| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Quadratic equation real roots |
| Difficulty | Moderate -0.8 This is a standard C1 discriminant question requiring routine application of b²-4ac > 0 for distinct real roots, followed by solving a straightforward quadratic inequality by factorization. Part (a) is essentially guided, and part (b) involves basic factorization (k+3)(k-1) > 0, making this easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac = (k-3)^2 - 4(3-2k)\) | M1 | For attempt to find \(b^2-4ac\) with one of \(b\) or \(c\) correct |
| \(k^2 - 6k + 9 - 4(3-2k) > 0\) or \((k-3)^2 - 12 + 8k > 0\) | M1 | For a correct inequality symbol and an attempt to expand |
| \(k^2 + 2k - 3 > 0\) | A1cso | No incorrect working seen |
| (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| \((k+3)(k-1)[=0]\) | M1 | For attempt to factorise or solve leading to \(k=\) (2 values) |
| Critical values \(k = 1\) or \(k = -3\) | A1 | |
| (choosing "outside" region) | M1 | For a method choosing the "outside" region; can follow through their critical values |
| \(k > 1\ \) or \(\ k < -3\) | A1 cao | Allow "," instead of "or"; \(\geq\) loses the final A1; \(1 < k < -3\) scores M1A0 unless a correct version is seen |
| (4 marks total) |
## Question 8:
### Part (a):
| $b^2 - 4ac = (k-3)^2 - 4(3-2k)$ | M1 | For attempt to find $b^2-4ac$ with one of $b$ or $c$ correct |
| $k^2 - 6k + 9 - 4(3-2k) > 0$ or $(k-3)^2 - 12 + 8k > 0$ | M1 | For a correct inequality symbol and an attempt to expand |
| $k^2 + 2k - 3 > 0$ | A1cso | No incorrect working seen |
|(3 marks total)|||
### Part (b):
| $(k+3)(k-1)[=0]$ | M1 | For attempt to factorise or solve leading to $k=$ (2 values) |
| Critical values $k = 1$ or $k = -3$ | A1 | |
| (choosing "outside" region) | M1 | For a method choosing the "outside" region; can follow through their critical values |
| $k > 1\ $ or $\ k < -3$ | A1 cao | Allow "," instead of "or"; $\geq$ loses the final A1; $1 < k < -3$ scores M1A0 unless a correct version is seen |
|(4 marks total)|||
---8. The equation $x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0$, where $k$ is a constant, has two distinct real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $k$ satisfies
$$k ^ { 2 } + 2 k - 3 > 0$$
\item Find the set of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2011 Q8 [7]}}