- In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
The curve \(C\) has equation
$$y = \frac { 16 } { 9 ( 3 x - k ) } \quad x \neq \frac { k } { 3 }$$
where \(k\) is a positive constant not equal to 3
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form in terms of \(k\).
The point \(P\) with \(x\) coordinate 1 lies on \(C\).
Given that the gradient of the curve at \(P\) is - 12 - find the two possible values of \(k\)
Given also that \(k < 3\)
- find the equation of the normal to \(C\) at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
- show, using algebraic integration that,
$$\int _ { 1 } ^ { 3 } \frac { 16 } { 9 ( 3 x - k ) } d x = \lambda \ln 10$$
where \(\lambda\) is a constant to be found.