| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Second derivative and nature determination |
| Difficulty | Easy -1.8 This is a straightforward differentiation exercise requiring only basic power rule application twice, followed by solving a simple linear equation. No chain rule is actually needed despite the topic label. This is significantly easier than average A-level questions as it involves pure mechanical differentiation with no problem-solving, application, or conceptual challenge. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = 4x^3 - 7x^2 + 5x - 10 \Rightarrow \frac{dy}{dx} = 12x^2 - 14x + 5\) | M1 | Award for \(x^3 \to x^2\) or \(x^2 \to x\) or \(5x \to 5\) or \(-10 \to 0\); indices may be unprocessed |
| \(\frac{dy}{dx} = 12x^2 - 14x + 5\) | A1 | Correct simplified expression; do not allow \(x^1\) for \(x\) or \(5x^0\) for 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{d^2y}{dx^2} = 24x - 14\) | A1ft | Correct simplified second derivative or follow through from first derivative; must be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(24x - 14 = 0 \Rightarrow x = \ldots\) | M1 | Sets second derivative of form \(ax + b\), \(a,b \neq 0\) equal to 0 and proceeds to value for \(x\); condone one slip in copying second derivative |
| \(x = \frac{7}{12}\) | A1 | Allow exact equivalents e.g. \(\frac{14}{24}\); not rounded decimals e.g. 0.583; allow \(0.58\dot{3}\) |
## Question 1:
**Part (a)(i):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 4x^3 - 7x^2 + 5x - 10 \Rightarrow \frac{dy}{dx} = 12x^2 - 14x + 5$ | M1 | Award for $x^3 \to x^2$ or $x^2 \to x$ or $5x \to 5$ or $-10 \to 0$; indices may be unprocessed |
| $\frac{dy}{dx} = 12x^2 - 14x + 5$ | A1 | Correct simplified expression; do not allow $x^1$ for $x$ or $5x^0$ for 5 |
**Part (a)(ii):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{d^2y}{dx^2} = 24x - 14$ | A1ft | Correct simplified second derivative or follow through from first derivative; must be simplified |
**Part (b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $24x - 14 = 0 \Rightarrow x = \ldots$ | M1 | Sets second derivative of form $ax + b$, $a,b \neq 0$ equal to 0 and proceeds to value for $x$; condone one slip in copying second derivative |
| $x = \frac{7}{12}$ | A1 | Allow exact equivalents e.g. $\frac{14}{24}$; not rounded decimals e.g. 0.583; allow $0.58\dot{3}$ |
---1.
$$y = 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 10$$
\begin{enumerate}[label=(\alph*)]
\item Find in simplest form
\begin{enumerate}[label=(\roman*)]
\item $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
\end{enumerate}\item Hence find the exact value of $x$ when $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 2 2024 Q1 [5]}}