| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| 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 +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (a) requires routine partial fraction decomposition with one linear and one irreducible quadratic factor. Part (b) involves expanding two terms using the binomial theorem for negative/fractional powers and collecting terms—straightforward application of techniques with no novel insight required. Slightly easier than average due to the mechanical nature of both parts. |
| 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 \(\frac{A}{x-2} + \frac{Bx+C}{2x^2+3}\) | B1 | If \(1 - \frac{A}{x-2} + \frac{Bx+C}{2x^2+3}\) or \(\frac{A}{x-2} + \frac{C}{2x^2+3}\) B0, then M1 A1 (for \(A=3\)) still possible. |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A = 3\), \(B = -1\) and \(C = 6\) | A1 | Allow all A marks obtained even if method would give errors if equations solved in a different order. |
| Obtain a second value | A1 | |
| Obtain the third value | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use correct method to find the first two terms of the expansion of \((x-2)^{-1}\), \(\left(1-\frac{1}{2}x\right)^{-1}\), \((2x^2+3)^{-1}\) or \(\left(1+\frac{2}{3}x^2\right)^{-1}\) | M1 | Symbolic binomial coefficients not sufficient for the M1 |
| Obtain correct unsimplified expansions, up to the term in \(x^2\), of each partial fraction | A1 FT, A1 FT | The FT is on \(A\), \(B\) and \(C\). \(-\frac{A}{2}\left[1-\left(-\frac{x}{2}\right)+\frac{(-1)(-2)}{2}\left(-\frac{x}{2}\right)^2+...\right]\); \(\frac{Bx+C}{3}\left[1-\frac{2x^2}{3}+...\right]\) |
| Extract the coefficient 3 correctly from \((2x^2+3)^{-1}\) with expansion to \(1\pm\frac{2}{3}x^2\), then multiply by \(Bx+C\) up to terms in \(x^2\), where \(BC\neq 0\) | M1 | \(\frac{C}{3}+\frac{Bx}{3}\pm\frac{C}{3}\left(\frac{2}{3}\right)x^2\) or \(\frac{1}{3}\left(C+Bx\pm C\left(\frac{2}{3}\right)x^2\right)\). Allow a slip in multiplication for M1. Allow miscopies in \(B\) and \(C\) from 7(a) |
| Obtain final answer \(\frac{1}{2}-\frac{13}{12}x-\frac{41}{24}x^2\) | A1 | Do not ISW |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{x-2} + \frac{Bx+C}{2x^2+3}$ | B1 | If $1 - \frac{A}{x-2} + \frac{Bx+C}{2x^2+3}$ or $\frac{A}{x-2} + \frac{C}{2x^2+3}$ B0, then M1 A1 (for $A=3$) still possible. |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A = 3$, $B = -1$ and $C = 6$ | A1 | Allow all A marks obtained even if method would give errors if equations solved in a different order. |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct method to find the first two terms of the expansion of $(x-2)^{-1}$, $\left(1-\frac{1}{2}x\right)^{-1}$, $(2x^2+3)^{-1}$ or $\left(1+\frac{2}{3}x^2\right)^{-1}$ | M1 | Symbolic binomial coefficients not sufficient for the M1 |
| Obtain correct unsimplified expansions, up to the term in $x^2$, of each partial fraction | A1 FT, A1 FT | The FT is on $A$, $B$ and $C$. $-\frac{A}{2}\left[1-\left(-\frac{x}{2}\right)+\frac{(-1)(-2)}{2}\left(-\frac{x}{2}\right)^2+...\right]$; $\frac{Bx+C}{3}\left[1-\frac{2x^2}{3}+...\right]$ |
| Extract the coefficient 3 correctly from $(2x^2+3)^{-1}$ with expansion to $1\pm\frac{2}{3}x^2$, then multiply by $Bx+C$ up to terms in $x^2$, where $BC\neq 0$ | M1 | $\frac{C}{3}+\frac{Bx}{3}\pm\frac{C}{3}\left(\frac{2}{3}\right)x^2$ or $\frac{1}{3}\left(C+Bx\pm C\left(\frac{2}{3}\right)x^2\right)$. Allow a slip in multiplication for M1. Allow miscopies in $B$ and $C$ from 7(a) |
| Obtain final answer $\frac{1}{2}-\frac{13}{12}x-\frac{41}{24}x^2$ | A1 | Do not ISW |
---7 Let $\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + 8 x - 3 } { ( x - 2 ) \left( 2 x ^ { 2 } + 3 \right) }$.
\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 2022 Q7 [10]}}