| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Reflection or vertical transformation |
| Difficulty | Standard +0.8 This is a multi-step Further Maths question requiring factorisation of a cubic (likely needing factor theorem), solving a cubic inequality, then applying two transformations and solving another inequality. While each individual step is standard, the combination of techniques and the transformation work in part (b) requires careful algebraic manipulation and understanding of function transformations, placing it moderately above average difficulty. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| 11(a)(i) | Obtains one factor of f(x) | 1.1a |
| Deduces the correct factorisation | 2.2a | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 11(a)(ii) | Deduces that x<5 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| solution to the inequality | 1.1b | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 11(b) | Obtains ±5.5 or ±12 |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(ii) | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(ii) | 2.2a | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| solution to the inequality | 1.1b | A1 |
| Subtotal | 3 | |
| Question total | 7 | |
| Q | Marking instructions | AO |
Question 11:
--- 11(a)(i) ---
11(a)(i) | Obtains one factor of f(x) | 1.1a | M1 | f(x)=( x−5 )( 2x+3 )2
Deduces the correct factorisation | 2.2a | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 11(a)(ii) ---
11(a)(ii) | Deduces that x<5 | 2.2a | M1 | If f 0,
x<−3,−3 < x<5
(𝑥𝑥)2< 2
Obtains a completely correct
solution to the inequality | 1.1b | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 11(b) ---
11(b) | Obtains ±5.5 or ±12
ft 7 + their critical values from
(a)(ii) | 2.2a | M1 | g(x)=0⇒ x=11,x=12
2
If g(x)≤0,
x=11 or x≥12
2
Obtains 5.5 and 12
ft 7 + their critical values from
(a)(ii) | 2.2a | A1F
Obtains a completely correct
solution to the inequality | 1.1b | A1
Subtotal | 3
Question total | 7
Q | Marking instructions | AO | Marks | Typical solutionThe function f is defined by
$$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Fully factorise $f(x)$
[2 marks]
\item Hence, solve the inequality $f(x) < 0$
[2 marks]
\end{enumerate}
\item The graph of $y = f(x)$ is translated by the vector $\begin{pmatrix} 7 \\ 0 \end{pmatrix}$
The new graph is then reflected in the $x$-axis, to give the graph of $y = g(x)$
Solve the inequality $g(x) \leq 0$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2023 Q11 [7]}}