| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative of simple polynomial (integer powers) |
| Difficulty | Easy -1.2 This is a straightforward C1 question requiring direct application of basic differentiation and integration rules for polynomial terms. Both parts involve routine manipulation (rewriting as powers of x) and applying standard formulas with no problem-solving or conceptual insight needed—easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{dy}{dx} = 6x^2 - 6x^{-3}\) | M1 A1 A1 | M1 for attempt to differentiate \(x^n \to x^{n-1}\); 1st A1 for \(6x^2\); 2nd A1 for \(-6x^{-3}\) or \(-\dfrac{6}{x^3}\). Inclusion of \(+c\) scores A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{2x^4}{4} + \dfrac{3x^{-1}}{-1}(+C)\) | M1 A1 | M1 for attempt to integrate \(x^n \to x^{n+1}\); 1st A1 for both \(x\) terms correct but unsimplified. Ignore \(+c\) here |
| \(\dfrac{x^4}{2} - 3x^{-1} + C\) | A1 | 2nd A1 for both terms simplified and \(+c\). Accept \(-\dfrac{3}{x}\) but NOT \(+-3x^{-1}\). Condone \(+c\) missing on final line if on unsimplified line |
# Question 3:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{dy}{dx} = 6x^2 - 6x^{-3}$ | M1 A1 A1 | M1 for attempt to differentiate $x^n \to x^{n-1}$; 1st A1 for $6x^2$; 2nd A1 for $-6x^{-3}$ or $-\dfrac{6}{x^3}$. Inclusion of $+c$ scores A0 |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{2x^4}{4} + \dfrac{3x^{-1}}{-1}(+C)$ | M1 A1 | M1 for attempt to integrate $x^n \to x^{n+1}$; 1st A1 for both $x$ terms correct but unsimplified. Ignore $+c$ here |
| $\dfrac{x^4}{2} - 3x^{-1} + C$ | A1 | 2nd A1 for both terms simplified and $+c$. Accept $-\dfrac{3}{x}$ but NOT $+-3x^{-1}$. Condone $+c$ missing on final line if on unsimplified line |
---3. Given that $y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0$, find
\begin{enumerate}[label=(\alph*)]
\item $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item $\int y \mathrm {~d} x$, simplifying each term.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2009 Q3 [6]}}