The curve $y=\mathrm{f}\left( x \right)$ passes through the point $\left( 0,81 \right)$ and has grad

Question

The curve $y=\mathrm{f}\left( x \right)$ passes through the point $\left( 0,81 \right)$ and has gradient given by $\frac{\mathrm{d}y}{\mathrm{d}x}={{\left( \frac{1}{3}y-15x \right)}^{\frac{1}{3}}}$. Find the first three non-zero terms in the Maclaurin series for $y$. [4] Let $\mathrm{g}\left( x \right)=\frac{4-3x+{{x}^{2}}}{\left( 1+x \right){{\left( 1-x \right)}^{2}}}$. Express $\mathrm{g}\left( x \right)$ 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 $\mathrm{g}\left( x \right)$, in ascending powers of $x$, is ${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+...+{{c}_{r}}{{x}^{r}}+...$ Find the values of ${{c}_{0}}$, ${{c}_{1}}$ and ${{c}_{2}}$ and show that ${{c}_{3}}=3$. [3] Express ${{c}_{r}}$ in terms of $r$. [1]

Accepted Answer

(a) $y=81+3x-\frac7{27}x^2+\cdots$; (b)(i) $g(x)= \frac2{1+x}+\frac1{1-x}+\frac1{(1-x)^2}$; (b)(iii) $c_r=2(-1)^r+r+2$

2019 RI P2 Q2

2019 RI P2 Q2

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