2019 NJC P1 Q4
(i)
Show that $\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}=\frac{p}{1-x}+\frac{q}{{{\left( 2+x \right)}^{2}}}$.
[1]
(ii)
Find the values of $p$, $q$ and $r$ such that the series expansion of
$\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}+r\ln \left( 1-3x \right)$
up to and including the term in ${{x}^{2}}$ is $11-3x+{{x}^{2}}$.
[4]
(iii)
Find the range of the values of $x$ for which the expansion is valid.
[2]
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