- A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = ( 2 - k x ) ^ { 5 }$$
and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)
- show that \(k = \frac { 2 } { 5 }\)
- Find the first three terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$
giving each term in simplest form.
Using the solution to part (b) and making your method clear,
- find the gradient of \(C\) at the point where \(x = 0\)