| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |linear| < |linear| |
| Difficulty | Standard +0.8 This question requires systematic case analysis of absolute value inequalities (splitting into regions based on sign changes at x=-1/2 and x=3), then solving multiple linear inequalities and combining solution sets. Part (ii) adds an optimization element requiring understanding of the solution set's geometry. While methodical, it demands careful organization and goes beyond routine absolute value exercises. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt process for finding both critical values | M1 | squaring both sides to obtain 3 terms on each side or considering 2 different linear eqns/inequalities |
| Obtain \(-4\) | A1 | — |
| Obtain \(\frac{2}{3}\) | A1 | — |
| Attempt process for solving inequality | M1 | table, sketch, ...; needs two critical values; implied by plausible answer |
| Obtain \(-4 \le x \le \frac{2}{3}\) | A1 5 | with \(\le\) and not \(<\) |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct process to find value of \(\left | x + 2 \right | \) using any value |
| Obtain \(2\frac{2}{3}\) or \(\frac{8}{3}\) | A1 2 | dependent on 5 marks awarded in part (i) |
## (i)
Attempt process for finding both critical values | M1 | squaring both sides to obtain 3 terms on each side or considering 2 different linear eqns/inequalities
Obtain $-4$ | A1 | —
Obtain $\frac{2}{3}$ | A1 | —
Attempt process for solving inequality | M1 | table, sketch, ...; needs two critical values; implied by plausible answer
Obtain $-4 \le x \le \frac{2}{3}$ | A1 5 | with $\le$ and not $<$
## (ii)
Use correct process to find value of $\left| x + 2 \right|$ using any value | M1 | ... whether part of answer to (i) or not
Obtain $2\frac{2}{3}$ or $\frac{8}{3}$ | A1 2 | dependent on 5 marks awarded in part (i)
---\begin{enumerate}[label=(\roman*)]
\item Solve the inequality $|2x + 1| \leqslant |x - 3|$. [5]
\item Given that $x$ satisfies the inequality $|2x + 1| \leqslant |x - 3|$, find the greatest possible value of $|x + 2|$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q5 [7]}}