| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Moderate -0.3 This is a structured, multi-part optimization question with extensive scaffolding. While it covers important calculus concepts (substitution, differentiation, second derivative test), each step is explicitly guided with 'show that' prompts that tell students exactly what answer to reach. The algebra is straightforward, and the techniques are standard C1 material requiring no novel insight—making it slightly easier than a typical A-level question. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = x(9-3x)^2\) | M1 | Attempt at \(V\) in terms of \(x\) (condone slip when rearranging formula for \(y = 9 - 3x\)) or \((9-3x)^2 = 81 - 54x + 9x^2\) |
| \(V = x(81 - 54x + 9x^2) = 81x - 54x^2 + 9x^3\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dV}{dx} = 81 - 108x + 27x^2\) | M1, A1, A1 | One term correct, Another correct, All correct (no \(+c\) etc) |
| \(= 27(x^2 - 4x + 3)\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \((x - 3)(x - 1)\) or \((27x - 81)(x - 1)\) etc \(\Rightarrow x = 1, 3\) | M1, A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2V}{dx^2} = -108 + 54x\) (condone one slip) | M1, A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 3 \Rightarrow \frac{d^2V}{dx^2} = 54; x = 1 \Rightarrow \frac{d^2V}{dx^2} = -54\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((x = 1)\) (gives maximum value) | E1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(V_{\max} = 36\) | B1 | 1 |
**3(a)**
| $V = x(9-3x)^2$ | M1 | Attempt at $V$ in terms of $x$ (condone slip when rearranging formula for $y = 9 - 3x$) or $(9-3x)^2 = 81 - 54x + 9x^2$ |
| $V = x(81 - 54x + 9x^2) = 81x - 54x^2 + 9x^3$ | A1 | 2 | AG; no errors in algebra |
**3(b)(i)**
| $\frac{dV}{dx} = 81 - 108x + 27x^2$ | M1, A1, A1 | One term correct, Another correct, All correct (no $+c$ etc) |
| $= 27(x^2 - 4x + 3)$ | A1 | 4 | CSO; all algebra and differentiation correct |
**3(b)(ii)**
| $(x - 3)(x - 1)$ or $(27x - 81)(x - 1)$ etc $\Rightarrow x = 1, 3$ | M1, A1 | 2 | "Correct" factors or correct use of formula SC: B1,B1 for $x = 1, x = 3$ found by inspection (provided no other values) |
**3(c)**
| $\frac{d^2V}{dx^2} = -108 + 54x$ (condone one slip) | M1, A1 | 2 | ft their $\frac{dV}{dx}$ (may have cancelled 27 etc) CSO; all differentiation correct |
**3(d)(i)**
| $x = 3 \Rightarrow \frac{d^2V}{dx^2} = 54; x = 1 \Rightarrow \frac{d^2V}{dx^2} = -54$ | B1 | 1 | ft their $\frac{d^2V}{dx^2}$ and their two $x$-values |
**3(d)(ii)**
| $(x = 1)$ (gives maximum value) | E1 | 1 | Provided their $\frac{d^2V}{dx^2} < 0$ |
**3(d)(iii)**
| $V_{\max} = 36$ | B1 | 1 | CAO |
---3 Two numbers, $x$ and $y$, are such that $3 x + y = 9$, where $x \geqslant 0$ and $y \geqslant 0$. It is given that $V = x y ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)$, and state the value of the integer $k$.
\item Hence find the two values of $x$ for which $\frac { \mathrm { d } V } { \mathrm {~d} x } = 0$.
\end{enumerate}\item Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ for each of the two values of $x$ found in part (b)(ii).
\item Hence determine the value of $x$ for which $V$ has a maximum value.
\item Find the maximum value of $V$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q3 [13]}}