| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| 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. The partial fractions decomposition with a repeated linear factor is routine A-level technique, and the subsequent binomial expansions of (1-x)^{-1} and (2-x)^{-2} are straightforward applications of the formula. While it requires careful algebraic manipulation and multiple steps, it follows a well-practiced procedure without requiring problem-solving insight, 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 the form \(\frac{A}{1-x} + \frac{B}{2-x} + \frac{C}{(2-x)^2}\) | B1 | |
| Use a correct method to determine a constant | M1 | |
| Obtain one of \(A = 2, B = -1, C = 3\) | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Use correct method to find the first two terms of the expansion of \((1-x)^{-1}, (2-x)^{-1}, (2-x)^{-2}, (1-\frac{1}{2}x)^{-1}\) or \((1-\frac{1}{2}x)^{-2}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1✓ + A1✓ + A1✓ | |
| Obtain final answer \(\frac{9}{4} + \frac{5}{2}x + \frac{39}{16}x^2\), or equivalent | A1 | [5] |
**(i)** State or imply the form $\frac{A}{1-x} + \frac{B}{2-x} + \frac{C}{(2-x)^2}$ | B1 |
Use a correct method to determine a constant | M1 |
Obtain one of $A = 2, B = -1, C = 3$ | A1 |
Obtain a second value | A1 |
Obtain a third value | A1 | [5]
**[The alternative form $\frac{A}{1-x} + \frac{Dx + E}{(2-x)^2}$, where $A = 2, D = 1, E = 1$ is marked B1M1A1A1A1 as above.]**
**(ii)** Use correct method to find the first two terms of the expansion of $(1-x)^{-1}, (2-x)^{-1}, (2-x)^{-2}, (1-\frac{1}{2}x)^{-1}$ or $(1-\frac{1}{2}x)^{-2}$ | M1 |
Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1✓ + A1✓ + A1✓ |
Obtain final answer $\frac{9}{4} + \frac{5}{2}x + \frac{39}{16}x^2$, or equivalent | A1 | [5]
**[Symbolic binomial coefficients, e.g. $\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ are not sufficient for M1. The ✓ is on $A, B, C$.]**
**[For the $A.D.E$ form of partial fractions, give M1 A1✓ A1✓ for the expansions then, if $D \neq 0$, M1 for multiplying out fully and A1 for the final answer.]**
**[In the case of an attempt to expand $(x^2 - 8x + 9)(1-x)^{-1}(2-x)^{-2}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]**9 Let $\mathrm { f } ( x ) = \frac { x ^ { 2 } - 8 x + 9 } { ( 1 - x ) ( 2 - x ) ^ { 2 } }$.\\
(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 $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2014 Q9 [10]}}