| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| 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 decomposition with binomial expansion. The partial fractions setup is routine with three linear factors, and the binomial expansions required are straightforward applications of (1+ax)^(-1). While it requires careful algebraic manipulation and multiple steps, it follows a well-practiced template with no novel insights needed, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement4.04a Line equations: 2D and 3D, cartesian and vector forms4.04f Line-plane intersection: find point4.04h Shortest distances: between parallel lines and between skew lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{1+2x}+\frac{B}{1-2x}+\frac{C}{2+x}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=-2\), \(B=1\) and \(C=4\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to find the first two terms of the expansion of \((1+2x)^{-1}\), \((1-2x)^{-1}\), \((2+x)^{-1}\) or \(\left(1+\frac{1}{2}x\right)^{-1}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1FT + A1FT + A1FT | The FT is on \(A\), \(B\) and \(C\) |
| Obtain final answer \(1+5x-\frac{7}{2}x^2\) | A1 | |
| Total | 5 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{1+2x}+\frac{B}{1-2x}+\frac{C}{2+x}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=-2$, $B=1$ and $C=4$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | |
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to find the first two terms of the expansion of $(1+2x)^{-1}$, $(1-2x)^{-1}$, $(2+x)^{-1}$ or $\left(1+\frac{1}{2}x\right)^{-1}$ | M1 | |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1FT + A1FT + A1FT | The FT is on $A$, $B$ and $C$ |
| Obtain final answer $1+5x-\frac{7}{2}x^2$ | A1 | |
| **Total** | **5** | |9 Let $\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q9 [10]}}