Find function constants from given conditions

6 The function f , where \(\mathrm { f } ( x ) = a \sin x + b\), is defined for the domain \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2\) and that \(\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8\),

1. find the values of \(a\) and \(b\),
2. find the values of \(x\) for which \(\mathrm { f } ( x ) = 0\), giving your answers in radians correct to 2 decimal places,
3. sketch the graph of \(y = \mathrm { f } ( x )\).

8 The function f is defined by \(\mathrm { f } ( x ) = a + b \cos 2 x\), for \(0 \leqslant x \leqslant \pi\). It is given that \(\mathrm { f } ( 0 ) = - 1\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 7\).

1. Find the values of \(a\) and \(b\).
2. Find the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

3 The function \(\mathrm { f } : x \mapsto a + b \cos x\) is defined for \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } ( 0 ) = 10\) and that \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1\), find

1. the values of \(a\) and \(b\),
2. the range of \(f\),
3. the exact value of \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).

4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).

1. Find the values of the constants \(a\) and \(b\).
   \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution.
   \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).

1. (i) Find the solutions of the equation \(\sin \left( 3 x - 15 ^ { \circ } \right) = \frac { 1 } { 2 }\), for which \(0 \leqslant x \leqslant 180 ^ { \circ }\)
   (6)
   (ii)

\begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = \sin ( a x - b ) , \text { where } a > 0,0 < b < \pi$$ The curve cuts the \(x\)-axis at the points \(P , Q\) and \(R\) as shown.
Given that the coordinates of \(P , Q\) and \(R\) are \(\left( \frac { \pi } { 10 } , 0 \right) , \left( \frac { 3 \pi } { 5 } , 0 \right)\) and \(\left( \frac { 11 \pi } { 10 } , 0 \right)\) respectively, find the values of \(a\) and \(b\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\). The endpoints of the curve are \(\mathrm { P } ( - \pi , 1 )\) and \(\mathrm { Q } ( \pi , 3 )\), and \(\mathrm { f } ( x ) = a + \sin b x\), where \(a\) and \(b\) are constants. \begin{figure}[h] \end{figure}

1. Using Fig. 9, show that \(a = 2\) and \(b = \frac { 1 } { 2 }\).
2. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,2 )\). Show that there is no point on the curve at which the gradient is greater than this.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range. Write down the gradient of \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 2,0 )\).
4. Find the area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\).