| 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 repeated linear factor) and binomial expansion. The partial fractions decomposition is routine, and the binomial expansions of (1+x)^{-2} and (3x+2)^{-1} require careful but straightforward application of the generalized binomial theorem. Slightly easier than average due to being a textbook-style exercise with clear steps. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks |
|---|---|
| State or imply partial fractions are of the form \(\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{3x+2}\) | B1 |
| Use any relevant method to obtain a constant | M1 |
| Obtain one of the values \(A = 1, B = 2, C = -3\) | A1 |
| Obtain a second value | A1 |
| Obtain the third value | A1 |
| [5] |
| Answer | Marks |
|---|---|
| Use correct method to obtain the first two terms of the expansion of \((x+1)^{-1}, (x+1)^{-2}, (3x+2)^{-1}\) or \((1 + \frac{3}{2}x)^{-1}\) | M1 |
| Obtain correct unsimplified expansion up to the term in \(x^2\) of each partial fraction | A1 V + A1 V + A1 V |
| Obtain answer \(\frac{3}{2} - \frac{11}{4}x + \frac{29}{8}x^2\), or equivalent | A1 |
| [5] |
**(i)**
| State or imply partial fractions are of the form $\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{3x+2}$ | B1 |
| Use any relevant method to obtain a constant | M1 |
| Obtain one of the values $A = 1, B = 2, C = -3$ | A1 |
| Obtain a second value | A1 |
| Obtain the third value | A1 |
| [5] |
**(ii)**
| Use correct method to obtain the first two terms of the expansion of $(x+1)^{-1}, (x+1)^{-2}, (3x+2)^{-1}$ or $(1 + \frac{3}{2}x)^{-1}$ | M1 |
| Obtain correct unsimplified expansion up to the term in $x^2$ of each partial fraction | A1 V + A1 V + A1 V |
| Obtain answer $\frac{3}{2} - \frac{11}{4}x + \frac{29}{8}x^2$, or equivalent | A1 |
| [5] |
[Symbolic binomial coefficients, e.g. $\begin{pmatrix} -1 \\ 1 \end{pmatrix}$, are not sufficient for the first M1. The f.t. is on $A, B, C$.]
[The form $\frac{Dx+E}{(x+1)^2} + \frac{C}{3x+2}$, where $D = 1, E = 3, C = -3$, is acceptable. In part (i) give B1M1A1A1A1.
In part (ii) give M1A1∨A1∨ for the expansions, and, if $DE \neq 0$, M1 for multiplying out fully and A1 for the final answer.]
[If $B$ or $C$ omitted from the form of fractions, give B0M1A0A0A0 in (i); M1A1∨A1∨ in (ii), max 4/10]
[If $D$ or $E$ 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 $(5x + 3)(x + 1)^{-2}(3x + 2)^{-1}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
[Allow use of Maclaurin, giving M1A1∨A1∨ for differentiating and obtaining $f(0) = \frac{3}{2}$ and $f'(0) = -\frac{11}{4}$, A1V for $f''(0) = \frac{29}{4}$, and A1 for the final answer (the f.t. is on $A, B, C$ if used).]8 (i) Express $\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 5 x + 3 } { ( x + 1 ) ^ { 2 } ( 3 x + 2 ) }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2009 Q8 [10]}}