1 The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2 \theta , y = \cos \theta$$ a) Find the Cartesian equation of \(C\).
b) Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac { \pi } { 3 }\), and at the point \(Q\), where \(\theta = \frac { \pi } { 2 }\). Write down the coordinates of \(P\) and \(Q\).
c) Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac { \pi } { 2 }\), $$\sin x + \cos x \geqslant 1$$ a) Given that \(f\) is a function,
i) state the condition for \(f ^ { - 1 }\) to exist,
ii) find \(f f ^ { - 1 } ( x )\).
b) The functions \(g\) and \(h\), are given by $$\begin{aligned} & g ( x ) = x ^ { 2 } - 1
& h ( x ) = \mathrm { e } ^ { x } + 1 \end{aligned}$$ i) Suggest a domain for \(g\) such that \(g ^ { - 1 }\) exists.
ii) Given the domain of \(h\) is ( \(- \infty , \infty\) ), find an expression for \(h ^ { - 1 } ( x )\) and sketch, using the same axes, the graphs of \(h ( x )\) and \(h ^ { - 1 } ( x )\). Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
iii) Determine an expression for \(g h ( x )\) in its simplest form. a) Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 360 ^ { \circ }\) satisfying $$8 \sin \theta - 15 \cos \theta - 7 = 0$$ c) Determine the greatest value and the least value of the expression $$\frac { 1 } { 8 \sin \theta - 15 \cos \theta + 23 }$$ Evaluate a) \(\int _ { 1 } ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x\).
b) \(\int _ { 0 } ^ { 1 } \frac { 2 + x } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\). The variable \(y\) satisfies the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 5 - 2 y , \quad \text { where } x \geqslant 0$$ Given that \(y = 1\) when \(x = 0\), find an expression for \(y\) in terms of \(x\). a) Differentiate the following functions with respect to \(x\), simplifying your answer wherever possible. i) \(e ^ { 3 \tan x }\),
ii) \(\frac { \sin 2 x } { x ^ { 2 } }\).
b) A function is defined implicitly by $$3 x ^ { 2 } y + y ^ { 2 } - 5 x = 5$$ Find the equation of the normal at the point (1, 2). By drawing suitable graphs, show that \(x - 1 = \cos x\) has only one root. Starting with \(x _ { 0 } = 1\), use the Newton-Raphson method to find the value of this root correct to two decimal places.