2019 RI P2 Q2
(a)
The curve $y=\text{f}\left( x \right)$ passes through the point $\left( 0,81 \right)$ and has gradient given by
$\frac{\text{d}y}{\text{d}x}={{\left( \frac{1}{3}y-15x \right)}^{\frac{1}{3}}}$.
Find the first three non-zero terms in the Maclaurin series for $y$.
[4]
(b)
Let $\text{g}\left( x \right)=\frac{4-3x+{{x}^{2}}}{\left( 1+x \right){{\left( 1-x \right)}^{2}}}$.
(i) Express $\text{g}\left( x \right)$ in the form $\frac{A}{1+x}+\frac{1}{1-x}+\frac{B}{{{\left( 1-x \right)}^{2}}}$ , where $A$ and $B$ are constants to be determined.
[2]
The expansion of $\text{g}\left( x \right)$, in ascending powers of $x$, is
${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+…+{{c}_{r}}{{x}^{r}}+…$
(ii) Find the values of ${{c}_{0}}$, ${{c}_{1}}$ and ${{c}_{2}}$ and show that ${{c}_{3}}=3$.
[3]
(iii) Express ${{c}_{r}}$ in terms of $r$.
[1]
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