| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Simplify then show identity |
| Difficulty | Standard +0.3 This is a straightforward C3 question requiring algebraic manipulation to combine fractions (factoring the quadratic denominator, finding common denominators), then differentiation using the quotient rule, followed by solving a quadratic equation. All techniques are standard and the question provides clear scaffolding with the target form given in part (a). |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02y Partial fractions: decompose rational functions1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(x^2 + 2x - 3 = (x+3)(x-1)\) | B1 | Correct factorisation |
| \(f(x) = \frac{x(x^2+2x-3)+3(x+3)-12}{(x+3)(x-1)}\) \(\left[= \frac{x^3+2x^2-3}{(x+3)(x-1)}\right]\) | M1A1 | Combining fractions over common denominator |
| \(= \frac{(x-1)(x^2+3x+3)}{(x-1)(x+3)}\) | M1 | Factorising numerator |
| \(= \frac{x^2+3x+3}{x+3}\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(f'(x) = \frac{(x+3)(2x+3)-(x^2+3x+3)}{(x+3)^2}\) \(\left[= \frac{x^2+6x+6}{(x+3)^2}\right]\) | M1 A2,1,0 | Quotient rule |
| Setting \(f'(x) = \frac{22}{25}\) and attempting to solve quadratic | M1 | |
| \(x = 2\) (only this solution) | A1 | (5) |
## Question 4:
### Part (a):
| Answer/Working | Marks | Notes |
|---|---|---|
| $x^2 + 2x - 3 = (x+3)(x-1)$ | B1 | Correct factorisation |
| $f(x) = \frac{x(x^2+2x-3)+3(x+3)-12}{(x+3)(x-1)}$ $\left[= \frac{x^3+2x^2-3}{(x+3)(x-1)}\right]$ | M1A1 | Combining fractions over common denominator |
| $= \frac{(x-1)(x^2+3x+3)}{(x-1)(x+3)}$ | M1 | Factorising numerator |
| $= \frac{x^2+3x+3}{x+3}$ | A1 | **(5)** |
### Part (b):
| Answer/Working | Marks | Notes |
|---|---|---|
| $f'(x) = \frac{(x+3)(2x+3)-(x^2+3x+3)}{(x+3)^2}$ $\left[= \frac{x^2+6x+6}{(x+3)^2}\right]$ | M1 A2,1,0 | Quotient rule |
| Setting $f'(x) = \frac{22}{25}$ and attempting to solve quadratic | M1 | |
| $x = 2$ (only this solution) | A1 | **(5)** |
**ALT (b):** $f(x) = x + \frac{3}{x+3}$, $f'(x) = 1 - \frac{3}{(x+3)^2}$
---4.
$$\mathrm { f } ( x ) = x + \frac { 3 } { x - 1 } - \frac { 12 } { x ^ { 2 } + 2 x - 3 } , x \in \mathbb { R } , x > 1$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 3 } { x + 3 }$.
\item Solve the equation $\mathrm { f } ^ { \prime } ( x ) = \frac { 22 } { 25 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q4 [10]}}