| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Find stationary points of standard polynomial |
| Difficulty | Moderate -0.3 This is a standard C1 curve sketching question requiring routine differentiation to find stationary points, second derivative test for classification, and interpretation of the sketch. While multi-part, each step follows textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks |
|---|---|
| \(f'(x) = 12x - 3x^2\) | M1 A1 |
| For SP: \(12x - 3x^2 = 0\) | |
| \(3x(4-x) = 0\) | M1 |
| \(x = 0, 4\) | |
| \(\therefore (0, 2), (4, 34)\) | A1 |
| Answer | Marks |
|---|---|
| \(f''(x) = 12 - 6x\) | M1 |
| \(f''(0) = 12\), \(f''(x) > 0\) \(\therefore (0, 2)\) minimum | A1 |
| \(f''(4) = -12\), \(f''(x) < 0\) \(\therefore (4, 34)\) maximum | A1 |
| Answer | Marks |
|---|---|
| Sketch showing correct cubic shape with maximum and minimum marked | B2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(2 < k < 34\) | B1 | (10) |
# Question 8:
## Part (i):
$f'(x) = 12x - 3x^2$ | M1 A1 |
For SP: $12x - 3x^2 = 0$ | |
$3x(4-x) = 0$ | M1 |
$x = 0, 4$ | |
$\therefore (0, 2), (4, 34)$ | A1 |
## Part (ii):
$f''(x) = 12 - 6x$ | M1 |
$f''(0) = 12$, $f''(x) > 0$ $\therefore (0, 2)$ minimum | A1 |
$f''(4) = -12$, $f''(x) < 0$ $\therefore (4, 34)$ maximum | A1 |
## Part (iii):
Sketch showing correct cubic shape with maximum and minimum marked | B2 |
## Part (iv):
$2 < k < 34$ | B1 | **(10)**
---\begin{enumerate}
\item $f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$.\\
(i) Find the coordinates of the stationary points of the curve $y = \mathrm { f } ( x )$.\\
(ii) Determine whether each stationary point is a maximum or minimum point.\\
(iii) Sketch the curve $y = \mathrm { f } ( x )$.\\
(iv) State the set of values of $k$ for which the equation $\mathrm { f } ( x ) = k$ has three solutions.
\item The points $P$ and $Q$ have coordinates $( 7,4 )$ and $( 9,7 )$ respectively.\\
\end{enumerate}
\hfill \mbox{\textit{OCR C1 Q8 [10]}}