Partial fractions with tangent equation

A question is this type if and only if it involves partial fractions, series expansion, and asks to find the equation of a tangent line to the curve at a specific point.

1 questions · Challenging +1.2

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Edexcel AEA 2003 June Q4
11 marks Challenging +1.2
4. $$f ( x ) = \frac { 1 - 3 x } { \left( 1 + 3 x ^ { 2 } \right) ( 1 - x ) ^ { 2 } } , x \neq 1$$
  1. Find the constants \(A , B , C\) and \(D\) such that $$\mathrm { f } ( x ) \equiv \frac { A x + B } { 1 + 3 x ^ { 2 } } + \frac { C } { 1 - x } + \frac { D } { ( 1 - x ) ^ { 2 } }$$
  2. Find a series expansion for \(\mathrm { f } ( x )\) in ascending powers of \(x\) ,up to and including the term in \(x ^ { 4 }\) .
  3. Find an equation of the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 0\) .