - 4x^2 - 13x - 6}\) | M1 | Attempts algebraic division of \(3x^3 - 4x^2 - 13x - 6\) by \(3x + 2\); must reach \(x^2 - 2x\pm\ldots\); note proceeding to three linear factors with no working is M0A0dM0A0 |
| \(x^2 - 2x - 3\ (= 0)\) or \(3x^2 - 6x - 9\ (= 0)\) | A1 | Correct quadratic factor |
| e.g. \((3x+2)(x-3)(x+1) = 0 \Rightarrow x = \ldots\) | dM1 | Attempts to solve resulting quadratic by factorising, formula or completing the square; dependent on previous M1 |
| \(x = -1,\ x = 3\) | A1cso | Only these two values; ignore any reference to \(x = -\frac{2}{3}\); withhold if one solution rejected |
## Question 5(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct positive reciprocal graph shape with both branches | M1 | Both branches anywhere on axes; condone poor curvature if no clear maximum/minimum intended |
| Fully correct graph with intersection labelled at $x = -2$ on graph | A1 | Condone labelled as $(0, -2)$; if two horizontal/vertical asymptotes drawn then mark cannot be scored |
## Question 5(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Asymptotes: $y = 3$, $x = 0$ | B1 | cao (not $x \neq 0$); must be correct for their graph; cannot be awarded for just stating 0 under $y$-axis and 3 on dashed horizontal line |
## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{6}{x} + 3 = 3x^2 - 4x - 10 \Rightarrow 6 + 3x = 3x^3 - 4x^2 - 10x$ | M1 | Sets $\frac{6}{x} + 3 = 3x^2 - 4x - 10$ and rearranges to form a cubic; terms need not be collected |
| $3x^3 - 4x^2 - 13x - 6 = 0$ | A1* | No errors seen; must have at least one stage of intermediate working |
## Question 5(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 2x \pm \ldots$ e.g. $3x + 2 \overline{)3x^3 - 4x^2 - 13x - 6}$ | M1 | Attempts algebraic division of $3x^3 - 4x^2 - 13x - 6$ by $3x + 2$; must reach $x^2 - 2x\pm\ldots$; note proceeding to three linear factors with no working is M0A0dM0A0 |
| $x^2 - 2x - 3\ (= 0)$ or $3x^2 - 6x - 9\ (= 0)$ | A1 | Correct quadratic factor |
| e.g. $(3x+2)(x-3)(x+1) = 0 \Rightarrow x = \ldots$ | dM1 | Attempts to solve resulting quadratic by factorising, formula or completing the square; dependent on previous M1 |
| $x = -1,\ x = 3$ | A1cso | Only these two values; ignore any reference to $x = -\frac{2}{3}$; withhold if one solution rejected |\begin{enumerate}
\item The curve $C _ { 1 }$ has equation
\end{enumerate}
$$y = \frac { 6 } { x } + 3$$
(a) (i) Sketch $C _ { 1 }$ stating the coordinates of any points where the curve cuts the coordinate axes.\\
(ii) State the equations of any asymptotes to the curve $C _ { 1 }$
The curve $C _ { 2 }$ has equation
$$y = 3 x ^ { 2 } - 4 x - 10$$
(b) Show that $C _ { 1 }$ and $C _ { 2 }$ intersect when
$$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$
Given that the $x$ coordinate of one of the points of intersection is $- \frac { 2 } { 3 }$\\
(c) use algebra to find the $x$ coordinates of the other points of intersection between $C _ { 1 }$ and $C _ { 2 }$\\
(Solutions relying on calculator technology are not acceptable.)
\hfill \mbox{\textit{Edexcel AS Paper 1 2024 Q5 [9]}}