| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary points coordinates |
| Difficulty | Standard +0.3 This is a straightforward stationary points question requiring differentiation, solving a quadratic, and using the second derivative test. Part (b) requires connecting stationary points to the number of roots, which is a standard application. The question is slightly easier than average as it involves routine techniques with clear guidance ('using your answers from part (a)'). |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(3x^2 - 12x + 9 = 0\) | M1 | Attempt differentiate & \(=0\); \(x(x-3)^2=0\) M1 tp at \(x=3\) A1 |
| \(x=3\) or \(1\) | A1 | Correct equation. May be implied by ans |
| \((3,0)\) and \((1,4)\) | A1f, A1 [4] | Allow when \(x=3, y=0\); when \(x=1, y=4\); cao. Must be paired |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch of "+ve" cubic with two SPs, roughly correct shape or just two TPs shown OR: \(f(1)=0\) and \(f(3)=0\), find \(k\) for each | M1 | Subst \(x=1\) & \(x=3\) into \(y=x^3-6x^2+9x\) ft (a); or identify \(k=-4\) and \(0\) |
| \(k>0\) or \(k<-4\) | A1f [2] | ft their (a); Correct ans: M1A1 |
## Question 4:
### Part (a):
$3x^2 - 12x + 9 = 0$ | **M1** | Attempt differentiate & $=0$; $x(x-3)^2=0$ M1 tp at $x=3$ A1
$x=3$ or $1$ | **A1** | Correct equation. May be implied by ans
$(3,0)$ and $(1,4)$ | **A1f, A1** [4] | Allow when $x=3, y=0$; when $x=1, y=4$; cao. Must be paired
### Part (b):
Sketch of "+ve" cubic with two SPs, roughly correct shape or just two TPs shown OR: $f(1)=0$ and $f(3)=0$, find $k$ for each | **M1** | Subst $x=1$ & $x=3$ into $y=x^3-6x^2+9x$ ft (a); or identify $k=-4$ and $0$
$k>0$ or $k<-4$ | **A1f** [2] | ft their (a); Correct ans: M1A1
---4
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the stationary points on the curve $y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$.
\item The equation $x ^ { 3 } - 6 x ^ { 2 } + 9 x + k = 0$ has exactly one real root.
Using your answers from part (a) or otherwise, find the range of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q4 [6]}}