| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Standard +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires routine partial fraction decomposition with one linear and one quadratic factor. Part (ii) applies standard binomial expansions to each fraction and collects terms—straightforward but requires careful algebra. Slightly above average due to the quadratic factor and term collection, but still a textbook exercise with no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply partial fractions are of the form \(\frac{A}{1-x} + \frac{Bx+C}{2+x^2}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain \(A = \frac{2}{3}, B = \frac{2}{3}\) and \(C = \frac{1}{3}\) | A1 + A1 + A1 | [5] |
| (ii) Use correct method to find first two terms of the expansion of \((1-x)^{-1}, (2+x^2)^{-1}\) or \((1 + \frac{1}{3}x^3)^{-1}\) | M1 | |
| Obtain complete unsimplified expansions up to \(x^2\) of each partial fraction e.g. \(\frac{2}{3}(1 + x + x^2)\) and \(\frac{1}{2}(\frac{1}{x} - \frac{1}{8}x^2)\) | A1√ + A1√ | |
| Carry out multiplication of \((2 + x^3)^{-1}\) by \((\frac{2}{3}x - \frac{1}{3})\), or equivalent, provided \(BC \neq 0\) | M1 | |
| Obtain answer \(\frac{1}{3} + x + \frac{3}{4}x^2\) | A1 | [5] |
**(i)** State or imply partial fractions are of the form $\frac{A}{1-x} + \frac{Bx+C}{2+x^2}$ | B1 |
Use a relevant method to determine a constant | M1 |
Obtain $A = \frac{2}{3}, B = \frac{2}{3}$ and $C = \frac{1}{3}$ | A1 + A1 + A1 | [5]
**(ii)** Use correct method to find first two terms of the expansion of $(1-x)^{-1}, (2+x^2)^{-1}$ or $(1 + \frac{1}{3}x^3)^{-1}$ | M1 |
Obtain complete unsimplified expansions up to $x^2$ of each partial fraction e.g. $\frac{2}{3}(1 + x + x^2)$ and $\frac{1}{2}(\frac{1}{x} - \frac{1}{8}x^2)$ | A1√ + A1√ |
Carry out multiplication of $(2 + x^3)^{-1}$ by $(\frac{2}{3}x - \frac{1}{3})$, or equivalent, provided $BC \neq 0$ | M1 |
Obtain answer $\frac{1}{3} + x + \frac{3}{4}x^2$ | A1 | [5]
[Symbolic binomial coefficients are not sufficient for the first M1. The f.t. is on A, B, C.]
[If B or C omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1∇A1∇ in (ii), max 4/10]
[In the case of an attempt to expand $(1+x)(1-x)^{-1}(2+x^3)^{-1}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
[Allow Maclaurin, giving M1A1∇A1∇ for differentiating and obtaining f(0) = $\frac{1}{3}$ and f'(0) = 1, A1√ for f''(0) = $\frac{5}{2}$, and A1 for the final answer (the f.t. is on A, B, C if used).]8 (i) Express $\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 1 + x } { ( 1 - x ) \left( 2 + x ^ { 2 } \right) }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2009 Q8 [10]}}