| 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 +0.3 This is a standard two-part question combining partial fractions with binomial expansion. Part (i) requires decomposing into the form A/(2+x) + B/(3-x) + C/(3-x)² which is routine. Part (ii) involves expanding each term using the binomial theorem for negative/fractional powers and collecting terms, which is a well-practiced technique. While it requires careful algebra across multiple steps, it follows a predictable pattern 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 \(\frac{A}{2+x} + \frac{B}{3-x} + \frac{C}{(3-x)^2}\) | B1 | |
| Use a correct method to obtain a constant | M1 | |
| Obtain one of \(A=2\), \(B=2\), \(C=-7\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form \(\frac{A}{2+x} + \frac{Dx+E}{(3-x)^2}\), where \(A=2\), \(D=-2\) and \(E=-1\), B1M1A1A1A1 |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to find the first two terms of the expansion of \((2+x)^{-1}\), \((3-x)^{-1}\) or \((3-x)^{-2}\), or equivalent, e.g. \(\left(1+\frac{1}{2}x\right)^{-1}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A1 A1 A1 | FT on \(A\), \(B\) and \(C\): \(1-\frac{x}{2}+\frac{x^2}{4}\) \(\frac{2}{3}\left(1+\frac{x}{3}+\frac{x^2}{9}\right)\) \(-\frac{7}{9}\left(1+\frac{2x}{3}+\frac{3x^2}{9}\right)\) |
| Obtain final answer \(\frac{8}{9} - \frac{43}{54}x + \frac{7}{108}x^2\) | A1 | |
| Total | 5 | For the \(A\), \(D\), \(E\) form: M1A1ftA1ft for expanded partial fractions, then if \(D\neq 0\), M1 for multiplying out fully, and A1 for final answer |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{2+x} + \frac{B}{3-x} + \frac{C}{(3-x)^2}$ | B1 | |
| Use a correct method to obtain a constant | M1 | |
| Obtain one of $A=2$, $B=2$, $C=-7$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form $\frac{A}{2+x} + \frac{Dx+E}{(3-x)^2}$, where $A=2$, $D=-2$ and $E=-1$, B1M1A1A1A1 |
| **Total** | **5** | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to find the first two terms of the expansion of $(2+x)^{-1}$, $(3-x)^{-1}$ or $(3-x)^{-2}$, or equivalent, e.g. $\left(1+\frac{1}{2}x\right)^{-1}$ | M1 | |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A1 A1 A1 | FT on $A$, $B$ and $C$: $1-\frac{x}{2}+\frac{x^2}{4}$ $\frac{2}{3}\left(1+\frac{x}{3}+\frac{x^2}{9}\right)$ $-\frac{7}{9}\left(1+\frac{2x}{3}+\frac{3x^2}{9}\right)$ |
| Obtain final answer $\frac{8}{9} - \frac{43}{54}x + \frac{7}{108}x^2$ | A1 | |
| **Total** | **5** | For the $A$, $D$, $E$ form: M1A1ftA1ft for expanded partial fractions, then if $D\neq 0$, M1 for multiplying out fully, and A1 for final answer |8 Let $f ( x ) = \frac { 16 - 17 x } { ( 2 + x ) ( 3 - x ) ^ { 2 } }$.\\
(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 $x ^ { 2 }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q8 [10]}}