| Exam Board | Edexcel |
|---|---|
| Module | FP1 AS (Further Pure 1 AS) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Standard +0.3 This is a straightforward rational inequality requiring standard algebraic manipulation: bringing terms to one side, finding a common denominator, factoring, and analyzing sign changes using critical points. While it requires care with sign analysis and understanding that multiplying by x(x+2) requires considering when this is positive/negative, this is a routine FP1 technique with no novel insight needed. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable |
| VIIIV SIHI NI JIIIM ION OC | VIIIV SHIL NI JIHM I I ON O O | VIIV SIHI NI JIIIM IONOO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{x} < \frac{x}{x+2}\) | ||
| \(\frac{(x+2)-x^2}{x(x+2)} < 0\) or \(x(x+2)^2 - x^3(x+2) < 0\) | M1 | Gathers terms on one side and puts over common denominator, or multiply by \(x^2(x+2)^2\) and gather terms on one side |
| \(\frac{x^2-x-2}{x(x+2)} > 0 \Rightarrow \frac{(x-2)(x+1)}{x(x+2)} > 0\) or \(x(x+2)(2-x)(x+1) < 0\) | M1 | Factorise numerator or find roots of numerator or factorise resulting equation into 4 factors |
| At least two correct critical values from \(-2, -1, 0, 2\) | A1 | At least 2 correct critical values found |
| All four correct critical values \(-2, -1, 0, 2\) | A1 | Exactly 4 correct critical values |
| \(\{x \in \mathbb{R}: x < -2\} \cup \{x \in \mathbb{R}: -1 < x < 0\} \cup \{x \in \mathbb{R}: x > 2\}\) | M1, A1 | M1: Deduces 2 "outsides" and "middle interval" required (by sketch, number line or other means). A1: Exactly 3 correct intervals; accept equivalent set notations e.g. \(\mathbb{R} - ([-2,-1]\cup[0,2])\) or \(\{x\in\mathbb{R}: x<-2 \text{ or } -1 |
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{x} < \frac{x}{x+2}$ | | |
| $\frac{(x+2)-x^2}{x(x+2)} < 0$ or $x(x+2)^2 - x^3(x+2) < 0$ | M1 | Gathers terms on one side and puts over common denominator, or multiply by $x^2(x+2)^2$ and gather terms on one side |
| $\frac{x^2-x-2}{x(x+2)} > 0 \Rightarrow \frac{(x-2)(x+1)}{x(x+2)} > 0$ or $x(x+2)(2-x)(x+1) < 0$ | M1 | Factorise numerator or find roots of numerator or factorise resulting equation into 4 factors |
| At least two correct critical values from $-2, -1, 0, 2$ | A1 | At least 2 correct critical values found |
| All four correct critical values $-2, -1, 0, 2$ | A1 | Exactly 4 correct critical values |
| $\{x \in \mathbb{R}: x < -2\} \cup \{x \in \mathbb{R}: -1 < x < 0\} \cup \{x \in \mathbb{R}: x > 2\}$ | M1, A1 | M1: Deduces 2 "outsides" and "middle interval" required (by sketch, number line or other means). A1: Exactly 3 correct intervals; accept equivalent set notations e.g. $\mathbb{R} - ([-2,-1]\cup[0,2])$ or $\{x\in\mathbb{R}: x<-2 \text{ or } -1<x<0 \text{ or } x>2\}$ |
---\begin{enumerate}
\item Use algebra to find the set of values of x for which
\end{enumerate}
$$\frac { 1 } { x } < \frac { x } { x + 2 }$$
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VIIIV SIHI NI JIIIM ION OC & VIIIV SHIL NI JIHM I I ON O O & VIIV SIHI NI JIIIM IONOO \\
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\hfill \mbox{\textit{Edexcel FP1 AS Q3 [6]}}