| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Open box from cut-corner sheet |
| Difficulty | Moderate -0.3 This is a standard C1 optimization problem with guided steps through differentiation, solving a quadratic, and using the second derivative test. While it requires multiple techniques, each step is routine and heavily scaffolded with no novel insight needed—slightly easier than the average A-level question due to the extensive guidance provided. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.07d 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 |
|---|---|
| Total: 15 |
| Answer | Marks | Guidance |
|---|---|---|
| \(k^2 + 10k + 25 - 12k^2 - 24k\) | M1 | Condone one slip |
| \(= -11k^2 - 14k + 25\) | A1 | No ISW here |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Real roots when "\(b^2 - 4ac \geq 0\)" | B1 | Non-negative discriminant (stated / used) |
| \((k + 5)^2 - 12k(k + 2)\) | M1 | Finding \(b^2 - 4ac\) in terms of k |
| \((k - 1)(11k + 25)\) attempted to be shown equal to \(11k^2 + 14k - 25\) | m1 | Or factorisation attempt |
| \(\Rightarrow -11k^2 - 14k + 25 \geq 0\) | A1 | Real roots condition correct and ... |
| \(\Rightarrow (k - 1)(11k + 25) \leq 0\) | A1 | AG (be convinced about inequality) |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| (Critical values) \(1\) and \(-\frac{25}{11}\) seen | B1 | + + |
| Sketch or sign diagram | M1 | __________ |
| \(\Rightarrow -\frac{25}{11} \leq k \leq 1\) | A1 | \(-\frac{25}{11}\) _____ \(1\) |
| Total: 3 |
| | **Total: 15**
## Question 7(a)
$k^2 + 10k + 25 - 12k^2 - 24k$ | M1 | Condone one slip
$= -11k^2 - 14k + 25$ | A1 | No ISW here
| | **Total: 2**
## Question 7(b)(i)
Real roots when "$b^2 - 4ac \geq 0$" | B1 | Non-negative discriminant (stated / used)
$(k + 5)^2 - 12k(k + 2)$ | M1 | Finding $b^2 - 4ac$ in terms of k
$(k - 1)(11k + 25)$ attempted to be shown equal to $11k^2 + 14k - 25$ | m1 | Or factorisation attempt
$\Rightarrow -11k^2 - 14k + 25 \geq 0$ | A1 | Real roots condition correct and ...
$\Rightarrow (k - 1)(11k + 25) \leq 0$ | A1 | AG (be convinced about inequality)
| | **Total: 5**
## Question 7(b)(ii)
(Critical values) $1$ and $-\frac{25}{11}$ seen | B1 | + +
Sketch or sign diagram | M1 | __________|__________|__________
$\Rightarrow -\frac{25}{11} \leq k \leq 1$ | A1 | $-\frac{25}{11}$ _____ $1$
| | **Total: 3**6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm .\\
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561}
A square of side $x \mathrm {~cm}$ is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height $x \mathrm {~cm}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of liquid the box can hold is given by
$$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
\item \begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } V } { \mathrm {~d} x }$.
\item Show that any stationary values of $V$ must occur when $x ^ { 2 } - 11 x + 18 = 0$.
\item Solve the equation $x ^ { 2 } - 11 x + 18 = 0$.
\item Explain why there is only one value of $x$ for which $V$ is stationary.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$.
\item Hence determine whether the stationary value is a maximum or minimum.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q6 [15]}}