Edexcel C34 2017 January — Question 3 9 marks

Exam BoardEdexcel
ModuleC34 (Core Mathematics 3 & 4)
Year2017
SessionJanuary
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeneralised Binomial Theorem and Partial Fractions
TypePartial fractions then binomial expansion
DifficultyStandard +0.3 This is a standard two-part question combining routine partial fractions decomposition with binomial expansion. Part (a) requires straightforward cover-up method or equating coefficients. Part (b) involves expanding two simple binomial terms (1-x)^{-1} and (3+2x)^{-1} using the standard formula, then multiplying and collecting terms—all mechanical processes covered extensively in C3/C4. No novel insight or complex manipulation required, making it slightly easier than average.
Spec1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1
3. (a) Express \(\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }\) in partial fractions.
(b) Hence, or otherwise, find the series expansion of $$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.
Question 3:
Part (a):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{9+11x}{(1-x)(3+2x)} = \frac{A}{1-x} + \frac{B}{3+2x}\) and attempt to find \(A\) or \(B\)M1 Use \(9+11x = A(3+2x)+B(1-x)\) with substitution or comparison
\(A = 4,\ B = -3\)A1, A1 (3) A1 one correct value, A1 both correct attached to correct fraction
Part (b):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((1-x)^{-1} = 1 + x + x^2 + x^3 + ...\)B1 Correct expansion, must be simplified
\((3+2x)^{-1} = \frac{1}{3}\times\left(1+(-1)\left(\frac{2}{3}x\right)+\frac{(-1)(-2)}{2}\left(\frac{2}{3}x\right)^2+\frac{(-1)(-2)(-3)}{6}\left(\frac{2}{3}x\right)^3...\right)\)B1 M1 B1: Factor of \(3^{-1}\) seen. M1: Binomial expansion with \(n=-1\) and term \(\left(\pm\frac{2}{3}x\right)\)
Attempts \(`4'\times(....) + `-3'\times(....)\)M1 Must attempt to combine both series using both coefficients
\(= 3 + \frac{14}{3}x + \frac{32}{9}x^2 + \frac{116}{27}x^3...\)A1, A1 (6) A1 two terms correct, A1 all four terms correct 
# Question 3:

## Part (a):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{9+11x}{(1-x)(3+2x)} = \frac{A}{1-x} + \frac{B}{3+2x}$ and attempt to find $A$ or $B$ | M1 | Use $9+11x = A(3+2x)+B(1-x)$ with substitution or comparison |
| $A = 4,\ B = -3$ | A1, A1 (3) | A1 one correct value, A1 both correct attached to correct fraction |

## Part (b):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(1-x)^{-1} = 1 + x + x^2 + x^3 + ...$ | B1 | Correct expansion, must be simplified |
| $(3+2x)^{-1} = \frac{1}{3}\times\left(1+(-1)\left(\frac{2}{3}x\right)+\frac{(-1)(-2)}{2}\left(\frac{2}{3}x\right)^2+\frac{(-1)(-2)(-3)}{6}\left(\frac{2}{3}x\right)^3...\right)$ | B1 M1 | B1: Factor of $3^{-1}$ seen. M1: Binomial expansion with $n=-1$ and term $\left(\pm\frac{2}{3}x\right)$ |
| Attempts $`4'\times(....) + `-3'\times(....)$ | M1 | Must attempt to combine both series using both coefficients |
| $= 3 + \frac{14}{3}x + \frac{32}{9}x^2 + \frac{116}{27}x^3...$ | A1, A1 (6) | A1 two terms correct, A1 all four terms correct |

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3. (a) Express $\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }$ in partial fractions.\\
(b) Hence, or otherwise, find the series expansion of

$$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$

in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.\\
Give each coefficient as a simplified fraction.\\

\hfill \mbox{\textit{Edexcel C34 2017 Q3 [9]}}
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