| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Compound inequality with double bound |
| Difficulty | Moderate -0.3 Part (i) is a routine linear double inequality requiring simple algebraic manipulation. Part (ii) is a standard quadratic inequality requiring rearrangement to standard form, factorization or quadratic formula, and correct interpretation of the solution set. Both are textbook exercises with no novel insight required, though part (ii) involves more steps and the quadratic inequality is slightly above average difficulty for C1, bringing the overall question slightly below average difficulty. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| i) \(5 - 3 < 6x < 14 - 3\); \(2 < 6x < 11\); \(\frac{1}{3} < x < \frac{11}{6}\) | M1, A1, A1 [3] | Attempt to solve two equations/inequalities each involving all 3 terms; 2, 11 seen from correct inequalities. www Award full marks if initially working with equations but final answer correct. Allow "\(\frac{1}{3} < x\) and \(x < \frac{11}{6}\)" or "\(\frac{1}{3} < x < \frac{11}{6}\)" or "\(\frac{11}{6} > x > \frac{1}{3}\)" but do not allow "\(\frac{11}{6}\)" but do not allow "\(x \leq -\frac{2}{3}, x \geq 5\)" or "\(x \leq -\frac{2}{3}\)" |
| ii) \(3x^2 - 13x - 10 \geq 0\); \((3x + 2)(x - 5) \geq 0\); \(x \leq -\frac{2}{3}, x \geq 5\) | M1*, M1dep*, A1, M1, A1 [5] | Expands and rearranges to collect all terms on one side. Correct method to find roots. \(-\frac{2}{3}, 5\) seen as roots. Chooses "outside region" for their roots of their quadratic. Do not allow strict inequalities for final mark. See guidance at end of mark scheme: e.g. \(-\frac{2}{3} \leq x \geq 5\) scores M1A0; Allow "\(x \leq -\frac{2}{3}, x \geq 5\)", "\(x \leq -\frac{2}{3}\)" or "\(x \leq -\frac{2}{3}\)" or \(x \geq 5\)" but do not allow "\(x \leq -\frac{2}{3}\)" |
**i)** $5 - 3 < 6x < 14 - 3$; $2 < 6x < 11$; $\frac{1}{3} < x < \frac{11}{6}$ | M1, A1, A1 [3] | Attempt to solve two equations/inequalities each involving all 3 terms; 2, 11 seen from correct inequalities. www Award full marks if initially working with equations but final answer correct. Allow "$\frac{1}{3} < x$ and $x < \frac{11}{6}$" or "$\frac{1}{3} < x < \frac{11}{6}$" or "$\frac{11}{6} > x > \frac{1}{3}$" but do not allow "$\frac{11}{6}$" but do not allow "$x \leq -\frac{2}{3}, x \geq 5$" or "$x \leq -\frac{2}{3}$"
**ii)** $3x^2 - 13x - 10 \geq 0$; $(3x + 2)(x - 5) \geq 0$; $x \leq -\frac{2}{3}, x \geq 5$ | M1*, M1dep*, A1, M1, A1 [5] | Expands and rearranges to collect all terms on one side. Correct method to find roots. $-\frac{2}{3}, 5$ seen as roots. Chooses "outside region" for their roots of their quadratic. Do not allow strict inequalities for final mark. See guidance at end of mark scheme: e.g. $-\frac{2}{3} \leq x \geq 5$ scores M1A0; Allow "$x \leq -\frac{2}{3}, x \geq 5$", "$x \leq -\frac{2}{3}$" or "$x \leq -\frac{2}{3}$" or $x \geq 5$" but do not allow "$x \leq -\frac{2}{3}$"Solve the following inequalities.
\begin{enumerate}[label=(\roman*)]
\item $5 < 6x + 3 < 14$ [3]
\item $x(3x - 13) \geqslant 10$ [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2014 Q5 [8]}}