| Exam Board | Edexcel |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Standard +0.8 This is a Further Maths FP1 question requiring factorization of two quadratics, finding a common denominator, combining fractions, solving the resulting inequality with sign analysis across multiple critical points, and careful handling of undefined points. More demanding than standard A-level inequalities but routine for FP1 students who have practiced rational inequalities. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{2x^2+3x+1-x^2-3x}{(2x-1)(2x+1)(x+3)}>0\) or \((x+1)(2x-1)(2x+1)^2(x+3)-x(2x-1)(2x+1)(x+3)^2>0\) | M1 | Gathers terms on one side over common denominator, or multiplies by \((2x+1)^2(2x-1)^2(x+3)^2\) |
| \(\frac{x^2+1}{(2x-1)(2x+1)(x+3)}>0\) or \((x+3)(2x-1)(2x+1)(x^2+1)>0\) | dM1 | Expands/simplifies numerator or factorises into 4 factors. Depends on previous M1 |
| All three critical values \(-3, -\frac{1}{2}, \frac{1}{2}\) | A1 | Correct critical values, no "extras"; ignore attempts to solve \(x^2+1=0\) |
| \(\left\{x\in\mathbb{R}:-3 | dM1, A1 | dM1: Deduces 1 "inside" and 1 "outside" interval needed with critical values in ascending order. A1: Exactly 2 correct intervals, equivalent notation acceptable |
## Question 2:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{2x^2+3x+1-x^2-3x}{(2x-1)(2x+1)(x+3)}>0$ or $(x+1)(2x-1)(2x+1)^2(x+3)-x(2x-1)(2x+1)(x+3)^2>0$ | M1 | Gathers terms on one side over common denominator, or multiplies by $(2x+1)^2(2x-1)^2(x+3)^2$ |
| $\frac{x^2+1}{(2x-1)(2x+1)(x+3)}>0$ or $(x+3)(2x-1)(2x+1)(x^2+1)>0$ | dM1 | Expands/simplifies numerator or factorises into 4 factors. Depends on previous M1 |
| All three critical values $-3, -\frac{1}{2}, \frac{1}{2}$ | A1 | Correct critical values, no "extras"; ignore attempts to solve $x^2+1=0$ |
| $\left\{x\in\mathbb{R}:-3<x<-\frac{1}{2}\right\}\cup\left\{x\in\mathbb{R}:x>\frac{1}{2}\right\}$ | dM1, A1 | dM1: Deduces 1 "inside" and 1 "outside" interval needed with critical values in ascending order. A1: Exactly 2 correct intervals, equivalent notation acceptable |
---\begin{enumerate}
\item Use algebra to determine the values of $x$ for which
\end{enumerate}
$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$
\hfill \mbox{\textit{Edexcel FP1 AS 2020 Q2 [5]}}