| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show discriminant inequality, then solve |
| Difficulty | Moderate -0.3 This is a standard discriminant question requiring routine application of b²-4ac ≥ 0 for real roots, solving a quadratic inequality, and finding when discriminant equals zero. All steps are textbook procedures with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| (a) real roots \(\therefore b^2 - 4ac \geq 0\) | M1 | |
| \((-k)^2 - [4 \times 4 \times (k-3)] \geq 0\) | ||
| \(k^2 - 16k + 48 \geq 0\) | A1 | |
| (b) \((k-4)(k-12) \geq 0\) | M1 | |
| M1 | ||
| \(k \leq 4\) or \(k \geq 12\) | A1 | |
| (c) \(k = 4\) | B1 | |
| \(4x^2 - 4x + 1 = 0\) | ||
| \((2x-1)^2 = 0\) | M1 | |
| \(x = \frac{1}{2}\) | A1 | (8) |
(a) real roots $\therefore b^2 - 4ac \geq 0$ | M1 |
$(-k)^2 - [4 \times 4 \times (k-3)] \geq 0$ | |
$k^2 - 16k + 48 \geq 0$ | A1 |
(b) $(k-4)(k-12) \geq 0$ | M1 |
| M1 |
$k \leq 4$ or $k \geq 12$ | A1 |
(c) $k = 4$ | B1 |
$4x^2 - 4x + 1 = 0$ | |
$(2x-1)^2 = 0$ | M1 |
$x = \frac{1}{2}$ | A1 | (8)7. Given that the equation
$$4 x ^ { 2 } - k x + k - 3 = 0$$
where $k$ is a constant, has real roots,
\begin{enumerate}[label=(\alph*)]
\item show that
$$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
\item find the set of possible values of $k$,
\item state the smallest value of $k$ for which the roots are equal and solve the equation when $k$ takes this value.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [8]}}