| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| 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 +0.3 This is a standard two-part question combining partial fractions (routine algebraic manipulation) with binomial expansion of simple linear denominators. Both techniques are core A-level skills with no novel insight required, 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 |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(A+\frac{B}{2+x}+\frac{C}{3-2x}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=2\), \(B=-4\) and \(C=6\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to find the first two terms of the expansion of \((2+x)^{-1}\) or \((3-2x)^{-1}\), or equivalent | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1ft+A1ft | The ft is on \(B\) and \(C\) |
| Add the value of \(A\) to the sum of the expansions | M1 | |
| Obtain final answer \(2+\frac{7}{3}x+\frac{7}{18}x^2\) | A1 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $A+\frac{B}{2+x}+\frac{C}{3-2x}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=2$, $B=-4$ and $C=6$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to find the first two terms of the expansion of $(2+x)^{-1}$ or $(3-2x)^{-1}$, or equivalent | M1 | |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1ft+A1ft | The ft is on $B$ and $C$ |
| Add the value of $A$ to the sum of the expansions | M1 | |
| Obtain final answer $2+\frac{7}{3}x+\frac{7}{18}x^2$ | A1 | |8 Let $\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }$.\\
(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 2019 Q8 [10]}}