| Exam Board | AQA |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve absolute value inequality |
| Difficulty | Standard +0.8 This question requires sketching a modulus function involving a quadratic (requiring identification of roots at 0 and 4, vertex at 2, and reflection of negative portions), then solving a modulus inequality by considering cases and solving resulting quadratics. While systematic, it demands careful case analysis, algebraic manipulation of quadratics, and geometric interpretation—more demanding than standard A-level but not requiring deep novel insight. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks |
|---|---|
| 15(a) | Draws curve with basically |
| Answer | Marks | Guidance |
|---|---|---|
| y-values. | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| four distinct points. | 1.1b | B1 |
| Value of 4 shown at x-intercept | 1.1b | B1 |
| Subtotal | 3 | |
| Q | Marking | |
| instructions | AO | Marks |
| Answer | Marks |
|---|---|
| 15(b) | Uses the modulus |
| Answer | Marks | Guidance |
|---|---|---|
| equations. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| approximations). | 1.1a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| approximations). | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| exact values. | 2.2a | A1 |
| Subtotal | 4 | |
| Question total | 7 | |
| Q | Marking Instructions | AO |
Question 15:
--- 15(a) ---
15(a) | Draws curve with basically
correct shape and no negative
y-values. | 1.1b | B1
Their graph intersects line at
four distinct points. | 1.1b | B1
Value of 4 shown at x-intercept | 1.1b | B1
Subtotal | 3
Q | Marking
instructions | AO | Marks | Typical solution
--- 15(b) ---
15(b) | Uses the modulus
function to obtain
two separate
quadratic
equations. | 3.1a | M1 | x2 – 4x = 5 – x
x2 – 3x – 5 = 0
3± 29
x =
2
4x – x2 = 5 – x
0 = x2 – 5x + 5
5± 5
x =
2
3– 29 5– 5 5+ 5 3+ 29
x < , < x < ,x>
2 2 2 2
Obtains four
correct x-values of
points of
intersection
(condone decimal
approximations). | 1.1a | A1
Uses their graph to
obtain at least one
subset of the
solution set
(condone decimal
approximations). | 2.2a | M1
Deduces a
completely correct
solution set with
exact values. | 2.2a | A1
Subtotal | 4
Question total | 7
Q | Marking Instructions | AO | Marks | Typical SolutionThe diagram shows the line $y = 5 - x$
\includegraphics{figure_15}
\begin{enumerate}[label=(\alph*)]
\item On the diagram above, sketch the graph of $y = |x^2 - 4x|$, including all parts of the graph where it intersects the line $y = 5 - x$
(You do not need to show the coordinates of the points of intersection.)
[3 marks]
\item Find the solution of the inequality
$$|x^2 - 4x| > 5 - x$$
Give your answer in an exact form.
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 2 2024 Q15 [7]}}