8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } ,
\mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$

1. Prove that the composite function gf is $$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
2. In the space provided on page 19, sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
3. Write down the range of gf.
4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.
   Turn over
   1. (a) By writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that
   $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
5. Given that \(\sin \theta = \frac { \sqrt { } 3 } { 4 }\), find the exact value of \(\sin 3 \theta\).
   2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$
6. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
7. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
8. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).
   3. The curve \(C\) has equation $$x = 2 \sin y .$$
9. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
10. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
11. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.
    4. (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
    (ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).
    5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-045_865_926_301_516}
    \end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$
12. Express the equation of the curve in the form \(y = R \sin ( x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
13. Find the values of \(x , 0 \leqslant x < 2 \pi\), for which \(y = 1\).
    1. The function \(f\) is defined by
    $$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$
14. Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of \(\mathrm { f } ^ { - 1 }\).
15. Write down the range of \(\mathrm { f } ^ { - 1 }\).
16. In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes. The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\). The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for \(k\).
17. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
18. Find the value of \(k\) to 3 decimal places. 7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$
19. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ - 2 , - 1 ]\).
20. Find the coordinates of the turning point on the graph of \(y = \mathrm { f } ( x )\).
21. Given that \(\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)\), find the values of the constants, \(a , b\) and \(c\).
22. In the space provided on page 21, sketch the graph of \(y = \mathrm { f } ( x )\).
23. Hence sketch the graph of \(y = | \mathrm { f } ( x ) |\).
    
    1. (i) Prove that
    $$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$
24. express arcsin \(x\) in terms of \(y\).
25. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).
    Turn over
    1. Find the exact solutions to the equations
    2. \(\ln x + \ln 3 = \ln 6\),
    3. \(\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4\).
       
    $$f ( x ) = \frac { 2 x + 3 } { x + 2 } - \frac { 9 + 2 x } { 2 x ^ { 2 } + 3 x - 2 } , \quad x > \frac { 1 } { 2 }$$
26. Show that \(\mathrm { f } ( x ) = \frac { 4 x - 6 } { 2 x - 1 }\).
27. Hence, or otherwise, find \(\mathrm { f } ^ { \prime } ( x )\) in its simplest form.
    3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$
28. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
29. Hence find the coordinates of the turning points of \(C\).
30. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
31. Determine the nature of each turning point of the curve \(C\).
    4. $$f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } - 1$$
32. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt { } \left( \frac { 1 } { 3 - x } \right)$$
33. Starting with \(x _ { 1 } = 0.6\), use the iteration $$\left. x _ { n + 1 } = \sqrt { ( } \frac { 1 } { 3 - x _ { n } } \right)$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving all your answers to 4 decimal places.
34. Show that \(x = 0.653\) is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
    5. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 }
    \mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$
35. Find the exact value of fg(4).
36. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.
37. Sketch the graph of \(y = | \mathrm { g } ( x ) |\). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the \(y\)-axis.
38. Find the exact values of \(x\) for which \(\left| \frac { 2 } { x - 3 } \right| = 3\).
    1. (a) Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
    2. Hence find the greatest value of \(( 3 \sin x + 2 \cos x ) ^ { 4 }\).
    3. Solve, for \(0 < x < 2 \pi\), the equation
    $$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.
    
