| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| 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 straightforward application of the discriminant condition for real roots (b² - 4ac > 0), followed by solving a simple quadratic inequality by factorization. Both parts are standard C1 techniques with no problem-solving insight required, making it easier than average but not trivial since it involves two connected steps. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(b^2 - 4ac > 0 \Rightarrow 16 - 4k(5-k) > 0\) or equiv. | M1, A1 | For attempting to use discriminant; \(> 0\) not required but substitution of \(a\), \(b\), \(c\) in correct formula required |
| \(k^2 - 5k + 4 > 0\) | A1cso | Allow any order of terms e.g. \(4 - 5k + k^2 > 0\) (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((k-4)(k-1) = 0\), \(k = 1\) or \(4\) | M1, A1 | For attempt to solve appropriate 3TQ; both critical values required |
| Choosing "outside" region | M1 | Diagram or table alone not sufficient; set must be narrowed down |
| \(\underline{k < 1}\) or \(\underline{k > 4}\) | A1 | For correct answer only; "\(1 > k > 4\)" is A0; use of \(\leq\) or \(\geq\) loses final mark (4 marks total) |
# Question 7:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $b^2 - 4ac > 0 \Rightarrow 16 - 4k(5-k) > 0$ or equiv. | M1, A1 | For attempting to use discriminant; $> 0$ not required but substitution of $a$, $b$, $c$ in correct formula required |
| $k^2 - 5k + 4 > 0$ | A1cso | Allow any order of terms e.g. $4 - 5k + k^2 > 0$ (3 marks total) |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(k-4)(k-1) = 0$, $k = 1$ or $4$ | M1, A1 | For attempt to solve appropriate 3TQ; both critical values required |
| Choosing "outside" region | M1 | Diagram or table alone not sufficient; set must be narrowed down |
| $\underline{k < 1}$ or $\underline{k > 4}$ | A1 | For correct answer only; "$1 > k > 4$" is A0; use of $\leq$ or $\geq$ loses final mark (4 marks total) |
---7. The equation $k x ^ { 2 } + 4 x + ( 5 - k ) = 0$, where $k$ is a constant, has 2 different real solutions for $x$.
\begin{enumerate}[label=(\alph*)]
\item Show that $k$ satisfies
$$k ^ { 2 } - 5 k + 4 > 0 .$$
\item Hence find the set of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2009 Q7 [7]}}