10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Find the equation of the curve.
- Find the coordinates of the other stationary point and determine its nature.
- Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646}
The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\). - Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
- Find the coordinates of \(A\), giving each coordinate in surd form.
- Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.