Edexcel C3 — Question 9

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
TopicExponential Equations & Modelling
9. (i) Find the exact solutions to the equations
  1. \(\ln ( 3 x - 7 ) = 5\),
  2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
    (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } ,
    \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
  3. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
  4. Find fg and state its range. \section*{6665/01} \section*{Advanced} \section*{Tuesday 15 June 2010 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit.
    1. (a) Show that
    $$\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta$$
  5. Hence find, for \(- 180 ^ { \circ } \leq \theta < 180 ^ { \circ }\), all the solutions of $$\frac { 2 \sin 2 \theta } { 1 + \cos 2 \theta } = 1 .$$ Give your answers to 1 decimal place.
    2. A curve \(C\) has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point \(P\) on \(C\) has \(x\)-coordinate 2 .
    Find an equation of the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. 3 $$\mathrm { f } ( x ) = 4 \operatorname { cosec } x - 4 x + 1 , \quad \text { where } x \text { is in radians. }$$
  6. Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) in the interval [1.2, 1.3].
  7. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 1 } { \sin x } + \frac { 1 } { 4 }$$
  8. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { \sin x _ { n } } + \frac { 1 } { 4 } , \quad x _ { 0 } = 1.25 ,$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  9. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, verify that \(\alpha = 1.291\) correct to 3 decimal places.
    4. The function \(f\) is defined by $$f : x | \rightarrow | 2 x - 5 \mid , \quad x \in \mathbb { R }$$
  10. Sketch the graph with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
  11. Solve \(\mathrm { f } ( x ) = 15 + x\). The function g is defined by $$\mathrm { g } : x \mid \rightarrow x ^ { 2 } - 4 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leq x \leq 5 .$$
  12. Find \(\mathrm { fg } ( 2 )\).
  13. Find the range of g .
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-28_455_730_292_1804} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) with the equation \(y = \left( 2 x ^ { 2 } - 5 x + 2 \right) \mathrm { e } ^ { - x }\).
  14. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
  15. Show that \(C\) crosses the \(x\)-axis at \(x = 2\) and find the \(x\)-coordinate of the other point where \(C\) crosses the \(x\)-axis.
  16. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  17. Hence find the exact coordinates of the turning points of \(C\).
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-29_515_565_278_445} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve with the equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
    The curve has a turning point at \(A ( 3 , - 4 )\) and also passes through the point \(( 0,5 )\).
  18. Write down the coordinates of the point to which \(A\) is transformed on the curve with equation
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
  19. Sketch the curve with equation \(y = \mathrm { f } ( | x | )\). On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the \(y\)-axis. The curve with equation \(y = \mathrm { f } ( x )\) is a translation of the curve with equation \(y = x ^ { 2 }\).
  20. Find \(\mathrm { f } ( x )\).
  21. Explain why the function f does not have an inverse.
    7. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 4 decimal places.
    1. Find the maximum value of \(2 \sin \theta - 1.5 \cos \theta\).
    2. Find the value of \(\theta\), for \(0 \leq \theta < \pi\), at which this maximum occurs. Tom models the height of sea water, \(H\) metres, on a particular day by the equation $$H = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leq t < 12 ,$$ where \(t\) hours is the number of hours after midday.
  23. Calculate the maximum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this maximum occurs.
  24. Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.
    (6)
    8. (a) Simplify fully $$\frac { 2 x ^ { 2 } + 9 x - 5 } { x ^ { 2 } + 2 x - 15 }$$ Given that $$\left( 2 x ^ { 2 } + 9 x - 5 \right) = 1 + \ln \left( x ^ { 2 } + 2 x - 15 \right) , \quad x \neq - 5 ,$$
  25. find \(x\) in terms of e. \section*{6665/01} \section*{Advanced Level} Mathematical Formulae (Pink) \section*{Monday 24 January 2011 - Morning} Time: 1 hour 30 minutes Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation or integration, or have retrievable mathematical formulae stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit.
    1. (a) Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
    2. Hence write down the minimum value of \(7 \cos x - 24 \sin x\).
    3. Solve, for \(0 \leq x < 2 \pi\), the equation
    $$7 \cos x - 24 \sin x = 10$$ giving your answers to 2 decimal places.
    2. (a) Express $$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form. Given that $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1 ,$$
  26. show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$
  27. Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).
    3. Find all the solutions of $$2 \cos 2 \theta = 1 - 2 \sin \theta$$ in the interval \(0 \leq \theta < 360 ^ { \circ }\).
    4. Joan brings a cup of hot tea into a room and places the cup on a table. At time \(t\) minutes after Joan places the cup on the table, the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are positive constants.
    Given that the initial temperature of the tea was \(90 ^ { \circ } \mathrm { C }\),
  28. find the value of \(A\). The tea takes 5 minutes to decrease in temperature from \(90 ^ { \circ } \mathrm { C }\) to \(55 ^ { \circ } \mathrm { C }\).
