| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| 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. Part (i) is routine A-level technique, and part (ii) requires applying the binomial theorem to each partial fraction term—straightforward but requiring careful algebraic manipulation. Slightly easier than average as it's a well-practiced question type with clear methodology. |
| Spec | 1.02y Partial fractions: decompose rational functions4.05c Partial fractions: extended to quadratic denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\dfrac{A}{1-x}+\dfrac{B}{2x+3}+\dfrac{C}{(2x+3)^2}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A=1\), \(B=-2\), \(C=5\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 | Alternative form \(\dfrac{A}{1-x}+\dfrac{Dx+E}{(2x+3)^2}\) where \(A=1\), \(D=-4\), \(E=-1\) scores B1M1A1A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to find the first two terms of the expansion of \((1-x)^{-1}\), \((1+\frac{2}{3}x)^{-1}\), \((2x+3)^{-1}\), \((1+\frac{2}{3}x)^{-2}\) or \((2x+3)^{-2}\) | M1 | |
| Obtain correct unsimplified expansions up to the term in \(x^2\) of each partial fraction | A3 FT | |
| Obtain final answer \(\dfrac{8}{9}+\dfrac{19}{27}x+\dfrac{13}{9}x^2\), or equivalent | A1 | |
| Total | 5 |
# Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\dfrac{A}{1-x}+\dfrac{B}{2x+3}+\dfrac{C}{(2x+3)^2}$ | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values $A=1$, $B=-2$, $C=5$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | Alternative form $\dfrac{A}{1-x}+\dfrac{Dx+E}{(2x+3)^2}$ where $A=1$, $D=-4$, $E=-1$ scores B1M1A1A1A1 |
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# Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to find the first two terms of the expansion of $(1-x)^{-1}$, $(1+\frac{2}{3}x)^{-1}$, $(2x+3)^{-1}$, $(1+\frac{2}{3}x)^{-2}$ or $(2x+3)^{-2}$ | M1 | |
| Obtain correct unsimplified expansions up to the term in $x^2$ of each partial fraction | A3 FT | |
| Obtain final answer $\dfrac{8}{9}+\dfrac{19}{27}x+\dfrac{13}{9}x^2$, or equivalent | A1 | |
| **Total** | **5** | |
---(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 2017 Q8 [10]}}