- (a) Use the binomial expansion to expand
$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) giving each coefficient as a fully simplified fraction.
$$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$
Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is
$$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$
(b) find the value of \(k\),
(c) find the value of \(m\).