| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| 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 application of differentiation requiring polynomial differentiation (routine), setting derivative > 0, and solving a quadratic inequality. All steps are standard AS-level techniques with no problem-solving insight needed, making it easier than average but not trivial. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| VIIIV SIHI NI JIIYM IONOO | VIUV SIHI NI JIIAM ION OO | VI4V SIHI NI JIIIM I ON OO |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 6x^2 - 4x - 2\) | M1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempts \(6x^2 - 4x - 2 \geq 0\) | M1 | 1.1b |
| \((6x + 2)(x - 1) \geq 0\) | M1 | 1.1b |
| \(x = -\frac{1}{3}, 1\) | A1 | 1.1b |
| Chooses outside region | M1 | 1.1b |
| \(\{x : x \leq -\frac{1}{3}\} \cup \{x : x \geq 1\}\) | A1 | 2.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Explains that \(OC\) is parallel to \(AB\) as \(8\mathbf{i} - 20\mathbf{j} = 4(2\mathbf{i} - 5\mathbf{j})\) | M1 | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Hence \(OABC\) is a trapezium. | A1 | 2.4 |
# Question 1(a)
$y = 2x^3 - 2x^2 - 2x + 8$
$\frac{dy}{dx} = 6x^2 - 4x - 2$ | M1 | A1 | 1.1b
(2 marks)
# Question 1(b)
Attempts $6x^2 - 4x - 2 \geq 0$ | M1 | 1.1b
$(6x + 2)(x - 1) \geq 0$ | M1 | 1.1b
$x = -\frac{1}{3}, 1$ | A1 | 1.1b
Chooses outside region | M1 | 1.1b
$\{x : x \leq -\frac{1}{3}\} \cup \{x : x \geq 1\}$ | A1 | 2.5
(4 marks)
**Notes:**
**Part (a):**
M1: Attempts to differentiate. Allow two correct terms unsimplified.
A1: $\frac{dy}{dx} = 6x^2 - 4x - 2$
**Part (b):**
M1: Attempts to find critical values of $\frac{dy}{dx} \geq 0$ or $\frac{dy}{dx} = 0$
A1: Correct critical values $x = -\frac{1}{3}, 1$
M1: Chooses the outside region
A1: $\{x : x \leq -\frac{1}{3}\} \cup \{x : x \geq 1\}$ or $\{x : x \in \mathbb{R}, x \leq -\frac{1}{3}$ or $x \geq 1\}$
Accept also $\{x : x \not\geq -\frac{1}{3}\} \cup \{x : x \not\leq 1\}$
---
# Question 1(b) — Vectors
Explains that $OC$ is parallel to $AB$ as $8\mathbf{i} - 20\mathbf{j} = 4(2\mathbf{i} - 5\mathbf{j})$ | M1 | 1.1b
As $OC = 4AB$ it is parallel to it and not the same length
Hence $OABC$ is a trapezium. | A1 | 2.4
(2 marks)
**Notes:**
**Part (a):**
M1: Attempts $\mathbf{AB} = \mathbf{OB} - \mathbf{OA}$ or equivalent. This may be implied by one correct component.
A1: $2\mathbf{i} - 5\mathbf{j}$
**Part (b):**
M1: Attempts to compare vectors $\mathbf{OC}$ and $\mathbf{AB}$ by considering their directions
A1: Fully explains why $OABC$ is a trapezium. (The candidate is required to state that $OC$ is parallel to $AB$ but not the same length as it.)\begin{enumerate}
\item A curve has equation
\end{enumerate}
$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
(b) Hence find the range of values of $x$ for which $y$ is increasing. Write your answer in set notation.
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VIIIV SIHI NI JIIYM IONOO & VIUV SIHI NI JIIAM ION OO & VI4V SIHI NI JIIIM I ON OO \\
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\hfill \mbox{\textit{Edexcel AS Paper 1 Q1 [6]}}