| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Curve from derivative information |
| Difficulty | Moderate -0.8 This is a straightforward C1 integration question requiring routine techniques: integrate a polynomial, use a point to find the constant, factorise a cubic (with verification provided), and sketch showing x-intercepts. All steps are standard procedures with no problem-solving insight required, making it easier than average but not trivial due to the multi-step nature. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials1.08b Integrate x^n: where n != -1 and sums |
(a) M1 for attempting to integrate, $x^n \to x^{n+1}$
$f(x) = \frac{6x^3}{3} - \frac{10x^2}{2} - 12x + C$
A1 for all x terms correct, need not be simplified. Ignore $+ C$ here.
M1 for some use of $x = 5$ and $f(5) = 65$ to form an equation in C based on their integration. There must be some visible attempt to use $x = 5$ and $f(5) = 65$. No $+C$ is M0.
$x = 5: 250 - 125 - 60 + C = 65$
$C = 0$
A1 for $C = 0$. This mark cannot be scored unless a suitable equation is seen.
(b) M1 for attempting to take out a correct factor or to verify. Allow usual errors on signs. They must get to the equivalent of one of the given partially factorised expressions or, if verifying, $x(2x^2 + 3x - 8x - 12)$ i.e. with no errors in signs.
$x(2x^2 - 5x - 12)$ or $(2x^2 + 3x)(x-4)$ or $(2x+3)(x^2-4x)$
$= x(2x+3)(x-4)$ (*)
A1cso for proceeding to printed answer with no incorrect working seen. Comment not required.
This mark is dependent upon a fully correct solution to part (a) so M1A1M0A0M1A0 for (a) \& (b). Will be common or M1A1M1A0M1A0. To score 2 in (b) they must score 4 in (a).
(c) B1 for positive $x^3$ shaped curve (with a max and a min) positioned anywhere.
B1 for any curve that passes through the origin (B0 if it only touches at the origin)
B1 for the two points clearly given as coords or values marked in appropriate places on9. The curve $C$ with equation $y = \mathrm { f } ( x )$ passes through the point $( 5,65 )$.
Given that $\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12$,
\begin{enumerate}[label=(\alph*)]
\item use integration to find $\mathrm { f } ( x )$.
\item Hence show that $\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )$.
\item In the space provided on page 17, sketch $C$, showing the coordinates of the points where $C$ crosses the $x$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2007 Q9 [9]}}