| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Transformation effect on key points |
| Difficulty | Standard +0.3 This is a standard C2 calculus question involving differentiation to find stationary points and a simple transformation. While it requires multiple steps (factorising, differentiating, solving f'(x)=0, classifying stationary points, and applying a horizontal stretch), all techniques are routine for this level. The transformation in part (iv) is straightforward. Slightly above average due to the multi-part nature and the transformation application, but no novel problem-solving required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(3x^2-6x-9\) | M1 | |
| use of their \(y'=0\) | M1 | |
| \(x=-1\) | A1 | |
| \(x=3\) | A1 | |
| valid method for determining nature of turning point | M1 | |
| max at \(x=-1\) and min at \(x=3\) | A1 | c.a.o. |
| Answer | Marks | Guidance |
|---|---|---|
| \(x(x^2-3x-9)\) | M1 | |
| \(\dfrac{3\pm\sqrt{45}}{2}\) or \((x-\frac{3}{2})^2=9+\frac{9}{4}\) | M1 | |
| \(0,\ \dfrac{3}{2}\pm\dfrac{\sqrt{45}}{2}\) o.e. | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| sketch of cubic with two turning points correct way up | G1 | |
| \(x\)-intercepts – negative, 0, positive shown | DG1 |
## Question 3:
**Part (i):**
$3x^2-6x-9$ | M1 |
use of their $y'=0$ | M1 |
$x=-1$ | A1 |
$x=3$ | A1 |
valid method for determining nature of turning point | M1 |
max at $x=-1$ and min at $x=3$ | A1 | c.a.o. | 6
**Part (ii):**
$x(x^2-3x-9)$ | M1 |
$\dfrac{3\pm\sqrt{45}}{2}$ or $(x-\frac{3}{2})^2=9+\frac{9}{4}$ | M1 |
$0,\ \dfrac{3}{2}\pm\dfrac{\sqrt{45}}{2}$ o.e. | A1 | | 3
**Part (iii):**
sketch of cubic with two turning points correct way up | G1 |
$x$-intercepts – negative, 0, positive shown | DG1 | | 2
---(i) Express $\mathrm { f } ( x )$ in factorised form.\\
(ii) Show that the equation of the curve may be written as $y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$.\\
(iii) Use calculus to show that, correct to 1 decimal place, the $x$-coordinate of the minimum point on the curve is 0.4 .
Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.\\
(iv) State, correct to 1 decimal place, the coordinates of the maximum point on the curve $y = \mathrm { f } ( 2 x )$.
\hfill \mbox{\textit{OCR MEI C2 Q3 [12]}}