| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions with quadratic factor |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with series expansion. Part (i) is routine A-level partial fractions with one linear and one irreducible quadratic factor. Part (ii) requires expanding each partial fraction term using binomial expansion, which is methodical but straightforward. The question tests standard techniques without requiring novel insight or complex problem-solving, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply partial fractions are of the form \(\frac{A}{x-2} + \frac{Bx+C}{x^{2}+3}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A = -1, B = 3, C = -1\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [5] |
| (ii) Use correct method to obtain the first two terms of the expansions of \((x-2)^{-1}\), \(\left(1-\frac{1}{2}x\right)^{-1}\), \((x^{2}+3)^{-1}\) or \(\left(1+\frac{1}{3}x^{2}\right)^{-1}\) | M1 | |
| Substitute correct unsimplified expansions up to the term in \(x^{2}\) into each partial fraction | A1*+A1* | |
| Multiply out fully by \(Bx + C\), where \(BC \neq 0\) | M1 | |
| Obtain final answer \(\frac{1}{6} + \frac{5}{4}x + \frac{17}{72}x^{2}\), or equivalent | A1 | [5] |
**(i)** State or imply partial fractions are of the form $\frac{A}{x-2} + \frac{Bx+C}{x^{2}+3}$ | B1 |
Use a relevant method to determine a constant | M1 |
Obtain one of the values $A = -1, B = 3, C = -1$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [5]
**(ii)** Use correct method to obtain the first two terms of the expansions of $(x-2)^{-1}$, $\left(1-\frac{1}{2}x\right)^{-1}$, $(x^{2}+3)^{-1}$ or $\left(1+\frac{1}{3}x^{2}\right)^{-1}$ | M1 |
Substitute correct unsimplified expansions up to the term in $x^{2}$ into each partial fraction | A1*+A1* |
Multiply out fully by $Bx + C$, where $BC \neq 0$ | M1 |
Obtain final answer $\frac{1}{6} + \frac{5}{4}x + \frac{17}{72}x^{2}$, or equivalent | A1 | [5]
[Symbolic binomial coefficients, e.g. $\binom{-1}{1}$ are not sufficient for the M1. The f.t. is on $A, B, C$.]
[In the case of an attempt to expand $\left(2x^{2}-7x-1\right)\left(x-2\right)^{-1}\left(x^{2}+3\right)^{-1}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
[If $B$ or $C$ omitted from the form of partial fractions, give B0M1A0A0A0 in (i); M1A1*A1*M1 in (ii)]Let $f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}$.
\begin{enumerate}[label=(\roman*)]
\item Express $f(x)$ in partial fractions. [5]
\item Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q7 [10]}}