| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two factors given |
| Difficulty | Moderate -0.3 This is a straightforward factor theorem application requiring students to substitute x = -1 and x = 3 to create simultaneous equations for p and q, then factorise. While it involves multiple steps and algebraic manipulation, it's a standard AS-level technique with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| 3(a) | Substitutes x = –1 or x = 3 into f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| eg f(x) ≡ (x + 1)( x – 3)(ax +b) | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| or obtains a = 2, b = 1 | 1.1b | A1 |
| Solves to find p and q CAO | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 3(b) | Uses inspection, division by |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Completes factorisation | 1.1b | A1 |
| Total | 5 | |
| Q | Marking Instructions | AO |
Question 3:
--- 3(a) ---
3(a) | Substitutes x = –1 or x = 3 into f(x)
to obtain one equation or uses
identity
eg f(x) ≡ (x + 1)( x – 3)(ax +b) | 3.1a | M1 | x = –1 –p – 3 + 8 + q = 0
x = 3 27p – 27 – 24 + q = 0
p = 2 and q = –3
Obtains two correct equations by
substitution method ACF
or obtains a = 2, b = 1 | 1.1b | A1
Solves to find p and q CAO | 1.1b | A1
--- 3(b) ---
3(b) | Uses inspection, division by
quadratic factor or repeated division
or finds third root x =
1
PI by (x + )
–2
1 | 1.1a | M1 | (x + 1)(x – 3) = x2 – 2x – 3
(x2 – 2x – 3)(2x + 1)
(x + 1)(x – 3)(2x + 1)
2
Completes factorisation | 1.1b | A1
Total | 5
Q | Marking Instructions | AO | Marks | Typical SolutionIt is given that $(x + 1)$ and $(x - 3)$ are two factors of $f(x)$, where
$$f(x) = px^3 - 3x^2 - 8x + q$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p$ and $q$.
[3 marks]
\item Fully factorise $f(x)$.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2019 Q3 [5]}}