| Exam Board | Edexcel |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Standard +0.3 This is a rational inequality requiring algebraic manipulation (multiplying by x², considering sign cases) and critical point analysis. While it's Further Maths (FP1), it's a straightforward first question requiring standard techniques without novel insight—slightly easier than average overall but appropriate difficulty for its context. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x(x-1) > \frac{x-1}{x}\) → \(\frac{x^2(x-1)-x-1}{x}>0\) or \(x^3(x-1)-x(x-1)>0\) | M1 | 2.1 |
| \(\frac{(x-1)^2(x+1)}{x}>0\) or \(x(x-1)^2(x+1)>0\) | M1 | 1.1b |
| Critical values \(0\) and \(1\) | A1 | 1.1b |
| All three critical values \(-1, 0, 1\) | A1 | 1.1b |
| \(\{x \in \mathbb{R}: x<-1\} \cup \{x \in \mathbb{R}: 0 | M1, A1 | 2.2a, 2.5 |
| Total: 6 marks |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x(x-1) > \frac{x-1}{x}$ → $\frac{x^2(x-1)-x-1}{x}>0$ or $x^3(x-1)-x(x-1)>0$ | M1 | 2.1 | Gathers terms on one side and puts over common denominator, or multiplies by $x^2$ and gathers terms on one side |
| $\frac{(x-1)^2(x+1)}{x}>0$ or $x(x-1)^2(x+1)>0$ | M1 | 1.1b | Factorises numerator into 3 factors or factorises into 4 factors |
| Critical values $0$ and $1$ | A1 | 1.1b | Identifies the critical values 0 and 1 |
| All three critical values $-1, 0, 1$ | A1 | 1.1b | All 3 correct critical values |
| $\{x \in \mathbb{R}: x<-1\} \cup \{x \in \mathbb{R}: 0<x<1\} \cup \{x \in \mathbb{R}: x>1\}$ | M1, A1 | 2.2a, 2.5 | Deduces 1 "inside" and 2 "outside" inequalities with critical values in ascending order; exactly 3 correct intervals using correct notation. Allow e.g. $\{x:x<-1\}\cup\{x:0<x<1\}\cup\{x:x>1\}$ |
| **Total: 6 marks** | | |\begin{enumerate}
\item Use algebra to determine the values of $x$ for which
\end{enumerate}
$$x ( x - 1 ) > \frac { x - 1 } { x }$$
giving your answer in set notation.
\hfill \mbox{\textit{Edexcel FP1 AS 2021 Q1 [6]}}