| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| 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 +0.8 This question requires decomposing a rational function with a repeated linear factor into partial fractions, then applying binomial expansion to multiple terms with negative/fractional powers. The repeated factor (1-x)² adds complexity to the partial fractions setup, and combining three separate binomial expansions requires careful coefficient collection. This is more demanding than routine single-expansion questions but remains a standard P3/Further Maths exercise. |
| 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 \(\frac{A}{3+x} + \frac{B}{1-x} + \frac{C}{(1-x)^2}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A = -3\), \(B = -1\), \(C = 2\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form \(\frac{A}{3+x} + \frac{Dx+E}{(1-x)^2}\), where \(A=-3\), \(D=1\) and \(E=1\), B1M1A1A1A1 as above. |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to find the first two terms of the expansion of \((3+x)^{-1}\), \((1+\frac{1}{3}x)^{-1}\), \((1-x)^{-1}\) or \((1-x)^{-2}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^3\) of each partial fraction | A1 | FT on A |
| A1 | FT on B | |
| A1 | FT on C | |
| Obtain final answer \(\frac{10}{3}x + \frac{44}{9}x^2 + \frac{190}{27}x^3\) | A1 | For the \(A\), \(D\), \(E\) form of fractions give M1A1ftA1ft for the expanded partial fractions, then, if \(D \neq 0\), M1 for multiplying out fully, and A1 for the final answer. |
| Total | 5 |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{3+x} + \frac{B}{1-x} + \frac{C}{(1-x)^2}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A = -3$, $B = -1$, $C = 2$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form $\frac{A}{3+x} + \frac{Dx+E}{(1-x)^2}$, where $A=-3$, $D=1$ and $E=1$, B1M1A1A1A1 as above. |
| **Total** | **5** | |
## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to find the first two terms of the expansion of $(3+x)^{-1}$, $(1+\frac{1}{3}x)^{-1}$, $(1-x)^{-1}$ or $(1-x)^{-2}$ | M1 | |
| Obtain correct unsimplified expansions up to the term in $x^3$ of each partial fraction | A1 | FT on A |
| | A1 | FT on B |
| | A1 | FT on C |
| Obtain final answer $\frac{10}{3}x + \frac{44}{9}x^2 + \frac{190}{27}x^3$ | A1 | For the $A$, $D$, $E$ form of fractions give M1A1ftA1ft for the expanded partial fractions, then, if $D \neq 0$, M1 for multiplying out fully, and A1 for the final answer. |
| **Total** | **5** | |9 Let $\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }$.\\
(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 $x ^ { 3 }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q9 [10]}}