| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| 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 (forming discriminant, expanding, factorising, 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 conditions |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = 144 - 4(k + 1)(k - 4)\) | M1 | Clear attempt at \(b^2 - 4ac\); Condone slip in one term of expression |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow k^2 - 3k - 40 \leq 0\) | B1 | Not just a statement, must involve \(k\); 3 marks |
| \(A1\) | AG (watch signs carefully) |
| Answer | Marks | Guidance |
|---|---|---|
| \((k - 8)(k + 5)\) | M1 | Factors attempt or formula |
| Critical points \(8\) and \(-5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(-5 \leq k \leq 8\) | M1, A1 | \(+\)ve \( |
## 7(a)
$b^2 - 4ac = 144 - 4(k + 1)(k - 4)$ | M1 | Clear attempt at $b^2 - 4ac$; Condone slip in one term of expression
Real roots when $b^2 - 4ac \geq 0$
$36 - (k^2 - 3k - 4) \geq 0$
$\Rightarrow k^2 - 3k - 40 \leq 0$ | B1 | Not just a statement, must involve $k$; 3 marks
$A1$ | AG (watch signs carefully)
## 7(b)
$(k - 8)(k + 5)$ | M1 | Factors attempt or formula
Critical points $8$ and $-5$ | A1 |
Sketch or sign diagram **correct, must have $8$ and $-5$**
$-5 \leq k \leq 8$ | M1, A1 | $+$ve $|$ $-$ve $|$ $+$ve with $-5$ and $8$ marked; 4 marks
A0 for $-5 < k < 8$ or two separate inequalities unless word AND used
**Total: 7**
---
# **GRAND TOTAL: 75**7 The quadratic equation $( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0$ has real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $k ^ { 2 } - 3 k - 40 \leqslant 0$.
\item Hence find the possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2007 Q7 [7]}}