- (a) Express \(1.5 \sin 2 x + 2 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate.
(b) Express \(3 \sin x \cos x + 4 \cos ^ { 2 } x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\) and \(c\) are constants to be found.
(c) Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos ^ { 2 } x\).
\begin{figure}[h]
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\caption{Figure 1}
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\end{figure}
Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = \frac { x ^ { 2 } + 1 } { ( 1 + x ) ( 3 - x ) } , 0 \leq x < 3$$
(a) Given that \(\mathrm { f } ( x ) = A + \frac { B } { 1 + x } + \frac { C } { 3 - x }\), find the values of the constants \(A , B\) and \(C\).
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
(b) Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p , q\) and \(r\) are rational constants to be found.