| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| 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 algebraic manipulation to decompose into partial fractions (with one linear and one quadratic factor). Part (ii) applies the binomial theorem to expand each fraction—straightforward but requiring careful bookkeeping through x³. This is slightly easier than average as it follows a predictable template with no novel insights required. |
| 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 \(\frac{A}{1+x} + \frac{Bx + C}{2 + x^2}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A = -2, B = 1, C = 4\) | 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 \((1 + x)^{-1}, \left(1 + \frac{1}{2}x^2\right)\) or \((2 + x^2)^{-1}\) in ascending powers of \(x\) | M1 | |
| Obtain correct unsimplified expansion up to the term in \(x^3\) of each partial fraction | A1√ & A1√ | |
| Multiply out fully by \(Bx + C\), where \(BC \neq 0\) | M1 | |
| Obtain final answer \(\frac{5}{2}x - 3x^2 + \frac{7}{4}x^3\), or equivalent | 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 one of the values $A = -2, B = 1, C = 4$ | 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 $(1 + x)^{-1}, \left(1 + \frac{1}{2}x^2\right)$ or $(2 + x^2)^{-1}$ in ascending powers of $x$ | M1 |
Obtain correct unsimplified expansion up to the term in $x^3$ of each partial fraction | A1√ & A1√ |
Multiply out fully by $Bx + C$, where $BC \neq 0$ | M1 |
Obtain final answer $\frac{5}{2}x - 3x^2 + \frac{7}{4}x^3$, or equivalent | A1 | [5]
[Symbolic binomial coefficients, e.g. $\binom{-1}{1}$, 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√ in (ii), max 4/10.]
[In the case of an attempt to expand $(5x - x^2)(1 + x)^{-1}(2 + x^2)^{-1}$, give M1A1A1 for the expansions, M1 for the multiplying out fully, and A1 for the final answer.]
[Allow use of Maclaurin, giving M1A1√A1√ for differentiating and obtaining $f(0) = 0$ and $f'(0) = \frac{5}{2}$, A1√ for $f''(0) = -6$, and A1 for $f'''(0) = \frac{21}{2}$ and the final answer (the f.t. is on $A, B, C$ if used).]
[For the identity $5x - x^2 = (2 + 2x + x^2 + x^3)(a + bx + cx^2 + dx^3)$ give M1A1; then M1A1 for using a relevant method to obtain two of $a = b = \frac{5}{2}, c = -3$ and $d = \frac{7}{4}$; then A1 for the final answer in series form.]8 (i) Express $\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
\hfill \mbox{\textit{CAIE P3 2011 Q8 [10]}}