Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as
$$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
The equation of a curve is \(y = x ^ { 3 } + a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants.
In the case where the curve has no stationary point, show that \(a ^ { 2 } < 3 b\).
In the case where \(a = - 6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).
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The diagram shows a pyramid \(O A B C X\). The horizontal square base \(O A B C\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(X D = 10\) units. The mid-point of \(O X\) is \(M\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
Express the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { A C }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
Use a scalar product to find angle \(M A C\).
(a) The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by \(S _ { n } = 32 n - n ^ { 2 }\). Find the first term and the common difference.
(b) A geometric progression in which all the terms are positive has sum to infinity 20 . The sum of the first two terms is 12.8 . Find the first term of the progression.