| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Moderate -0.8 This is a straightforward AS-level question requiring routine differentiation of a polynomial and solving a quadratic inequality. Both parts involve standard techniques with no problem-solving insight needed, making it easier than average but not trivial since part (b) requires correct inequality reasoning. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 2x^2 - 7x - 4\) | M1 | Decreases the power of \(x\) by one for at least one term; \(x^n \to x^{n-1}\); allow \(5 \to 0\) |
| Correct derivative \(2x^2 - 7x - 4\) | A1 | Fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Attempts to solve \(\frac{dy}{dx} = 2x^2 - 7x - 4 \ldots 0\), e.g. \((2x+1)(x-4)=0\) leading to two values of \(x\) | M1 | Sets \(\frac{dy}{dx}\ldots 0\) (equality or inequality); proceeds to find two values from a 3TQ |
| Correct critical values \(x = -\frac{1}{2}, 4\) | A1 | May come from calculator or sketch |
| Chooses inside region for their critical values | dM1 | Dependent on previous M1 |
| \(-\frac{1}{2} < x < 4\) or \(-\frac{1}{2} \leq x \leq 4\) | A1 | Also accept \(x \in \left(-\frac{1}{2}, 4\right)\) or \(x \in \left[-\frac{1}{2}, 4\right]\); condone \(x > -\frac{1}{2}\) and \(x < 4\) |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 2x^2 - 7x - 4$ | M1 | Decreases the power of $x$ by one for at least one term; $x^n \to x^{n-1}$; allow $5 \to 0$ |
| Correct derivative $2x^2 - 7x - 4$ | A1 | Fully correct |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Attempts to solve $\frac{dy}{dx} = 2x^2 - 7x - 4 \ldots 0$, e.g. $(2x+1)(x-4)=0$ leading to two values of $x$ | M1 | Sets $\frac{dy}{dx}\ldots 0$ (equality or inequality); proceeds to find two values from a 3TQ |
| Correct critical values $x = -\frac{1}{2}, 4$ | A1 | May come from calculator or sketch |
| Chooses inside region for their critical values | dM1 | Dependent on previous M1 |
| $-\frac{1}{2} < x < 4$ **or** $-\frac{1}{2} \leq x \leq 4$ | A1 | Also accept $x \in \left(-\frac{1}{2}, 4\right)$ or $x \in \left[-\frac{1}{2}, 4\right]$; condone $x > -\frac{1}{2}$ and $x < 4$ |
---\begin{enumerate}
\item A curve has equation
\end{enumerate}
$$y = \frac { 2 } { 3 } x ^ { 3 } - \frac { 7 } { 2 } x ^ { 2 } - 4 x + 5$$
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ writing your answer in simplest form.\\
(b) Hence find the range of values of $x$ for which $y$ is decreasing.
\hfill \mbox{\textit{Edexcel AS Paper 1 2023 Q1 [6]}}