| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| 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) requires routine algebraic manipulation to decompose into A/(3-x) + (Bx+C)/(1+x²). Part (ii) applies standard binomial expansions to each term. While it requires multiple steps and careful algebra, both techniques are core A-level procedures with no novel insight needed, 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/Working | Mark | Guidance |
| State or imply form \(\frac{A}{3-x} + \frac{Bx+C}{1+x^2}\) | B1 | |
| Use relevant method to determine a constant | M1 | |
| Obtain \(A = 6\) | A1 | |
| Obtain \(B = -2\) | A1 | |
| Obtain \(C = 1\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct method to obtain first two terms of expansion of \((3-x)^{-1}\) or \(\left(1-\frac{1}{3}x\right)^{-1}\) or \((1+x^2)^{-1}\) | M1 | |
| Obtain \(\frac{A}{3}\left(1+\frac{1}{3}x+\frac{1}{9}x^2+\frac{1}{27}x^3\right)\) | A1 | |
| Obtain \((Bx+C)(1-x^2)\) | A1 | |
| Obtain sufficient terms of product \((Bx+C)(1-x^2)\), \(B,C \neq 0\) and add the two expansions | M1 | |
| Obtain final answer \(3-\frac{4}{3}x-\frac{7}{9}x^2+\frac{56}{27}x^3\) | A1 | [5] |
# Question 9:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply form $\frac{A}{3-x} + \frac{Bx+C}{1+x^2}$ | B1 | |
| Use relevant method to determine a constant | M1 | |
| Obtain $A = 6$ | A1 | |
| Obtain $B = -2$ | A1 | |
| Obtain $C = 1$ | A1 | [5] |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct method to obtain first two terms of expansion of $(3-x)^{-1}$ or $\left(1-\frac{1}{3}x\right)^{-1}$ or $(1+x^2)^{-1}$ | M1 | |
| Obtain $\frac{A}{3}\left(1+\frac{1}{3}x+\frac{1}{9}x^2+\frac{1}{27}x^3\right)$ | A1 | |
| Obtain $(Bx+C)(1-x^2)$ | A1 | |
| Obtain sufficient terms of product $(Bx+C)(1-x^2)$, $B,C \neq 0$ and add the two expansions | M1 | |
| Obtain final answer $3-\frac{4}{3}x-\frac{7}{9}x^2+\frac{56}{27}x^3$ | A1 | [5] |
---9 (i) Express $\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }$ in partial fractions.\\
(ii) Hence obtain the expansion of $\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
\hfill \mbox{\textit{CAIE P3 2012 Q9 [10]}}