| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Quadratic equation real roots |
| Difficulty | Moderate -0.3 This is a standard C1 discriminant question requiring students to apply b²-4ac ≥ 0 for real roots, then factorise and solve a quadratic inequality. All steps are routine textbook procedures with no novel insight required, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = 4 - 4(k-1)(2k-3)\) | M1 | (or seen in formula) condone one slip |
| Real roots when \(b^2 - 4ac \geqslant 0\) | E1 | must involve \(f(k) \geqslant 0\) (usually M1 must be earned) |
| \(4 - 4(2k^2 - 5k + 3) \geqslant 0\) | ||
| \(\Rightarrow -2k^2 + 5k - 3 + 1 \geqslant 0\) | at least one step of working justifying \(\leqslant 0\) | |
| \(\Rightarrow 2k^2 - 5k + 2 \leqslant 0\) | A1 | AG |
| Answer | Marks |
|---|---|
| \((2k-1)(k-2)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (Critical values) \(\dfrac{1}{2}\) and \(2\) | B1\(\checkmark\) | ft their factors or correct values seen on diagram, sketch or inequality or stated |
| Sign diagram showing \(+\) — \(+\) with \(\dfrac{1}{2}\) and \(2\) | M1 | use of sketch / sign diagram |
| \(\Rightarrow 0.5 \leqslant k \leqslant 2\) | A1 | M1A0 for \(0.5 < k < 2\) or \(k \geqslant 0.5\), \(k \leqslant 2\) |
# Question 7(a):
$b^2 - 4ac = 4 - 4(k-1)(2k-3)$ | M1 | (or seen in formula) condone one slip
Real roots when $b^2 - 4ac \geqslant 0$ | E1 | must involve $f(k) \geqslant 0$ (usually M1 must be earned)
$4 - 4(2k^2 - 5k + 3) \geqslant 0$ | |
$\Rightarrow -2k^2 + 5k - 3 + 1 \geqslant 0$ | | at least one step of working justifying $\leqslant 0$
$\Rightarrow 2k^2 - 5k + 2 \leqslant 0$ | A1 | AG
**Total: 3 marks**
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# Question 7(b)(i):
$(2k-1)(k-2)$ | B1 |
**Total: 1 mark**
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# Question 7(b)(ii):
(Critical values) $\dfrac{1}{2}$ and $2$ | B1$\checkmark$ | ft their factors or correct values seen on diagram, sketch or inequality or stated
Sign diagram showing $+$ — $+$ with $\dfrac{1}{2}$ and $2$ | M1 | use of sketch / sign diagram
$\Rightarrow 0.5 \leqslant k \leqslant 2$ | A1 | M1A0 for $0.5 < k < 2$ or $k \geqslant 0.5$, $k \leqslant 2$
**Total: 3 marks**7 The quadratic equation
$$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$
where $k$ is a constant, has real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $2 k ^ { 2 } - 5 k + 2 \leqslant 0$.
\item \begin{enumerate}[label=(\roman*)]
\item Factorise $2 k ^ { 2 } - 5 k + 2$.
\item Hence, or otherwise, solve the quadratic inequality
$$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q7 [7]}}