| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| 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. The partial fractions decomposition with a repeated linear factor is routine A-level technique, and the binomial expansion of each term up to x² requires only straightforward application of the formula. While it involves multiple steps, each step follows well-practiced procedures without requiring problem-solving insight or novel approaches. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\dfrac{A}{1-x}+\dfrac{B}{2+3x}+\dfrac{C}{(2+3x)^2}\) | B1 | |
| Use a correct method for finding a coefficient | M1 | |
| Obtain one of \(A=1,\ B=-1,\ C=6\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | In the form \(\dfrac{A}{1-x}+\dfrac{Dx+E}{(2+3x)^2}\), where \(A=1,\ D=-3\) and \(E=4\) can score B1 M1 A1 A1 A1 as above |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use a correct method to find the first two terms of the expansion of \((1-x)^{-1}\), \((2+3x)^{-1}\), \(\left(1+\frac{3}{2}x\right)^{-1}\), \((2+3x)^{-2}\) or \(\left(1+\frac{3}{2}x\right)^{-2}\) | M1 | Symbolic coefficients not sufficient for M1; shown with \(A=1\), \(B=1\), \(C=6\) expansions in guidance |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1 FT | \((1+x+x^2)+\left(-\frac{1}{2}+\left(\frac{3}{4}\right)x-\left(\frac{9}{8}\right)x^2\right)\) |
| A1 FT | \(+\left(\frac{6}{4}-\left(\frac{18}{4}\right)x+\left(\frac{81}{8}\right)x^2\right)\) [FT is on \(A\), \(B\), \(C\)] | |
| A1 FT | \(\left(1-\frac{1}{2}+\frac{6}{4}\right)+\left(1+\frac{3}{4}-\frac{18}{4}\right)x+\left(1-\frac{9}{8}+\frac{81}{8}\right)x^2\) | |
| Obtain final answer \(2-\dfrac{11}{4}x+10x^2\), or equivalent | A1 | Allow unsimplified fractions; if using \(D=-3\), \(E=4\) form, FT is on \(A\), \(D\), \(E\) |
| 5 |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\dfrac{A}{1-x}+\dfrac{B}{2+3x}+\dfrac{C}{(2+3x)^2}$ | B1 | |
| Use a correct method for finding a coefficient | M1 | |
| Obtain one of $A=1,\ B=-1,\ C=6$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | In the form $\dfrac{A}{1-x}+\dfrac{Dx+E}{(2+3x)^2}$, where $A=1,\ D=-3$ and $E=4$ can score B1 M1 A1 A1 A1 as above |
| | **5** | |
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use a correct method to find the first two terms of the expansion of $(1-x)^{-1}$, $(2+3x)^{-1}$, $\left(1+\frac{3}{2}x\right)^{-1}$, $(2+3x)^{-2}$ or $\left(1+\frac{3}{2}x\right)^{-2}$ | M1 | Symbolic coefficients not sufficient for M1; shown with $A=1$, $B=1$, $C=6$ expansions in guidance |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1 FT | $(1+x+x^2)+\left(-\frac{1}{2}+\left(\frac{3}{4}\right)x-\left(\frac{9}{8}\right)x^2\right)$ |
| | A1 FT | $+\left(\frac{6}{4}-\left(\frac{18}{4}\right)x+\left(\frac{81}{8}\right)x^2\right)$ [FT is on $A$, $B$, $C$] |
| | A1 FT | $\left(1-\frac{1}{2}+\frac{6}{4}\right)+\left(1+\frac{3}{4}-\frac{18}{4}\right)x+\left(1-\frac{9}{8}+\frac{81}{8}\right)x^2$ |
| Obtain final answer $2-\dfrac{11}{4}x+10x^2$, or equivalent | A1 | Allow unsimplified fractions; if using $D=-3$, $E=4$ form, FT is on $A$, $D$, $E$ |
| | **5** | |9 Let $\mathrm { f } ( x ) = \frac { 8 + 5 x + 12 x ^ { 2 } } { ( 1 - x ) ( 2 + 3 x ) ^ { 2 } }$.
\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]}}