| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Combined linear and quadratic inequalities |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing routine manipulation of linear and quadratic inequalities. Part (a) requires simple algebraic rearrangement, part (b) involves factorizing a quadratic and applying standard inequality rules, and part (c) combines the results using set intersection. All techniques are standard textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(6x + 3 > 5 - 2x \Rightarrow 8x > 2\) | M1 | Multiply out and collect terms (allow one slip, allow \(=\) here) |
| \(x > \frac{1}{4}\) or \(0.25\) or \(\frac{2}{8}\) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((2x-1)(x-3)\ (> 0)\) | M1 | Attempting to factorise 3TQ \(\to x = \ldots\) |
| Critical values \(x = \frac{1}{2},\ 3\) (both) | A1 | |
| Choosing "outside" region | M1 | 2nd M1 for choosing outside region |
| \(x > 3\) or \(x < \frac{1}{2}\) | A1 f.t. (4) | f.t. their critical values; N.B. \(x>3, x>\frac{1}{2}\) is M0A0; penalise \(p < x < q\) where \(p < q\) by final A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x > 3\) or \(\frac{1}{4} < x < \frac{1}{2}\) | B1f.t. B1f.t. (2) | f.t. answers from (a) and (b); 1st B1 correct f.t. to infinite region; 2nd B1 correct f.t. to finite region; penalise \(\leq\) or \(\geq\) once only at first offence |
## Question 6:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $6x + 3 > 5 - 2x \Rightarrow 8x > 2$ | M1 | Multiply out and collect terms (allow one slip, allow $=$ here) |
| $x > \frac{1}{4}$ or $0.25$ or $\frac{2}{8}$ | A1 (2) | |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(2x-1)(x-3)\ (> 0)$ | M1 | Attempting to factorise 3TQ $\to x = \ldots$ |
| Critical values $x = \frac{1}{2},\ 3$ (both) | A1 | |
| Choosing "outside" region | M1 | 2nd M1 for choosing outside region |
| $x > 3$ or $x < \frac{1}{2}$ | A1 f.t. (4) | f.t. their critical values; N.B. $x>3, x>\frac{1}{2}$ is M0A0; penalise $p < x < q$ where $p < q$ by final A1 |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x > 3$ or $\frac{1}{4} < x < \frac{1}{2}$ | B1f.t. B1f.t. (2) | f.t. answers from (a) and (b); 1st B1 correct f.t. to infinite region; 2nd B1 correct f.t. to finite region; penalise $\leq$ or $\geq$ once only at first offence |
---6. Find the set of values of $x$ for which
\begin{enumerate}[label=(\alph*)]
\item $3 ( 2 x + 1 ) > 5 - 2 x$,
\item $2 x ^ { 2 } - 7 x + 3 > 0$,
\item both $3 ( 2 x + 1 ) > 5 - 2 x$ and $2 x ^ { 2 } - 7 x + 3 > 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q6 [8]}}