Find and simplify the first four terms in the expansion of \(( 1 + 4 x ) ^ { 7 }\) in ascending powers of \(x\).
In the expansion of
$$( 3 + a x ) ( 1 + 4 x ) ^ { 7 }$$
the coefficient of \(x ^ { 2 }\) is 1001 . Find the value of \(a\).
(a) Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), indicating the coordinates of any points where the curve meets the axes.
(b) Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(- 180 ^ { \circ }\) and \(180 ^ { \circ }\).