Solutions from graphical analysis

5
\includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).

1. Find the values of the positive integers \(a , b\) and \(c\).
2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
   1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
   2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\)
      The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .

9
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-14_558_963_260_589} The function \(\mathrm { f } : x \mapsto p \sin ^ { 2 } 2 x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. In terms of \(p\) and \(q\), state the range of f .
2. State the number of solutions of the following equations.
   (a) \(\mathrm { f } ( x ) = p + q\)
   (b) \(\mathrm { f } ( x ) = q\)
   (c) \(\mathrm { f } ( x ) = \frac { 1 } { 2 } p + q\)
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\mathrm { f } ( x ) = 4\), showing all necessary working.

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$\mathrm { f } ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence find the set of values of \(x\) for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$

6.
\includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

8. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$f ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence, using set notation, find the set of values of \(x\) for which $$f ( - x ) \geq 6 \text { and } x < 0$$