2019 RI P2 Q2

Timothy Gan

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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Published: 17th January 2023

Written by

Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.

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