| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Moderate -0.3 This is a standard rational inequality requiring sign analysis and critical point identification. While it involves algebraic manipulation (bringing to common form, finding critical points from numerator and denominator) and careful consideration of sign changes, it's a routine Further Maths technique with no conceptual surprises. Slightly easier than average due to being a single-part question with straightforward algebra. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Critical values \(-2\) and \(-5\) | B1, B1 | Seen anywhere in solution. Both correct B1B1; one correct B1B0 |
| \(\frac{x}{x+2} - \frac{2}{x+5} < 0\) leading to \(\frac{x^2+3x-4}{(x+2)(x+5)} < 0\) | ||
| \(\frac{(x+4)(x-1)}{(x+2)(x+5)} < 0\) | M1 | Attempt single fraction and factorise numerator or use quad formula |
| Critical values \(-4\) and \(1\) | A1 | Correct critical values. May be seen on graph or number line |
| \(-5 < x < -4,\ -2 < x < 1\) or \((-5,-4) \cup (-2,1)\) | dM1A1, A1 | dM1: Attempt interval inequality using one of \(-2\) or \(-5\) with another cv. A1,A1: Correct intervals. One correct scores A1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Critical values \(-2\) and \(-5\) | B1, B1 | Seen anywhere in solution |
| \(\frac{x}{x+2} < \frac{2}{x+5} \Rightarrow x(x+5)^2(x+2) < 2(x+2)^2(x+5)\) | ||
| \(\Rightarrow (x+5)(x+2)\big[x(x+5)-2(x+2)\big] < 0\) | ||
| \(\Rightarrow (x+5)(x+2)\big[(x-1)(x+4)\big] < 0\) | M1 | Multiply by \((x+5)^2(x+2)^2\) and attempt to factorise a quartic or use quad formula |
| Critical values \(-4\) and \(1\) | A1 | Correct critical values |
| \(-5 < x < -4,\ -2 < x < 1\) or \((-5,-4)\cup(-2,1)\) | dM1A1, A1 | dM1: Attempt interval using one of \(-2\) or \(-5\) with another cv. A1,A1: Correct intervals. One correct scores A1A0 |
## Question 1:
$$\frac{x}{x+2} < \frac{2}{x+5}$$
| Answer/Working | Mark | Guidance |
|---|---|---|
| Critical values $-2$ and $-5$ | B1, B1 | Seen anywhere in solution. Both correct B1B1; one correct B1B0 |
| $\frac{x}{x+2} - \frac{2}{x+5} < 0$ leading to $\frac{x^2+3x-4}{(x+2)(x+5)} < 0$ | | |
| $\frac{(x+4)(x-1)}{(x+2)(x+5)} < 0$ | M1 | Attempt single fraction and factorise numerator or use quad formula |
| Critical values $-4$ and $1$ | A1 | Correct critical values. May be seen on graph or number line |
| $-5 < x < -4,\ -2 < x < 1$ or $(-5,-4) \cup (-2,1)$ | dM1A1, A1 | dM1: Attempt interval inequality using one of $-2$ or $-5$ with another cv. A1,A1: Correct intervals. One correct scores A1A0 |
**ALT method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Critical values $-2$ and $-5$ | B1, B1 | Seen anywhere in solution |
| $\frac{x}{x+2} < \frac{2}{x+5} \Rightarrow x(x+5)^2(x+2) < 2(x+2)^2(x+5)$ | | |
| $\Rightarrow (x+5)(x+2)\big[x(x+5)-2(x+2)\big] < 0$ | | |
| $\Rightarrow (x+5)(x+2)\big[(x-1)(x+4)\big] < 0$ | M1 | Multiply by $(x+5)^2(x+2)^2$ and attempt to factorise a quartic or use quad formula |
| Critical values $-4$ and $1$ | A1 | Correct critical values |
| $-5 < x < -4,\ -2 < x < 1$ or $(-5,-4)\cup(-2,1)$ | dM1A1, A1 | dM1: Attempt interval using one of $-2$ or $-5$ with another cv. A1,A1: Correct intervals. One correct scores A1A0 |
*Note: Any solutions with no algebra (e.g. sketch graph followed by critical values with no working) scores max B1B1*
---\begin{enumerate}
\item Using algebra, find the set of values of $x$ for which
\end{enumerate}
$$\frac { x } { x + 2 } < \frac { 2 } { x + 5 }$$
\hfill \mbox{\textit{Edexcel F2 2015 Q1 [7]}}