Single unknown from one coefficient condition

3. (a) Expand $$\frac { 1 } { ( 2 - 5 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. Given that the binomial expansion of \(\frac { 2 + k x } { ( 2 - 5 x ) ^ { 2 } } , | x | < \frac { 2 } { 5 }\), is $$\frac { 1 } { 2 } + \frac { 7 } { 4 } x + A x ^ { 2 } + \ldots$$ (b) find the value of the constant \(k\),
(c) find the value of the constant \(A\).

8. $$f ( x ) = ( 8 - 3 x ) ^ { \frac { 4 } { 3 } } \quad 0 < x < \frac { 8 } { 3 }$$

1. Show that the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) is $$A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } + \ldots$$ where \(A\) and \(B\) are constants to be found.
2. Use proof by contradiction to prove that the curve with equation $$y = 8 + 8 x - \frac { 15 } { 2 } x ^ { 2 }$$ does not intersect the curve with equation $$y = A - 8 x + \frac { x ^ { 2 } } { 2 } + B x ^ { 3 } \quad 0 < x < \frac { 8 } { 3 }$$ where \(A\) and \(B\) are the constants found in part (a).
   (Solutions relying on calculator technology are not acceptable.)

3

1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).