| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative after algebraic simplification (fractional/mixed powers) |
| Difficulty | Easy -1.2 This is a straightforward application of standard differentiation and integration rules for polynomials with fractional powers. Both parts require only direct recall of power rule formulas with no problem-solving or conceptual challenges—simpler 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 |
| \(4x^3 + 3x^{-\frac{1}{2}}\) | M1A1A1 (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{x^5}{5} + 4x^2 + C\) | M1A1A1 (3 marks) |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4x^3 + 3x^{-\frac{1}{2}}$ | M1A1A1 (3 marks) | |
**Notes:**
- M1: for $x^n \to x^{n-1}$, i.e. $x^3$ or $x^{-\frac{1}{2}}$ seen
- 1st A1: for $4x^3$ **or** $6 \times \frac{1}{2} \times x^{-\frac{1}{2}}$ (o.e.) — ignore any $+c$ for this mark
- 2nd A1: for simplified terms i.e. **both** $4x^3$ **and** $3x^{-\frac{1}{2}}$, or $\frac{3}{\sqrt{x}}$, and no $+c$ — note $\frac{3}{1}x^{-\frac{1}{2}}$ is A0
- Apply ISW here and award marks when first seen
---
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{x^5}{5} + 4x^2 + C$ | M1A1A1 (3 marks) | |
**Notes:**
- M1: for $x^n \to x^{n+1}$ applied to $y$ only, so $x^5$ or $x^{\frac{3}{2}}$ seen. Do not award for integrating answer to part (a)
- 1st A1: for $\dfrac{x^5}{5}$ or $\dfrac{6x^{\frac{3}{2}}}{\frac{3}{2}}$ (or better) — allow $\frac{1}{5}x^5$ here but not for 2nd A1
- 2nd A1: for fully correct and simplified answer with $+C$ — allow $\left(\frac{1}{5}\right)x^5$. If $+C$ appears earlier but not on a line where 2nd A1 could be scored, then A0
**Total: 6 marks**Given that $y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }$, find in their simplest form
\begin{enumerate}[label=(\alph*)]
\item $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item $\int y \mathrm {~d} x$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2012 Q1 [6]}}