| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve quadratic inequality |
| Difficulty | Moderate -0.8 This is a straightforward quadratic inequality requiring rearrangement to standard form, factorization or quadratic formula, and identification of the solution region. It's a routine P1 exercise with clear steps and no conceptual challenges, making it easier than average but not trivial since it requires proper method and sign analysis. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4x^2 - 3x + 7 \geqslant 4x + 9 \Rightarrow 4x^2 - 7x - 2 \geqslant 0\) | Gather terms to one side | |
| \((4x+1)(x-2) = 0 \Rightarrow x = ...\) or via quadratic formula or completing the square | M1 | Solves 3TQ (not \(4x^2-3x+7=0\)) to get at least one critical value. Note: \((x+\frac{1}{4})(x-2)=0\) scores M0 |
| \(x = -\frac{1}{4}, \quad 2\) | A1 | Both correct critical values. Allow unsimplified single fractions e.g. \(-\frac{2}{8}, \frac{16}{8}\) |
| \(x \leqslant -\frac{1}{4}\), \(\quad x \geqslant 2\) | M1 | Chooses outside region for their 2 critical values. Allow \(<\) and \(>\) |
| \(x \leqslant -\frac{1}{4}\) or \(x \geqslant 2\) | A1 | Correct solution only — depends on all previous marks. Do not allow \(-\frac{1}{4} \geqslant x \geqslant 2\) or \(x \leqslant -\frac{1}{4} \cap x \geqslant 2\) |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4x^2 - 3x + 7 \geqslant 4x + 9 \Rightarrow 4x^2 - 7x - 2 \geqslant 0$ | | Gather terms to one side |
| $(4x+1)(x-2) = 0 \Rightarrow x = ...$ or via quadratic formula or completing the square | **M1** | Solves 3TQ (not $4x^2-3x+7=0$) to get at least one critical value. Note: $(x+\frac{1}{4})(x-2)=0$ scores M0 |
| $x = -\frac{1}{4}, \quad 2$ | **A1** | Both correct critical values. Allow unsimplified single fractions e.g. $-\frac{2}{8}, \frac{16}{8}$ |
| $x \leqslant -\frac{1}{4}$, $\quad x \geqslant 2$ | **M1** | Chooses outside region for their 2 critical values. Allow $<$ and $>$ |
| $x \leqslant -\frac{1}{4}$ or $x \geqslant 2$ | **A1** | Correct solution only — depends on all previous marks. Do **not** allow $-\frac{1}{4} \geqslant x \geqslant 2$ or $x \leqslant -\frac{1}{4} \cap x \geqslant 2$ |
---\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
\end{enumerate}
Solve the inequality
$$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$
\hfill \mbox{\textit{Edexcel P1 2023 Q1 [4]}}