| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two factors given |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem with two given factors, requiring systematic substitution and solving simultaneous equations. The multi-part structure guides students through standard steps (apply factor theorem, solve for constants, factorize, identify roots), making it easier than average despite being worth several marks. No novel insight or complex algebraic manipulation is required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((f(4)=)\ 2\times4^3-3a\times4^2+4b+8a=0\) | M1 | Attempts \(f(4)=0\) leading to equation in \(a\) and \(b\) only. Condone slips. Attempts using algebraic division score M0A0 |
| \(128+4b=40a \Rightarrow 32+b=10a\) | A1* | Simplifies and rearranges to given answer with no errors. Must be at least one intermediate stage. The \(=0\) must be correctly seen at some point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(2)=2\times2^3-3a\times2^2+2b+8a=0 \Rightarrow 8+b=2a\) | M1 | Attempts \(f(2)=0\) to form another equation in \(a\) and \(b\). Condone slips substituting in 2 |
| Solve simultaneously \(\Rightarrow a=\ldots\) or \(\Rightarrow b=\ldots\) | dM1 | Attempts to solve equations simultaneously to find value for \(a\) or \(b\). Dependent on previous M mark |
| \(a=3\) or \(b=-2\) or \(k=3\) | A1 | Sight of \(a=3\) or \(b=-2\) or \(k=3\) scores first 3 marks BUT answers with no working — send to review |
| \((f(x)=)(2x+3)(x-4)(x-2)\) | A1 | All on one line. Stating values of \(a\), \(b\) and \(k\) alone does not score this mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3\) | B1 | Listing the actual roots only is B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(12\) | B1ft | Follow through on their \(2x+k \Rightarrow -\frac{3k}{2}\) if \(k<-8\) |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(f(4)=)\ 2\times4^3-3a\times4^2+4b+8a=0$ | M1 | Attempts $f(4)=0$ leading to equation in $a$ and $b$ only. Condone slips. Attempts using algebraic division score M0A0 |
| $128+4b=40a \Rightarrow 32+b=10a$ | A1* | Simplifies and rearranges to given answer with no errors. Must be at least one intermediate stage. The $=0$ must be correctly seen at some point |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(2)=2\times2^3-3a\times2^2+2b+8a=0 \Rightarrow 8+b=2a$ | M1 | Attempts $f(2)=0$ to form another equation in $a$ and $b$. Condone slips substituting in 2 |
| Solve simultaneously $\Rightarrow a=\ldots$ or $\Rightarrow b=\ldots$ | dM1 | Attempts to solve equations simultaneously to find value for $a$ or $b$. Dependent on previous M mark |
| $a=3$ or $b=-2$ or $k=3$ | A1 | Sight of $a=3$ or $b=-2$ or $k=3$ scores first 3 marks BUT answers with no working — send to review |
| $(f(x)=)(2x+3)(x-4)(x-2)$ | A1 | All on one line. Stating values of $a$, $b$ and $k$ alone does not score this mark |
## Part (c)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3$ | B1 | Listing the actual roots only is B0 |
## Part (c)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $12$ | B1ft | Follow through on their $2x+k \Rightarrow -\frac{3k}{2}$ if $k<-8$ |
---\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.
$$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$
where $a$ and $b$ are constants.\\
Given that ( $x - 4$ ) is a factor of $\mathrm { f } ( x )$,\\
(a) use the factor theorem to show that
$$10 a = 32 + b$$
Given also that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$,\\
(b) express $\mathrm { f } ( x )$ in the form
$$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$
where $k$ is a constant to be found.\\
(c) Hence,\\
(i) state the number of real roots of the equation $\mathrm { f } ( x ) = 0$\\
(ii) write down the largest root of the equation $\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0$
\hfill \mbox{\textit{Edexcel AS Paper 1 2024 Q2 [8]}}