| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find curve equation from derivative (straightforward integration + point) |
| Difficulty | Easy -1.2 This is a straightforward integration question requiring only the power rule for standard polynomials, followed by substituting a point to find the constant of integration, then evaluating at another point. It's a routine C1 exercise with no problem-solving insight needed, making it easier than average. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \([f(x) =] \frac{3x^3}{3} - \frac{3x^2}{2} + 5x [+c]\) or \(\left\{x^3 - \frac{3}{2}x^2 + 5x(+c)\right\}\) | M1A1 | M1 for attempt to integrate \(x^n \to x^{n+1}\); A1 all correct, possibly unsimplified, ignore \(+c\) |
| \(10 = 8 - 6 + 10 + c\) | M1 | For using \(x=2\) and \(f(2)=10\) to form linear equation in \(c\); allow sign errors; must substitute into a changed expression |
| \(c = -2\) | A1 | |
| \(f(1) = 1 - \frac{3}{2} + 5\ "-2" = \frac{5}{2}\) | A1ft | For \(\frac{9}{2} + c\); follow through numerical \(c\ (\neq 0)\); dependent on 1st M1 and 1st A1 only |
## Question 7:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[f(x) =] \frac{3x^3}{3} - \frac{3x^2}{2} + 5x [+c]$ or $\left\{x^3 - \frac{3}{2}x^2 + 5x(+c)\right\}$ | M1A1 | M1 for attempt to integrate $x^n \to x^{n+1}$; A1 all correct, possibly unsimplified, ignore $+c$ |
| $10 = 8 - 6 + 10 + c$ | M1 | For using $x=2$ and $f(2)=10$ to form linear equation in $c$; allow sign errors; must substitute into a changed expression |
| $c = -2$ | A1 | |
| $f(1) = 1 - \frac{3}{2} + 5\ "-2" = \frac{5}{2}$ | A1ft | For $\frac{9}{2} + c$; follow through numerical $c\ (\neq 0)$; dependent on 1st M1 and 1st A1 only |
---\begin{enumerate}
\item A curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 2,10 )$. Given that
\end{enumerate}
$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$
find the value of $\mathrm { f } ( 1 )$.\\
\hfill \mbox{\textit{Edexcel C1 2012 Q7 [5]}}