| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| 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 discriminant problem requiring students to apply b²-4ac ≥ 0 for real roots, then solve a quadratic inequality. While it involves multiple steps (finding discriminant, simplifying, solving inequality), these are routine C1 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = 16k^2 - 36(k + 1)\) | M1 | |
| Real roots: discriminant \(\geq 0\) \(\Rightarrow 16k^2 - 36k - 36 \geq 0\) \(\Rightarrow 4k^2 - 9k - 9 \geq 0\) | B1 | |
| A1 | 3 | AG (watch signs) |
| Answer | Marks | Guidance |
|---|---|---|
| \((4k + 3)(k - 3)\) | M1 | |
| critical points \((k =) -\frac{3}{4}\), \(3\) | A1 | |
| sketch | M1 | |
| \(k \geq 3\), \(k \leq -\frac{3}{4}\) | A1 | 4 |
**8(a)**
| $b^2 - 4ac = 16k^2 - 36(k + 1)$ | M1 | | Condone one slip |
| Real roots: discriminant $\geq 0$ $\Rightarrow 16k^2 - 36k - 36 \geq 0$ $\Rightarrow 4k^2 - 9k - 9 \geq 0$ | B1 | | |
| | A1 | 3 | AG (watch signs) |
**8(b)**
| $(4k + 3)(k - 3)$ | M1 | | Or correct use of formula (unsimplified) |
| critical points $(k =) -\frac{3}{4}$, $3$ | A1 | | Not in a form involving surds Values may be seen in inequalities etc |
| sketch | M1 | | Or sign diagram |
| $k \geq 3$, $k \leq -\frac{3}{4}$ | A1 | 4 | NMS full marks Condone use of word "and" but final answer in a form such as $3 \leq k \leq -\frac{3}{4}$ scores A0 |
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# Summary
**TOTAL: 75 marks**8 The quadratic equation $( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0$ has real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $4 k ^ { 2 } - 9 k - 9 \geqslant 0$.
\item Hence find the possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q8 [7]}}