a) State or imply partial fractions of the form $\frac{A}{{1 - 2x}} + \frac{B}{{2 + x}} + \frac{C}{{{{\left( {2 + x} \right)}^2}}}$

Use any relevant method to determine a constant

Obtain one of the values $A = 1$, $B = 1$, $C =  - 2$

Obtain a second value

Obtain the third value

[The form $\frac{A}{{1 - 2x}} + \frac{{Dx + E}}{{{{\left( {2 + x} \right)}^2}}}$, where $A = 1$, $D = 1$, $E = 0$, is acceptable scoring B1M1A1A1A1 as above.]

b) Use correct method to obtain the first two terms of the expansion of ${\left( {1 - 2x} \right)^{ - 1}}$, ${\left( {2 + x} \right)^{ - 1}}$, ${\left( {2 + x} \right)^{ - 2}}$, ${\left( {1 + \frac{1}{2}x} \right)^{ - 1}}$, or ${\left( {1 + \frac{1}{2}x} \right)^{ - 2}}$

Obtain correct unsimplified expansions up to the term in ${x^2}$ of each partial fraction A1√ + A1√ + A1√

Obtain answer $1 + \frac{9}{4}x + \frac{{15}}{4}{x^2}$, or equivalent

[Symbolic binomial coefficients, e.g. $\left( \begin{gathered}
   - 1 \hfill \\
  \,\,1 \hfill \\ 
\end{gathered}  \right)$, are not sufficient for the M1. The f.t. is on $A$, $B$, $C$.]

[For the $A$, $D$, $E$ form of partial fractions, give M1A1√A1√ for the expansions then, if $D \ne 0$, M1 for multiplying out fully and A1 for the final answer.]

[In the case of an attempt to expand $\left( {4 + 5x - {x^2}} \right){\left( {1 - 2x} \right)^{ - 1}}{\left( {2 + x} \right)^{ - 2}}$, give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]