Find coordinates of turning points

2

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575} The diagram shows the curve \(y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)\) where \(k\) is a positive constant and \(x\) is measured in radians. The curve crosses the \(x\)-axis at point \(A\) and \(B\) is a minimum point. Find the coordinates of \(A\) and \(B\).
2. Find the exact value of \(t\) that satisfies the equation $$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}

5. \begin{figure}[h] \end{figure} Figure 2 shows a plot of part of the curve with equation \(y = \cos 2 x\) with \(x\) being measured in radians. The point \(P\), shown on Figure 2, is a minimum point on the curve.

1. State the coordinates of \(P\). A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
2. Sketch, on Diagram 1, the curve with equation \(y = \sin x\)
3. Hence, or otherwise, deduce the number of solutions of the equation
   1. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 20 \pi\)
   2. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 21 \pi\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447} \captionsetup{labelformat=empty} \caption{
      Diagram 1}\}
      \end{figure} \textbackslash section*\{Diagram 1

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C _ { 1 }\) with equation \(y = 3 \sin x\), where \(x\) is measured in degrees. The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 3.

1. State
   1. the coordinates of \(P\),
   2. the coordinates of \(Q\). A different curve \(C _ { 2 }\) has equation \(y = 3 \sin x + k\), where \(k\) is a constant.
      The curve \(C _ { 2 }\) has a maximum \(y\) value of 10
      The point \(R\) is the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
2. State the coordinates of \(R\). Figure 3

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation \(y = 4 \cos x ^ { \circ }\) The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 1.

1. State
   1. the coordinates of \(P\),
   2. the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 4 \cos x ^ { \circ } + k\), where \(k\) is a constant.
      Curve \(C _ { 2 }\) has a minimum \(y\) value of - 1
      The point \(R\) is the maximum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
2. State the coordinates of \(R\).

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = A \cos ( x - 30 ) ^ { \circ }$$ where \(A\) is a constant. The point \(P\) is a minimum point on the curve and has coordinates \(( 30 , - 3 )\) as shown in Figure 4.

1. Write down the value of \(A\). The point \(Q\) is shown in Figure 4 and is a maximum point.
2. Find the coordinates of \(Q\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = 5 \cos x$$ with \(x\) being measured in degrees.
The point \(P\), shown in Figure 2, is a minimum point on \(C _ { 1 }\)

1. State the coordinates of \(P\) The point \(Q\) lies on a different curve \(C _ { 2 }\)
   Given that point \(Q\)
   • is a maximum point on the curve
   • is the maximum point with the smallest \(x\) coordinate, \(x > 0\)
   • find the coordinates of \(Q\) when
     1. \(C _ { 2 }\) has equation \(y = 5 \cos x - 2\)
     2. \(C _ { 2 }\) has equation \(y = - 5 \cos x\)

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 12 \sin x$$ where \(x\) is measured in radians.
The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)

1. Find the coordinates of \(P\). The curve \(C _ { 2 }\) has equation $$y = 12 \sin x + k$$ where \(k\) is a constant.
   Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3
2. find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate. The curve \(C _ { 3 }\) has equation $$y = 12 \sin ( x + B )$$ where \(B\) is a positive constant.
   Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate,
3. find
   1. the value of \(A\),
   2. the smallest possible value of \(B\).

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point \(Q\) and the point \(R ( k , 0 )\) lie on the curve and are shown in Figure 2.

1. State
   1. the coordinates of \(Q\),
   2. the value of \(k\).
2. Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for \(p\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where \(n\) is a constant.
The curve \(C _ { 1 }\) cuts the positive \(x\)-axis for the first time at point \(P ( 270,0 )\), as shown in Figure 4.

1. 1. State the value of \(n\)
   2. State the period of \(C _ { 1 }\) The point \(Q\), shown in Figure 4, is a minimum point of \(C _ { 1 }\)
2. State the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 2 \sin x ^ { \circ } + k\), where \(k\) is a constant.
   The point \(R \left( a , \frac { 12 } { 5 } \right)\) and the point \(S \left( - a , - \frac { 3 } { 5 } \right)\), both lie on \(C _ { 2 }\)
   Given that \(a\) is a constant less than 90
3. find the value of \(k\).

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1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_376_764_276_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_371_766_959_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
   (a) another solution of the equation \(2 \sin x = k\),
   (b) one solution of the equation \(2 \sin x = - k\).
3. Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).