| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Discriminant for real roots condition |
| Difficulty | Moderate -0.3 This is a straightforward C1 completing the square question with standard techniques. Part (a) requires routine algebraic manipulation to complete the square and solve, while part (b) involves a simple inequality from the discriminant condition. The presence of parameter k adds minor complexity but this remains a typical textbook exercise requiring no novel insight. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks |
|---|---|
| \((x + 2k)^2 = 4k^2 + k\) | M1 |
| A1 | |
| \(x + 2k = \pm\sqrt{4k^2 + k}\) | M1 |
| \(x = -2k \pm \sqrt{4k^2 + k}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(k(4k+1) < 0\), critical values: \(-\frac{1}{4}, 0\) | M1 | |
| A1 | ||
| M1 | ||
| \(\therefore -\frac{1}{4} < k < 0\) | A1 | (8 marks) |
(a) $(x + 2k)^2 - (2k)^2 - k = 0$
$(x + 2k)^2 = 4k^2 + k$ | M1 |
| A1 |
$x + 2k = \pm\sqrt{4k^2 + k}$ | M1 |
$x = -2k \pm \sqrt{4k^2 + k}$ | A1 |
(b) no real roots if $4k^2 + k < 0$
$k(4k+1) < 0$, critical values: $-\frac{1}{4}, 0$ | M1 |
| A1 |
| M1 |
$\therefore -\frac{1}{4} < k < 0$ | A1 | (8 marks)\begin{enumerate}[label=(\alph*)]
\item By completing the square, find in terms of the constant $k$ the roots of the equation
$$x^2 + 4kx - k = 0.$$ [4]
\item Hence find the set of values of $k$ for which the equation has no real roots. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q6 [8]}}