| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Standard partial fractions with series expansion |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions decomposition with series expansion. While it requires multiple techniques (partial fractions, then binomial expansion of each term), both are routine A-level procedures with no novel problem-solving required. The 5-mark allocation suggests moderate length but straightforward execution, placing it slightly above average difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions4.05c Partial fractions: extended to quadratic denominators |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply the form \(\frac{A}{3-2x} + \frac{Bx + C}{x^2 + 4}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A = 3\), \(B = -1\), \(C = -2\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [5] |
| (ii) Use correct method to find the first two terms of the expansion of \((3-2x)^{-1}\), \((1 - \frac{2}{3}x)^{-1}\), \((4 + x^2)^{-1}\) or \((1 + \frac{1}{4}x^2)^{-1}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1✱+A1✱ | |
| Multiply out up to the term in \(x^2\) by \(Bx + C\), where \(BC \neq 0\) | M1 | |
| Obtain final answer \(\frac{1}{2} + \frac{5}{12}x + \frac{41}{72}x^2\), or equivalent | A1 | [5] |
**(i)** State or imply the form $\frac{A}{3-2x} + \frac{Bx + C}{x^2 + 4}$ | B1 |
Use a relevant method to determine a constant | M1 |
Obtain one of the values $A = 3$, $B = -1$, $C = -2$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [5]
**(ii)** Use correct method to find the first two terms of the expansion of $(3-2x)^{-1}$, $(1 - \frac{2}{3}x)^{-1}$, $(4 + x^2)^{-1}$ or $(1 + \frac{1}{4}x^2)^{-1}$ | M1 |
Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1✱+A1✱ |
Multiply out up to the term in $x^2$ by $Bx + C$, where $BC \neq 0$ | M1 |
Obtain final answer $\frac{1}{2} + \frac{5}{12}x + \frac{41}{72}x^2$, or equivalent | A1 | [5]
[Symbolic coefficients, e.g. $\begin{pmatrix}-1\\2\end{pmatrix}$ are not sufficient for the first M1. The f.t. is on $A$, $B$, $C$.]
[In the case of an attempt to expand $(5x^2 + x + 6)(3-2x)^{-1}(x^2 + 4)^{-1}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.](i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$. [5]
\hfill \mbox{\textit{CAIE P3 2015 Q8 [10]}}