| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem and Partial Fractions |
| Type | Partial fractions then binomial expansion |
| Difficulty | Standard +0.8 This question requires decomposing a rational function with both linear and irreducible quadratic factors into partial fractions (non-routine setup), then applying binomial expansion to multiple terms. The irreducible quadratic denominator significantly increases algebraic complexity beyond standard partial fractions questions, and combining this with careful binomial expansion of several terms requires solid technique and attention to detail. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\dfrac{A}{2+x} + \dfrac{Bx+C}{4+x^2}\) | B1 | |
| Use 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to obtain first two terms of expansion of \((1+\frac{1}{2}x)^{-1}\), \((2+x)^{-1}\), \((1+\frac{1}{4}x^2)^{-1}\) or \((4+x^2)^{-1}\) | M1 | |
| Obtain correct unsimplified expansions up to term in \(x^2\) of each partial fraction | A1+A1 | f.t. on \(A\), \(B\), \(C\) |
| Multiply out up to term in \(x^2\) by \(Bx+C\), where \(BC \neq 0\) | M1 | |
| Obtain final answer \(\dfrac{3}{4}x - \dfrac{1}{2}x^2\) | A1 | |
| [Symbolic binomial coefficients e.g. \({}_{-1}C_2\) not sufficient for first M1. f.t. is on \(A\), \(B\), \(C\).] | ||
| [If attempt to expand \(x(6-x)(2+x)^{-1}(4+x^2)^{-1}\), give M1A1A1 for expansions, M1 for multiplying out fully, A1 for final answer.] |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\dfrac{A}{2+x} + \dfrac{Bx+C}{4+x^2}$ | B1 | |
| Use 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 | |
**Total: 5**
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## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to obtain first two terms of expansion of $(1+\frac{1}{2}x)^{-1}$, $(2+x)^{-1}$, $(1+\frac{1}{4}x^2)^{-1}$ or $(4+x^2)^{-1}$ | M1 | |
| Obtain correct unsimplified expansions up to term in $x^2$ of each partial fraction | A1+A1 | f.t. on $A$, $B$, $C$ |
| Multiply out up to term in $x^2$ by $Bx+C$, where $BC \neq 0$ | M1 | |
| Obtain final answer $\dfrac{3}{4}x - \dfrac{1}{2}x^2$ | A1 | |
| [Symbolic binomial coefficients e.g. ${}_{-1}C_2$ not sufficient for first M1. f.t. is on $A$, $B$, $C$.] | | |
| [If attempt to expand $x(6-x)(2+x)^{-1}(4+x^2)^{-1}$, give M1A1A1 for expansions, M1 for multiplying out fully, A1 for final answer.] | | |
**Total: 5**
---9 Let $\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }$.\\
(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 2017 Q9 [10]}}