| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| 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 testing basic differentiation and integration of power functions. Both parts require only direct application of standard rules (power rule) after rewriting the fraction as a negative power. No problem-solving or conceptual insight needed—pure routine manipulation. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dy}{dx} = 4x + 18x^{-4}\) | M1 A1 | M1: \(x^2 \to x\) or \(x^{-3} \to x^{-4}\) |
| (b) \(\frac{2x^3}{3} - \frac{6x^{-2}}{-2} + C\) | M1 A1 A1 | M1: \(x^2 \to x^3\) or \(x^{-3} \to x^{-2}\) or \(+C\) |
| \(\left(= \frac{2x^3}{3} + 3x^{-2} + C\right)\) |
| Answer | Marks |
|---|---|
| Guidance: In both parts, accept any correct version, simplified or not. Accept \(4x'\) for \(4x\). + C in part (a) instead of part (b): Penalise only once, so if otherwise correct scores M1 A0, M1 A1 A1. |
**(a)** $\frac{dy}{dx} = 4x + 18x^{-4}$ | M1 A1 | M1: $x^2 \to x$ or $x^{-3} \to x^{-4}$
**(b)** $\frac{2x^3}{3} - \frac{6x^{-2}}{-2} + C$ | M1 A1 A1 | M1: $x^2 \to x^3$ or $x^{-3} \to x^{-2}$ or $+C$
$\left(= \frac{2x^3}{3} + 3x^{-2} + C\right)$ | |
**Total: 5 marks**
| Guidance: In both parts, accept any correct version, simplified or not. Accept $4x'$ for $4x$. + C in part (a) instead of part (b): Penalise only once, so if otherwise correct scores M1 A0, M1 A1 A1.
---4. Given that $y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0$,
\begin{enumerate}[label=(\alph*)]
\item find $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
\item find $\int y \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2006 Q4 [5]}}