| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2014 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.2 This is a straightforward Core 1/2 question testing basic differentiation and integration of standard power functions. Students need only rewrite terms in index form (x^{-2} and x^{1/2}), then apply power rule mechanically. No problem-solving or conceptual insight required—pure routine manipulation that's easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07l Derivative of ln(x): and related functions1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = -16x^{-3} - 2x^{-\frac{1}{2}} + 3\) or \(f'(x) = -\frac{16}{x^3} - \frac{2}{\sqrt{x}} + 3\) | M1 A1 A1 | M1: Attempt to differentiate – power reduced \(x^n \to x^{n-1}\) or \(3x\) becomes \(3\) |
| [3] | A1: two correct terms (of the three shown), may be unsimplified. A1: fully correct and simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int f(x)dx = -8x^{-1} - \frac{4x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{3x^2}{2} - x + (c)\) | M1 A1 A1 | M1: Attempt to integrate, one power increased \(x^n \to x^{n+1}\). A1: Two of four terms correct unsimplified. A1: Three terms correct unsimplified |
| \(\int f(x)dx = -8x^{-1} - \frac{8x^{\frac{3}{2}}}{3} + \frac{3x^2}{2} - x + c\) or \(\frac{-8}{x} - \frac{8x\sqrt{x}}{3} + \frac{3x^2}{2} - x + c\) | A1 | A1: All correct simplified with constant – allow \(-1x\) for \(-x\). N.B. Integrating answer to part (a) is M0 |
| [4] |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = -16x^{-3} - 2x^{-\frac{1}{2}} + 3$ or $f'(x) = -\frac{16}{x^3} - \frac{2}{\sqrt{x}} + 3$ | M1 A1 A1 | M1: Attempt to differentiate – power reduced $x^n \to x^{n-1}$ or $3x$ becomes $3$ |
| | [3] | A1: two correct terms (of the three shown), may be unsimplified. A1: fully correct and **simplified** |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int f(x)dx = -8x^{-1} - \frac{4x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{3x^2}{2} - x + (c)$ | M1 A1 A1 | M1: Attempt to integrate, one power increased $x^n \to x^{n+1}$. A1: Two of four terms correct unsimplified. A1: Three terms correct unsimplified |
| $\int f(x)dx = -8x^{-1} - \frac{8x^{\frac{3}{2}}}{3} + \frac{3x^2}{2} - x + c$ or $\frac{-8}{x} - \frac{8x\sqrt{x}}{3} + \frac{3x^2}{2} - x + c$ | A1 | A1: All correct **simplified** with constant – allow $-1x$ for $-x$. N.B. Integrating answer to part (a) is M0 |
| | [4] | |
---2.
$$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$
Giving your answers in their simplest form, find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { f } ^ { \prime } ( x )$
\item $\int \mathrm { f } ( x ) \mathrm { d } x$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2014 Q2 [7]}}