| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve absolute value inequality |
| Difficulty | Standard +0.8 This FP1 question requires systematic case analysis of the absolute value inequality, solving multiple quadratic inequalities, and carefully combining solution sets—significantly more demanding than routine A-level pure maths inequalities but standard for Further Maths students who have practiced this technique. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.02l Modulus function: notation, relations, equations and inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Complete method to form equations: Method 1: \(x^2 - 2x = x \Rightarrow x^2 - 3x = 0\) and \(x^2 - 2x = -x \Rightarrow x^2 - x = 0\); Method 2: \((x^2-2x)^2 = x^2 \Rightarrow x^4 - 4x^3 + 3x^2 = 0\) | M1 | Both equations attempted (M1); allow inequalities in place of equals |
| Solves equations to find at least all non-zero critical values | dM1 | Dependent on M1; both equations must be solved in Method 1 |
| \(x = 0, 1, 3\) | A1 | Correct critical values including 0 |
| \(x = 0,\ 1 \leq x \leq 3\) | A1 | Correct range; accept set/interval notation e.g. \(\{0\}\cup\{x:1\leq x \leq 3\}\) or \([0]\cup[1,3]\) |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Complete method to form equations: **Method 1:** $x^2 - 2x = x \Rightarrow x^2 - 3x = 0$ and $x^2 - 2x = -x \Rightarrow x^2 - x = 0$; **Method 2:** $(x^2-2x)^2 = x^2 \Rightarrow x^4 - 4x^3 + 3x^2 = 0$ | M1 | Both equations attempted (M1); allow inequalities in place of equals |
| Solves equations to find at least all non-zero critical values | dM1 | Dependent on M1; both equations must be solved in Method 1 |
| $x = 0, 1, 3$ | A1 | Correct critical values including 0 |
| $x = 0,\ 1 \leq x \leq 3$ | A1 | Correct range; accept set/interval notation e.g. $\{0\}\cup\{x:1\leq x \leq 3\}$ or $[0]\cup[1,3]$ |
---\begin{enumerate}
\item Use algebra to determine the values of $x$ for which
\end{enumerate}
$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$
\hfill \mbox{\textit{Edexcel FP1 2024 Q2 [4]}}