| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Simplify then solve or apply |
| Difficulty | Standard +0.3 This is a standard C3 algebraic fractions question requiring common denominator manipulation and completing the square. Part (a) involves routine algebraic manipulation (5 marks suggests multiple steps but no novel insight). Part (b) requires recognizing that a quadratic with negative discriminant has no real roots—a standard technique. Slightly above average difficulty due to the algebraic complexity, but follows predictable C3 patterns without requiring creative problem-solving. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02y Partial fractions: decompose rational functions |
| Answer | Marks |
|---|---|
| (a) \(= \frac{(x-10)(2x-1) - (x-8)(x+4)}{(x-3)(x+4)(2x-1)}\) | M1 A1 |
| \(= \frac{x^2-17x+42}{(x-3)(x+4)(2x-1)}\) | A1 |
| \(= \frac{(x-14)(x-3)}{(x-3)(x+4)(2x-1)} = \frac{x-14}{(x+4)(2x-1)}\) | M1 A1 |
| (b) \(\frac{x-14}{(x+4)(2x-1)} = 1\), \(x - 14 = 2x^2 + 7x - 4\) | M1 |
| \(x^2 + 3x + 5 = 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(b^2 - 4ac < 0 \therefore\) no real roots | M1 A1 | (9 marks) |
**(a)** $= \frac{(x-10)(2x-1) - (x-8)(x+4)}{(x-3)(x+4)(2x-1)}$ | M1 A1 |
$= \frac{x^2-17x+42}{(x-3)(x+4)(2x-1)}$ | A1 |
$= \frac{(x-14)(x-3)}{(x-3)(x+4)(2x-1)} = \frac{x-14}{(x+4)(2x-1)}$ | M1 A1 |
**(b)** $\frac{x-14}{(x+4)(2x-1)} = 1$, $x - 14 = 2x^2 + 7x - 4$ | M1 |
$x^2 + 3x + 5 = 0$ | A1 |
$b^2 - 4ac = 9 - 20 = -11$
$b^2 - 4ac < 0 \therefore$ no real roots | M1 A1 | (9 marks)\begin{enumerate}[label=(\alph*)]
\item Express
$$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)}$$
as a single fraction in its simplest form. [5]
\item Hence, show that the equation
$$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)} = 1$$
has no real roots. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q4 [9]}}