| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show discriminant inequality, then solve |
| Difficulty | Moderate -0.8 This is a straightforward discriminant question requiring standard application of b²-4ac ≥ 0 for real roots, followed by routine inequality solving and identification of equal roots case. All steps are textbook procedures with no novel insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac \geq 0\), \((5k)^2 - 8k \geq 0\), \(k(25k-8) \geq 0\) | M1, A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Critical values: \(k = 0\), \(k = \frac{8}{25}\) | B1 B1 | |
| \(k \leq 0\), \(k \geq \frac{8}{25}\) | M1 A1 ft | A1 requires \(\leq\) and \(\geq\) (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = 0\) and \(k = \frac{8}{25}\) | B1 | Clearly seen as solutions for (c) (1 mark) |
## Question 5:
### Part (a)
$b^2 - 4ac \geq 0$, $(5k)^2 - 8k \geq 0$, $k(25k-8) \geq 0$ | M1, A1 | (2 marks)
### Part (b)
Critical values: $k = 0$, $k = \frac{8}{25}$ | B1 B1 |
$k \leq 0$, $k \geq \frac{8}{25}$ | M1 A1 ft | A1 requires $\leq$ and $\geq$ (4 marks)
### Part (c)
$k = 0$ and $k = \frac{8}{25}$ | B1 | Clearly seen as solutions for (c) (1 mark)
---5. The equation $x ^ { 2 } + 5 k x + 2 k = 0$, where $k$ is a constant, has real roots.
\begin{enumerate}[label=(\alph*)]
\item Prove that $k ( 25 k - 8 ) \geq 0$.
\item Hence find the set of possible values of $k$.
\item Write down the values of $k$ for which the equation $x ^ { 2 } + 5 k x + 2 k = 0$ has equal roots.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q5 [7]}}