    1. (a) Prove that
    $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$
39. On the axes on page 20, sketch the graph of \(y = 2 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
40. Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 3 ,$$ giving your answers to 1 decimal place.
    \includegraphics[max width=\textwidth, alt={}, center]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-063_899_1253_315_347}
    8. The amount of a certain type of drug in the bloodstream \(t\) hours after it has been taken is given by the formula $$x = D \mathrm { e } ^ { - \frac { 1 } { 8 } t } ,$$ where \(x\) is the amount of the drug in the bloodstream in milligrams and \(D\) is the dose given in milligrams. A dose of 10 mg of the drug is given.
41. Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours.
42. Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places. No more doses of the drug are given. At time \(T\) hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg .
43. Find the value of \(T\). \end{table} Paper Reference(s) \section*{6665/01} \section*{Edexcel GCE } \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-079_97_309_495_1636}
    \end{figure} $$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .
44. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
45. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.
    2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),
46. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
47. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
48. 1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
    2. Find the smallest positive value of \(x\) for which this maximum value occurs.
       3. \begin{figure}[h]
       \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-083_623_977_207_479} \captionsetup{labelformat=empty} \caption{Figure 1}
       \end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
       The graph consists of two line segments that meet at the point \(P\).
       The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of
49. \(y = | f ( x ) |\),
50. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
51. find the coordinates of the points \(P , Q\) and \(R\),
52. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).
    4. The function \(f\) is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$
53. Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
54. Find the range of f.
55. Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function. The function \(g\) is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R } .$$
56. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).
    5. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
57. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.
    6. (a) Differentiate with respect to \(x\),
    1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
    2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
58. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
59. Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).
    7. $$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$
60. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between \(x = 1.4\) and \(x = 1.45\)
61. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
62. Starting with \(x _ { 0 } = 1.43\), use the iteration $$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
63. By choosing a suitable interval, show that \(\alpha = 1.435\) is correct to 3 decimal places. \end{table} Turn over
    advancing learning, changing lives
    1. (a) Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(x = 2\) on the curve with equation
    $$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$
64. Differentiate \(\frac { \sin 2 x } { x ^ { 2 } }\) with respect to \(x\).
    2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$
65. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
66. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-095_767_913_246_511} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , \quad 1 < x < 9\).
    The points \(T ( 3,5 )\) and \(S ( 7,2 )\) are turning points on the graph.
    Sketch, on separate diagrams, the graphs of
67. \(y = 2 \mathrm { f } ( x ) - 4\),
68. \(y = | \mathrm { f } ( x ) |\). Indicate on each diagram the coordinates of any turning points on your sketch.
    4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.
    5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R }
    & \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } \end{aligned}$$
69. Write down the range of g.
70. Show that the composite function fg is defined by $$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
71. Write down the range of fg.
72. Solve the equation \(\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)\).
    6. (a) (i) By writing \(3 \theta = ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (ii) Hence, or otherwise, for \(0 < \theta < \frac { \pi } { 3 }\), solve $$8 \sin ^ { 3 } \theta - 6 \sin \theta + 1 = 0 .$$ Give your answers in terms of \(\pi\).
73. Using \(\sin ( \theta - \alpha ) = \sin \theta \cos \alpha - \cos \theta \sin \alpha\), or otherwise, show that $$\sin 15 ^ { \circ } = \frac { 1 } { 4 } ( \sqrt { } 6 - \sqrt { } 2 )$$ 7. $$f ( x ) = 3 x e ^ { x } - 1$$ The curve with equation \(y = \mathrm { f } ( x )\) has a turning point \(P\).
74. Find the exact coordinates of \(P\). The equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 0.25\) and \(x = 0.3\)
75. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } }$$ with \(x _ { 0 } = 0.25\) to find, to 4 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
76. By choosing a suitable interval, show that a root of \(\mathrm { f } ( x ) = 0\) is \(x = 0.2576\) correct to 4 decimal places.
    8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
77. Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
78. Calculate the minimum temperature of the warehouse as given by this model.
79. Find the value of \(t\) when this minimum temperature occurs.
    \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}
    \end{table} Paper Reference(s) \section*{6665/01} \section*{Edexcel GCE } \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-105_99_311_495_1635}
    \end{figure} $$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$ is used.
80. Taking \(x _ { 0 } = 2.5\), find the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give your answers to 3 decimal places where appropriate.
81. Show that \(\alpha = 2.359\) correct to 3 decimal places.
    2. (a) Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
82. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$
    1. Rabbits were introduced onto an island. The number of rabbits, \(P , t\) years after they were introduced is modelled by the equation
    $$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$
83. Write down the number of rabbits that were introduced to the island.
84. Find the number of years it would take for the number of rabbits to first exceed 1000.
85. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\).
86. Find \(P\) when \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).
    4. (i) Differentiate with respect to \(x\)
87. \(x ^ { 2 } \cos 3 x\)
88. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\)
    (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-111_721_1217_237_397} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
    The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation
89. \(y = | f ( x ) |\),
90. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
91. state the range of f ,
92. find \(\mathrm { f } ^ { - 1 } ( x )\),
93. write down the domain of \(\mathrm { f } ^ { - 1 }\).
    1. (a) Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that
    $$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x
    & C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$
94. Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$
95. Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
96. Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.
    7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$
97. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
98. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
99. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)
    8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
100. Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.
     \end{table} Turn over
     advancing learning, changing lives
     1. Express
     $$\frac { x + 1 } { 3 x ^ { 2 } - 3 } - \frac { 1 } { 3 x + 1 }$$ as a single fraction in its simplest form.
     2. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x - 11$$
101. Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as $$x = \sqrt { } \left( \frac { 3 x + 11 } { x + 2 } \right) , \quad x \neq - 2 .$$ The equation \(\mathrm { f } ( x ) = 0\) has one positive root \(\alpha\). The iterative formula \(x _ { n + 1 } = \sqrt { } \left( \frac { 3 x _ { n } + 11 } { x _ { n } + 2 } \right)\) is used to find an approximation to \(\alpha\).
102. Taking \(x _ { 1 } = 0\), find, to 3 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
103. Show that \(\alpha = 2.057\) correct to 3 decimal places.
     3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
104. Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.
     4. (i) Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
     (ii) Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
     5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.
     6. \begin{figure}[h]
     \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-124_380_574_269_722} \captionsetup{labelformat=empty} \caption{Figure 1}
     \end{figure} Figure 1 shows a sketch of the graph of \(y = \mathrm { f } ( x )\).
     The graph intersects the \(y\)-axis at the point \(( 0,1 )\) and the point \(A ( 2,3 )\) is the maximum turning point. Sketch, on separate axes, the graphs of
     (i) \(y = \mathrm { f } ( - x ) + 1\),
     (ii) \(y = \mathrm { f } ( x + 2 ) + 3\),
     (iii) \(y = 2 \mathrm { f } ( 2 x )\). On each sketch, show the coordinates of the point at which your graph intersects the \(y\)-axis and the coordinates of the point to which \(A\) is transformed.
     1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).
     Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
105. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
106. Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures. 8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).