| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Differentiation of Simplified Fractions |
| Difficulty | Standard +0.3 This is a slightly above-average C3 question requiring algebraic manipulation to combine fractions with a common denominator, followed by straightforward quotient rule differentiation. The algebra is moderately involved but follows standard techniques, and the 'hence' structure guides students through the approach. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses common denominator: \(\frac{15(x-1)-2x(3x+4)+14}{(3x+4)(x-1)}\) | M1 | Combines two or three fractions into single fraction with correct use of common denominator |
| \(= \frac{-6x^2+7x-1}{(3x+4)(x-1)}\) | A1 | Correct answer with collected terms giving three term quadratic numerator |
| \(= \frac{-(6x-1)(x-1)}{(3x+4)(x-1)}\) | M1 | Factorises their quadratic following usual rules in numerator |
| \(= \frac{(1-6x)}{(3x+4)}\) or \(\frac{(-6x+1)}{(3x+4)}\) or \(-\frac{(6x-1)}{(3x+4)}\) | A1 (4) | cao, may be written in different ways |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f'(x) = \frac{(3x+4)\times(-6)-(1-6x)\times 3}{(3x+4)^2}\) | M1 A1ft | Applies quotient rule correctly; A1ft follows through their answer to (a) |
| \(= \frac{-27}{(3x+4)^2}\) | A1cao (3) | Must be correct simplified form; accept \(\frac{-27}{9x^2+24x+16}\) as alternative |
# Question 2:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses common denominator: $\frac{15(x-1)-2x(3x+4)+14}{(3x+4)(x-1)}$ | M1 | Combines two or three fractions into single fraction with correct use of common denominator |
| $= \frac{-6x^2+7x-1}{(3x+4)(x-1)}$ | A1 | Correct answer with collected terms giving three term quadratic numerator |
| $= \frac{-(6x-1)(x-1)}{(3x+4)(x-1)}$ | M1 | Factorises their quadratic following usual rules in numerator |
| $= \frac{(1-6x)}{(3x+4)}$ or $\frac{(-6x+1)}{(3x+4)}$ or $-\frac{(6x-1)}{(3x+4)}$ | A1 (4) | cao, may be written in different ways |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f'(x) = \frac{(3x+4)\times(-6)-(1-6x)\times 3}{(3x+4)^2}$ | M1 A1ft | Applies quotient rule correctly; A1ft follows through their answer to (a) |
| $= \frac{-27}{(3x+4)^2}$ | A1cao (3) | Must be correct simplified form; accept $\frac{-27}{9x^2+24x+16}$ as alternative |
---2.
$$f ( x ) = \frac { 15 } { 3 x + 4 } - \frac { 2 x } { x - 1 } + \frac { 14 } { ( 3 x + 4 ) ( x - 1 ) } , \quad x > 1$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ as a single fraction in its simplest form.
\item Hence, or otherwise, find $\mathrm { f } ^ { \prime } ( x )$, giving your answer as a single fraction in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2014 Q2 [7]}}