| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(algebraic transform) = 0 |
| Difficulty | Standard +0.3 Part (a) is a straightforward cubic factorization (factor out x, then solve quadratic). Part (b) requires recognizing the substitution u = (y-2)² to transform it into the form from (a), then solving u = x² for y. This is a standard 'hence' question testing algebraic manipulation and substitution, slightly above average due to the multi-step nature and the (y-2)² substitution, but still a routine AS-level exercise. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(3x^3-17x^2-6x=0 \Rightarrow x(3x^2-17x-6)=0\) | M1 | Factorises out or cancels by \(x\) to form a quadratic |
| \(\Rightarrow x(3x+1)(x-6)=0\) | dM1 | Attempt to find \(x\); may be awarded for factorisation or use of quadratic formula |
| \(x = 0, -\dfrac{1}{3}, 6\) | A1 | All three correct and no extras |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Attempts to solve \((y-2)^2 = n\) where \(n\) is any solution from (a) | M1 | At least one stage of working must be seen; e.g. \((y-2)^2=0 \Rightarrow y=2\) |
| Two of \(2,\ 2\pm\sqrt{6}\) | A1ft | Follow through on \((y-2)^2=n \Rightarrow y=2\pm\sqrt{n}\) where \(n\) is a positive solution from (a) |
| All three of \(2,\ 2\pm\sqrt{6}\) | A1 | All three and no extra solutions; requires M1A1 scored |
# Question 6:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $3x^3-17x^2-6x=0 \Rightarrow x(3x^2-17x-6)=0$ | M1 | Factorises out or cancels by $x$ to form a quadratic |
| $\Rightarrow x(3x+1)(x-6)=0$ | dM1 | Attempt to find $x$; may be awarded for factorisation or use of quadratic formula |
| $x = 0, -\dfrac{1}{3}, 6$ | A1 | All three correct and no extras |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Attempts to solve $(y-2)^2 = n$ where $n$ is any solution from (a) | M1 | At least one stage of working must be seen; e.g. $(y-2)^2=0 \Rightarrow y=2$ |
| Two of $2,\ 2\pm\sqrt{6}$ | A1ft | Follow through on $(y-2)^2=n \Rightarrow y=2\pm\sqrt{n}$ where $n$ is a positive solution from (a) |
| All three of $2,\ 2\pm\sqrt{6}$ | A1 | All three and no extra solutions; requires M1A1 scored |
---\begin{enumerate}
\item In this question you should show all stages of your working.
\end{enumerate}
Solutions relying on calculator technology are not acceptable.\\
(a) Using algebra, find all solutions of the equation
$$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
(b) Hence find all real solutions of
$$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q6 [6]}}