| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Applied differentiation |
| Type | Find curve equation from derivative |
| Difficulty | Standard +0.3 This is a straightforward integration problem with three conditions to find three unknowns. Students must integrate a simple expression involving powers of x, apply the stationary point condition (f'(4)=0), and use two boundary conditions. All steps are routine AS-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(f'(4) = 0 \Rightarrow 16 + 2a + b = 0\) | M1 | Key step setting up equation in \(a\) and \(b\). Condone slips. |
| Integrates \(f'(x) = 4x + a\sqrt{x} + b \Rightarrow \{f(x) =\} 2x^2 + \frac{2}{3}ax^{\frac{3}{2}} + bx\ \{+c\}\) | M1, A1ft | Award for \(x^n \to x^{n+1}\) or \(b \to bx\). Allow ft on \(b\) in terms of \(a\). May be left unsimplified but indices must be processed. |
| Deduces that \(c = -5\) | B1 | Must be the constant term in \(f(x)\). Note deducing \(b=-5\) is B0. |
| Full and complete method using \(f'(4)=0\) and \(f(4)=3\) to find values of \(a\) and \(b\). Note: \(a=-15\) and \(b=14\) | ddM1 | Dependent on both previous M marks. Must include \(f'(4)=0\) giving \(16+2a+b=0\) and \(f(4)=3\) giving \(32+\frac{16}{3}a+4b-5=3\) |
| \(\{f(x)=\}\ 2x^2 - 10x^{\frac{3}{2}} + 14x - 5\) | A1 | Exact simplified equivalent accepted, e.g. use of \(x\sqrt{x}\) in place of \(x^{\frac{3}{2}}\) |
## Question 16:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $f'(4) = 0 \Rightarrow 16 + 2a + b = 0$ | M1 | Key step setting up equation in $a$ and $b$. Condone slips. |
| Integrates $f'(x) = 4x + a\sqrt{x} + b \Rightarrow \{f(x) =\} 2x^2 + \frac{2}{3}ax^{\frac{3}{2}} + bx\ \{+c\}$ | M1, A1ft | Award for $x^n \to x^{n+1}$ or $b \to bx$. Allow ft on $b$ in terms of $a$. May be left unsimplified but indices must be processed. |
| Deduces that $c = -5$ | B1 | Must be the constant term in $f(x)$. Note deducing $b=-5$ is B0. |
| Full and complete method using $f'(4)=0$ and $f(4)=3$ to find values of $a$ and $b$. Note: $a=-15$ and $b=14$ | ddM1 | Dependent on **both** previous M marks. Must include $f'(4)=0$ giving $16+2a+b=0$ and $f(4)=3$ giving $32+\frac{16}{3}a+4b-5=3$ |
| $\{f(x)=\}\ 2x^2 - 10x^{\frac{3}{2}} + 14x - 5$ | A1 | Exact simplified equivalent accepted, e.g. use of $x\sqrt{x}$ in place of $x^{\frac{3}{2}}$ |
---\begin{enumerate}
\item A curve has equation $y = \mathrm { f } ( x ) , x \geqslant 0$
\end{enumerate}
Given that
\begin{itemize}
\item $\mathrm { f } ^ { \prime } ( x ) = 4 x + a \sqrt { x } + b$, where $a$ and $b$ are constants
\item the curve has a stationary point at $( 4,3 )$
\item the curve meets the $y$-axis at - 5\\
find $\mathrm { f } ( x )$, giving your answer in simplest form.
\end{itemize}
\hfill \mbox{\textit{Edexcel AS Paper 1 2023 Q16 [6]}}