| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Find stationary points of standard polynomial |
| Difficulty | Moderate -0.3 This is a structured multi-part question covering standard C2 topics (factorization from roots, expanding brackets, differentiation for stationary points, and function transformations). Parts (i)-(iii) are routine textbook exercises requiring straightforward application of techniques. Part (iv) adds mild challenge with the horizontal stretch transformation, but overall this is slightly easier than average due to its highly scaffolded nature and standard methods throughout. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \((x + 5)(x - 2)(x + 2)\) | 2 | M1 for \(a(x + 5)(x - 2)(x + 2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([(x + 2)](x^2 + 3x - 10)\) | M1 | for correct expansion of one pair of their brackets |
| \(x^3 + 3x^2 - 10x + 2x^2 + 6x - 20\) o.e. | M1 | for clear expansion of correct factors – accept given answer from \((x + 5)(x^2 - 4)\) as first step |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y' = 3x^2 + 10x - 4\) | M2 | |
| \(\frac{dy}{dx} = 3x^2 + 10x - 4 = 0\) s.o.i. | M1 | M1 if one error or M1 for substitution of 0.4 if trying to obtain 0, and A1 for correct demonstration of sign change |
| \(x = 0.36\ldots\) from formula o.e. | A1 | |
| \((-3.7, 12.6)\) | B1+1 | |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| \((-1.8, 12.6)\) | B1+1 | accept \((-1.9, 12.6)\) or f.t. (\(\frac{1}{2}\) their max \(x\), their max \(y\)) |
| 2 |
### Part i
$(x + 5)(x - 2)(x + 2)$ | 2 | M1 for $a(x + 5)(x - 2)(x + 2)$
### Part ii
$[(x + 2)](x^2 + 3x - 10)$ | M1 | for correct expansion of one pair of their brackets
$x^3 + 3x^2 - 10x + 2x^2 + 6x - 20$ o.e. | M1 | for clear expansion of correct factors – accept given answer from $(x + 5)(x^2 - 4)$ as first step
| 2 |
### Part iii
$y' = 3x^2 + 10x - 4$ | M2 |
$\frac{dy}{dx} = 3x^2 + 10x - 4 = 0$ s.o.i. | M1 | M1 if one error or M1 for substitution of 0.4 if trying to obtain 0, and A1 for correct demonstration of sign change
$x = 0.36\ldots$ from formula o.e. | A1 |
$(-3.7, 12.6)$ | B1+1 |
| 6 |
### Part iv
$(-1.8, 12.6)$ | B1+1 | accept $(-1.9, 12.6)$ or f.t. ($\frac{1}{2}$ their max $x$, their max $y$)
| 2 |\includegraphics{figure_11}
Fig. 11 shows a sketch of the cubic curve $y = \text{f}(x)$. The values of $x$ where it crosses the $x$-axis are $-5$, $-2$ and $2$, and it crosses the $y$-axis at $(0, -20)$.
\begin{enumerate}[label=(\roman*)]
\item Express f($x$) in factorised form. [2]
\item Show that the equation of the curve may be written as $y = x^3 + 5x^2 - 4x - 20$. [2]
\item 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. [6]
\item State, correct to 1 decimal place, the coordinates of the maximum point on the curve $y = \text{f}(2x)$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2008 Q11 [12]}}