| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Improper fraction partial fractions |
| Difficulty | Standard +0.3 This is a standard C4 partial fractions question with an improper fraction requiring polynomial division first, followed by routine binomial expansion. The algebraic manipulation is straightforward with no conceptual surprises, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(2x^2 + 5x - 10 = A(x-1)(x+2) + B(x+2) + C(x-1)\) | |
| \(x \to 1\): \(-3 = 3B \Rightarrow B = -1\) | M1 A1 | |
| \(x \to -2\): \(-12 = -3C \Rightarrow C = 4\) | A1 | |
| \(A = 2\) | B1 | |
| (b) | \(\frac{2x^2 + 5x - 10}{(x-1)(x+2)} = 2 + (1-x)^{-1} + 2\left(1 + \frac{x}{2}\right)^{-1}\) | M1 |
| \((1-x)^{-1} = 1 + x + x^2 + \ldots\) | B1 | |
| \(\left(1 + \frac{x}{2}\right)^{-1} = 1 - \frac{x}{2} + \frac{x^2}{4} + \ldots\) | B1 | |
| \(\frac{2x^2 + 5x - 10}{(x-1)(x+2)} = (2 + 1 + 2) + (1-1)x + \left(1 + \frac{1}{2}\right)x^2 + \ldots = 5 + \ldots\) | M1 | |
| \(= \ldots + \frac{3}{2}x^2 + \ldots\) | ft their \(A - B + \frac{1}{2}C\); 0x stated or implied | |
| A1 ft A1 A1 |
(a) | $2x^2 + 5x - 10 = A(x-1)(x+2) + B(x+2) + C(x-1)$ | | |
| $x \to 1$: $-3 = 3B \Rightarrow B = -1$ | M1 A1 | |
| $x \to -2$: $-12 = -3C \Rightarrow C = 4$ | A1 | |
| $A = 2$ | B1 | |
(b) | $\frac{2x^2 + 5x - 10}{(x-1)(x+2)} = 2 + (1-x)^{-1} + 2\left(1 + \frac{x}{2}\right)^{-1}$ | M1 | |
| $(1-x)^{-1} = 1 + x + x^2 + \ldots$ | B1 | |
| $\left(1 + \frac{x}{2}\right)^{-1} = 1 - \frac{x}{2} + \frac{x^2}{4} + \ldots$ | B1 | |
| $\frac{2x^2 + 5x - 10}{(x-1)(x+2)} = (2 + 1 + 2) + (1-1)x + \left(1 + \frac{1}{2}\right)x^2 + \ldots = 5 + \ldots$ | M1 | |
| $= \ldots + \frac{3}{2}x^2 + \ldots$ | | ft their $A - B + \frac{1}{2}C$; 0x stated or implied |
| | A1 ft A1 A1 | |
---5.
$$\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) } \equiv A + \frac { B } { x - 1 } + \frac { C } { x + 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $A , B$ and $C$.
\item Hence, or otherwise, expand $\frac { 2 x ^ { 2 } + 5 x - 10 } { ( x - 1 ) ( x + 2 ) }$ in ascending powers of $x$, as far as the term in $x ^ { 2 }$. Give each coefficient as a simplified fraction.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2010 Q5 [11]}}