| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Moderate -0.5 This is a straightforward rational inequality requiring standard algebraic manipulation: bringing to common denominator, considering critical points x=0 and x=2, and testing intervals. While it requires care with sign changes, it's a routine technique covered early in Further Maths with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance Notes |
| \(\frac{1}{x-2} - \frac{2}{x} > 0 \Rightarrow \frac{4-x}{x(x-2)} > 0\) | M1 | Collect to one side and attempt common denominator of \(x(x-2)\) |
| \(x = 0,\ 2,\ 4\) | B1, A1 | B1 for 0 and 2, A1 for 4 |
| \(x < 0,\ 2 < x < 4\) | M1 | For critical values \(\alpha, \beta, \gamma\) in ascending order, attempts \(x < \alpha\) and \(\beta < x < \gamma\), condoning mixed open/closed inequalities; or for one of \(x<0\) or \(2 |
| \(x < 0,\ 2 < x < 4\) or \((-\infty, 0)\) or \([-\infty, 0),\ (2,4)\) | A1 | Correct inequalities. Ignore connector between inequalities e.g. allow "or", "and", "," etc. but not \(\cap\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance Notes |
| \(x^2(x-2) - 2x(x-2)^2 > 0\) then \(x(x-2)(4-x) > 0\) | M1 | Multiply by \(x^2(x-2)^2\) and attempt to factorise taking out factor of \(x(x-2)\) |
| \(x = 0,\ 2,\ 4\) | B1, A1 | B1 for 0 and 2, A1 for 4 |
| \(x < 0,\ 2 < x < 4\) | M1 | As above |
| \(x < 0,\ 2 < x < 4\) | A1 | Correct inequalities |
# Question 1:
| Working/Answer | Marks | Guidance Notes |
|---|---|---|
| $\frac{1}{x-2} - \frac{2}{x} > 0 \Rightarrow \frac{4-x}{x(x-2)} > 0$ | M1 | Collect to one side and attempt common denominator of $x(x-2)$ |
| $x = 0,\ 2,\ 4$ | B1, A1 | B1 for 0 **and** 2, A1 for 4 |
| $x < 0,\ 2 < x < 4$ | M1 | For critical values $\alpha, \beta, \gamma$ in ascending order, attempts $x < \alpha$ and $\beta < x < \gamma$, condoning mixed open/closed inequalities; or for **one of** $x<0$ **or** $2<x<4$ |
| $x < 0,\ 2 < x < 4$ or $(-\infty, 0)$ or $[-\infty, 0),\ (2,4)$ | A1 | Correct inequalities. Ignore connector between inequalities e.g. allow "or", "and", "," etc. but not $\cap$ |
**Alternative 1:** Multiply by $x^2(x-2)^2$
| Working/Answer | Marks | Guidance Notes |
|---|---|---|
| $x^2(x-2) - 2x(x-2)^2 > 0$ then $x(x-2)(4-x) > 0$ | M1 | Multiply by $x^2(x-2)^2$ and attempt to factorise taking out factor of $x(x-2)$ |
| $x = 0,\ 2,\ 4$ | B1, A1 | B1 for 0 **and** 2, A1 for 4 |
| $x < 0,\ 2 < x < 4$ | M1 | As above |
| $x < 0,\ 2 < x < 4$ | A1 | Correct inequalities |
---\begin{enumerate}
\item Use algebra to find the set of values of $x$ for which
\end{enumerate}
$$\frac { 1 } { x - 2 } > \frac { 2 } { x }$$
\hfill \mbox{\textit{Edexcel F2 2018 Q1 [5]}}