| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find range for no real roots |
| Difficulty | Moderate -0.8 This is a straightforward discriminant question requiring students to apply b²-4ac < 0 for no real roots, then solve a quadratic inequality. It's a standard P1/AS-level exercise with clear steps (set up discriminant, form inequality, solve), making it easier than average but not trivial since it requires careful handling of the inequality and the constraint k≠0. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(b^2 - 4ac = (6k)^2 - 4 \times k \times 5\) | M1 | Attempts to use \(b^2 - 4ac\) with \(b=6k\), \(a=k\), \(c=5\). May be part of quadratic formula. Condone if "6" isn't squared e.g. \(6k^2 - 4 \times k \times 5\) but \(k\) must be squared |
| \(b^2 - 4ac = (6k)^2 - 4 \times k \times 5 \ldots 0 \Rightarrow k\ldots\) | dM1 | Dependent on M1. Setting \(b^2 - 4ac \ldots 0\) leading to non-zero value for \(k\) from equation of form \(\alpha k^2 - \beta k \ldots 0\). Condone any of "=", "<", ">" etc. for "…" |
| \(k < \frac{5}{9}\) | A1 | Upper limit for \(k\) of \(\frac{5}{9}\) (not just value). Condone \(k \leqslant \frac{5}{9}\). Allow exact equivalents e.g. \(\frac{10}{18}\). Allow \(0.\dot{5}\). Condone use of \(x\) |
| \(0 < k < \frac{5}{9}\) | A1 | Allow inequalities on separate lines e.g. \(k < \frac{5}{9}\), \(k > 0\) |
## Question 4:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $b^2 - 4ac = (6k)^2 - 4 \times k \times 5$ | M1 | Attempts to use $b^2 - 4ac$ with $b=6k$, $a=k$, $c=5$. May be part of quadratic formula. Condone if "6" isn't squared e.g. $6k^2 - 4 \times k \times 5$ but $k$ must be squared |
| $b^2 - 4ac = (6k)^2 - 4 \times k \times 5 \ldots 0 \Rightarrow k\ldots$ | dM1 | Dependent on M1. Setting $b^2 - 4ac \ldots 0$ leading to non-zero value for $k$ from equation of form $\alpha k^2 - \beta k \ldots 0$. Condone any of "=", "<", ">" etc. for "…" |
| $k < \frac{5}{9}$ | A1 | Upper limit for $k$ of $\frac{5}{9}$ (not just value). Condone $k \leqslant \frac{5}{9}$. Allow exact equivalents e.g. $\frac{10}{18}$. Allow $0.\dot{5}$. Condone use of $x$ |
| $0 < k < \frac{5}{9}$ | A1 | Allow inequalities on separate lines e.g. $k < \frac{5}{9}$, $k > 0$ |
---\begin{enumerate}
\item Given that the equation\\
$k x ^ { 2 } + 6 k x + 5 = 0 \quad$ where $k$ is a non zero constant has no real roots, find the range of possible values for $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2023 Q4 [4]}}