Sketch multiple graphs and count intersections

5

1. Sketch, on the same diagram, the curves \(y = \sin 2 x\) and \(y = \cos x - 1\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Hence state the number of solutions, in the interval \(0 \leqslant x \leqslant 2 \pi\), of the equations
   (a) \(2 \sin 2 x + 1 = 0\),
   (b) \(\sin 2 x - \cos x + 1 = 0\).

6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).

1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).

4

1. Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2 x\), for the interval \(0 \leqslant x \leqslant \pi\).
2. Hence state the number of solutions of the equation \(2 \sin x = \cos 2 x\) in the interval \(0 \leqslant x \leqslant \pi\).

4

1. Sketch the curve \(y = 2 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. By adding a suitable straight line to your sketch, determine the number of real roots of the equation $$2 \pi \sin x = \pi - x$$ State the equation of the straight line.

3

1. Sketch, on a single diagram, the graphs of \(y = \cos 2 \theta\) and \(y = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Write down the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(0 \leqslant \theta \leqslant 2 \pi\).
3. Deduce the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(10 \pi \leqslant \theta \leqslant 20 \pi\).

5

1. Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Verify that \(x = 30 ^ { \circ }\) is a root of the equation \(\sin x = \cos 2 x\), and state the other root of this equation for which \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
3. Hence state the set of values of \(x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), for which \(\sin x < \cos 2 x\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of

• the curve with equation \(y = \tan x\)
• the straight line l with equation \(y = \pi x\)
  in the interval \(- \pi < x < \pi\)
  1. State the period of \(\tan x\)
  2. Write down the number of roots of the equation
     1. \(\tan x = ( \pi + 2 ) x\) in the interval \(- \pi < x < \pi\)
     2. \(\tan x = \pi x\) in the interval \(- 2 \pi < x < 2 \pi\)
     3. \(\tan x = \pi x\) in the interval \(- 100 \pi < x < 100 \pi\)

9. \begin{figure}[h] \end{figure} Figure 3 shows a plot of the curve with equation \(y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)

1. State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
2. On Diagram 1, sketch and label the curves
   1. \(y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
   2. \(y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
3. Hence find the number of solutions of the equation
   1. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 2160 ^ { \circ }\)
   2. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 1980 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
      \section*{Diagram 1}

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line \(l\), shown in Figure 4, is an asymptote to \(y = \tan x\)

1. State an equation for \(l\). A copy of Figure 4, labelled Diagram 1, is shown on the next page.
2. 1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
   2. Hence, giving a reason, state the number of solutions of the equation
3. State the number of solutions of the equation \(\tan x = \frac { 1 } { x } + 1\) in the region
   1. \(0 \leqslant x \leqslant 40 \pi\)
   2. \(- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi\) $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1 \\ & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the graph of the curve with equation \(y = \sin x\) Given that \(\sin \alpha = p\), where \(0 < \alpha < 90 ^ { \circ }\)

1. state, in terms of \(p\), the value of
   1. \(2 \sin \left( 180 ^ { \circ } - \alpha \right)\)
   2. \(\sin \left( \alpha - 180 ^ { \circ } \right)\)
   3. \(3 + \sin \left( 180 ^ { \circ } + \alpha \right)\) A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
2. sketch the graph of \(y = \sin 2 x\)
3. Hence find, in terms of \(\alpha\), the \(x\) coordinates of any points in the interval \(0 < x < 180 ^ { \circ }\) where $$\sin 2 x = p$$
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
   \section*{Diagram 1}

1. (i) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
   (ii) Hence state how many solutions exist to the equation

$$\sin 2 x = \tan \frac { x } { 2 } ,$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.

4. (a) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
(b) Hence state how many solutions exist to the equation $$\sin 2 x = \tan \frac { x } { 2 }$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.