Function properties and inverses

5 The function f is such that \(\mathrm { f } ( x ) = 2 \sin ^ { 2 } x - 3 \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Express \(\mathrm { f } ( x )\) in the form \(a + b \cos ^ { 2 } x\), stating the values of \(a\) and \(b\).
2. State the greatest and least values of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) + 1 = 0\).

11 The function f : \(x \mapsto 4 - 3 \sin x\) is defined for the domain \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. The function \(\mathrm { g } : x \mapsto 4 - 3 \sin x\) is defined for the domain \(\frac { 1 } { 2 } \pi \leqslant x \leqslant A\).
4. State the largest value of \(A\) for which g has an inverse.
5. For this value of \(A\), find the value of \(\mathrm { g } ^ { - 1 } ( 3 )\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

1 Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\),
2. \(\tan \theta\),
3. \(\sin ( \theta + \pi )\).

8 The function f : \(x \mapsto 5 + 3 \cos \left( \frac { 1 } { 2 } x \right)\) is defined for \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 7\), giving your answer correct to 2 decimal places.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Explain why f has an inverse.
4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

10 The function f is defined by \(\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Solve the equation \(\mathrm { f } ( x ) + 4 = 0\), giving your answer correct to 1 decimal place.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).

5 The function f is such that \(\mathrm { f } ( x ) = a - b \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), where \(a\) and \(b\) are positive constants. The maximum value of \(\mathrm { f } ( x )\) is 10 and the minimum value is - 2 .

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

1 Given that \(\cos x = p\), where \(x\) is an acute angle in degrees, find, in terms of \(p\),

1. \(\sin x\),
2. \(\tan x\),
3. \(\tan \left( 90 ^ { \circ } - x \right)\).

10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
4. State the largest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

4 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 - 3 \cos x \quad \text { for } 0 \leqslant x \leqslant 2 \pi \\ & \mathrm {~g} : x \mapsto \frac { 1 } { 2 } x \quad \text { for } 0 \leqslant x \leqslant 2 \pi \end{aligned}$$

1. Solve the equation \(\operatorname { fg } ( x ) = 1\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).

3 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).

3 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).

4. $$\mathrm { f } ( x ) = \frac { 4 } { 2 + \sin x ^ { \circ } }$$

1. State the maximum value of \(\mathrm { f } ( x )\) and the smallest positive value of \(x\) for which \(\mathrm { f } ( x )\) takes this value.
2. Solve the equation \(\mathrm { f } ( x ) = 3\) for \(0 \leq x \leq 360\), giving your answers to 1 decimal place.

4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).

3 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
[0pt] [3]

1 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).

10 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ } .$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).

8 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.

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\hline \end{tabular} \end{center} 1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ),

1. find the value of \(p\). Given also that \(q = 3\), and \(p\) has the value found in part (a),
2. find the value of the remainder.
   2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}
   \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.
3. Write down the value of \(a\) and the value of \(b\).
4. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
5. Find, to 3 significant figures, the length of \(P T\).
   3. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
6. Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
   4. The first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + p x ) ^ { n }\), are \(1 - 18 x + 36 p ^ { 2 } x ^ { 2 }\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\).
   (7 marks)
   5. Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
7. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
8. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.
   6. Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
9. \(\log _ { 2 } ( 16 x )\),
10. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
11. Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
    7. The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
12. Sketch \(C\).
13. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
14. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi\), $$\cos \left( x + \frac { \pi } { 4 } \right) = 0.5$$ giving your answers in terms of \(\pi\).

3 In this question you must show detailed reasoning. You are given that \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Explain why \(\tan 690 ^ { \circ } = - \frac { 1 } { \sqrt { 3 } }\).

3 State the interval for which \(\sin x\) is a decreasing function for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) [0pt] [2 marks]

1 It is given that \(\tan \theta ^ { \circ } = k\), where \(k\) is a constant.
Find \(\tan ( \theta + 180 ) ^ { \circ }\) Circle your answer. \(- k\) \(- \frac { 1 } { k }\) \(\frac { 1 } { k }\) \(k\)

2 It is given that \(\sin \theta = \frac { 4 } { 5 }\) and \(90 ^ { \circ } < \theta < 180 ^ { \circ }\) Find the value of \(\cos \theta\) Circle your answer.
[0pt] [1 mark] \(- \frac { 3 } { 4 }\) \(- \frac { 3 } { 5 }\) \(\frac { 3 } { 5 }\) \(\frac { 3 } { 4 }\)

2 Which one of the following equations has no real solutions?
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] $$\begin{aligned} & \cot x = 0 \\ & \ln x = 0 \\ & | x + 1 | = 0 \\ & \sec x = 0 \end{aligned}$$ □
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