| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Quadratic equation real roots |
| Difficulty | Moderate -0.3 This is a standard C1 discriminant question requiring students to apply b²-4ac < 0 for no real roots, then solve a quadratic inequality. The algebra is straightforward and the method is routine textbook material, making it slightly easier than average but still requiring correct execution of multiple steps. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(b^2 - 4ac < 0 \Rightarrow 4^2 - 4(p-1)(p-5) < 0\) or \(0 > 4^2 - 4(p-1)(p-5)\) or \(4^2 < 4(p-1)(p-5)\) or \(4(p-1)(p-5) > 4^2\) | M1 | Attempts to use \(b^2 - 4ac\) with at least two of \(a\), \(b\) or \(c\) correct. May be in quadratic formula. No \(x\) terms needed. Inequality sign not needed for M1. |
| \(4 < p^2 - 6p + 5\) | A1 | Correct un-simplified inequality that is not the given answer |
| \(p^2 - 6p + 1 > 0\) | A1* | Correct solution with no errors, includes expansion of \((p-1)(p-5)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(p^2 - 6p + 1 = 0 \Rightarrow p = \ldots\) | M1 | Attempt to solve \(p^2 - 6p + 1 = 0\) leading to 2 solutions. Must use quadratic formula or completing the square (not factorising) |
| \(p = 3 \pm 2\sqrt{2}\) or \(p = \frac{6 \pm \sqrt{32}}{2}\) | A1 | Any equivalent correct expression. Discriminant must be single number, not e.g. \(36-4\) |
| \(p < 3 - \sqrt{8}\) or \(p > 3 + \sqrt{8}\) | M1 | Chooses outside region — not dependent on previous method mark |
| \(p < 3-\sqrt{8}\), \(p > 3+\sqrt{8}\) or equivalent e.g. \((-\infty, 3-\sqrt{8}) \cup (3+\sqrt{8}, \infty)\) | A1 | Allow ",", "or" or space between answers. Do not allow "and". Apply ISW if necessary. |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b^2 - 4ac < 0 \Rightarrow 4^2 - 4(p-1)(p-5) < 0$ or $0 > 4^2 - 4(p-1)(p-5)$ or $4^2 < 4(p-1)(p-5)$ or $4(p-1)(p-5) > 4^2$ | M1 | Attempts to use $b^2 - 4ac$ with at least two of $a$, $b$ or $c$ correct. May be in quadratic formula. No $x$ terms needed. Inequality sign not needed for M1. |
| $4 < p^2 - 6p + 5$ | A1 | Correct un-simplified inequality that is not the given answer |
| $p^2 - 6p + 1 > 0$ | A1* | Correct solution with no errors, includes expansion of $(p-1)(p-5)$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p^2 - 6p + 1 = 0 \Rightarrow p = \ldots$ | M1 | Attempt to solve $p^2 - 6p + 1 = 0$ leading to 2 solutions. Must use quadratic formula or completing the square (not factorising) |
| $p = 3 \pm 2\sqrt{2}$ or $p = \frac{6 \pm \sqrt{32}}{2}$ | A1 | Any equivalent correct expression. Discriminant must be single number, not e.g. $36-4$ |
| $p < 3 - \sqrt{8}$ or $p > 3 + \sqrt{8}$ | M1 | Chooses outside region — not dependent on previous method mark |
| $p < 3-\sqrt{8}$, $p > 3+\sqrt{8}$ or equivalent e.g. $(-\infty, 3-\sqrt{8}) \cup (3+\sqrt{8}, \infty)$ | A1 | Allow ",", "or" or space between answers. Do not allow "and". Apply ISW if necessary. |
> Note: $p > 3 \pm \sqrt{8}$ scores M1A1M0A0. $3+\sqrt{8} < p < 3-\sqrt{8}$ scores M1A0.
---\begin{enumerate}
\item The equation
\end{enumerate}
$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$
has no real roots.\\
(a) Show that $p$ satisfies $p ^ { 2 } - 6 p + 1 > 0$\\
(b) Hence find the set of possible values of $p$.\\
\hfill \mbox{\textit{Edexcel C1 2015 Q5 [7]}}