  29. Show that \(k = \frac { 1 } { 5 } \ln 2\).
  30. Find the rate at which the temperature of the tea is decreasing at the instant when \(t = 10\). Give your answer, in \({ } ^ { \circ } \mathrm { C }\) per minute, to 3 decimal places.
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-31_417_705_294_1813} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 1.
  31. Write down the coordinates of \(A\) and the coordinates of \(B\).
  32. Find \(\mathrm { f } ^ { \prime } ( x )\).
  33. Show that the \(x\)-coordinate of \(Q\) lies between 3.5 and 3.6
  34. Show that the \(x\)-coordinate of \(Q\) is the solution of $$x = \frac { 8 } { 1 + \ln x }$$ To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } }$$ is used.
  35. Taking \(x _ { 0 } = 3.55\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give your answers to 3 decimal places.
    6. The function f is defined by $$\mathrm { f } : x \mapsto \frac { 3 - 2 x } { x - 5 } , \quad x \in \mathbb { R } , \quad x \neq 5 .$$
  36. Find \(\mathrm { f } ^ { - 1 } ( x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-32_585_714_482_392} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The function g has domain \(- 1 \leq x \leq 8\), and is linear from ( \(- 1 , - 9\) ) to ( 2,0 ) and from ( 2,0 ) to \(( 8,4 )\). Figure 2 shows a sketch of the graph of \(y = \mathrm { g } ( x )\).
  37. Write down the range of g.
  38. Find \(\mathrm { gg } ( 2 )\).
  39. Find \(\mathrm { fg } ( 8 )\).
  40. On separate diagrams, sketch the graph with equation
    1. \(y = | \mathrm { g } ( x ) |\),
    2. \(y = g ^ { - 1 } ( x )\). Show on each sketch the coordinates of each point at which the graph meets or cuts the axes.
  41. State the domain of the inverse function \(\mathrm { g } ^ { - 1 }\).
    7. The curve \(C\) has equation $$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$
  42. Show that $$\frac { d y } { d x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
  43. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.
    8. Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \cos x ) = - \sin x ,$$
  44. show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \sec x \tan x\). Given that $$x = \sec 2 y$$
  45. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  46. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). \section*{END} \section*{Advanced Level} Mathematical Formulae (Pink) \section*{Thursday 16 June 2011 - Afternoon} Time: 1 hour 30 minutes Items included with question papers
    Nil Calculators may NOT be used in this examination. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    There are 8 questions in this question paper. The total mark for this paper is 75. You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit.
    1. Differentiate with respect to \(x\)
    2. \(\ln \left( x ^ { 2 } + 3 x + 5 \right)\),
    3. \(\frac { \cos x } { x ^ { 2 } }\).
    $$\mathrm { f } ( x ) = 2 \sin \left( x ^ { 2 } \right) + x - 2 , \quad 0 \leq x < 2 \pi .$$
  47. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 0.75\) and \(x = 0.85\). The equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = [ \arcsin ( 1 - 0.5 x ) ] ^ { \frac { 1 } { 2 } }\).
  48. Use the iterative formula $$x _ { n + 1 } = \left[ \arcsin \left( 1 - 0.5 x _ { n } \right) \right] ^ { \frac { 1 } { 2 } } , \quad x _ { 0 } = 0.8 ,$$ to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 5 decimal places.
  49. Show that \(\alpha = 0.80157\) is correct to 5 decimal places.
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-34_291_442_283_529} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
    The graph consists of two line segments that meet at the point \(R ( 4 , - 3 )\), as shown in Figure 1.
    Sketch, on separate diagrams, the graphs of
  50. \(y = 2 \mathrm { f } ( x + 4 )\),
  51. \(y = | \mathrm { f } ( - x ) |\). On each diagram, show the coordinates of the point corresponding to \(R\).
    4. The function f is defined by $$f : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$
  52. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  53. Find the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R } .$$
  54. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  55. Find the range of fg.
    5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t } ,$$ where \(k\) and \(p\) are positive constants.
    When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.
  56. Write down the value of \(p\).
  57. Show that \(k = \frac { 1 } { 4 } \ln 3\).
  58. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).
    6. (a) Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , \quad n \in \mathbb { Z } .$$
  59. Hence, or otherwise,
    1. show that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\),
    2. solve, for \(0 < x < 360 ^ { \circ }\), $$\operatorname { cosec } 4 x - \cot 4 x = 1$$ 7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$
  60. Show that $$\mathrm { f } ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The point \(P \left( - 1 , - \frac { 5 } { 2 } \right)\) lies on \(C\).
  61. Find an equation of the normal to \(C\) at \(P\).
    8. (a) Express \(2 \cos 3 x - 3 \sin 3 x\) in the form \(R \cos ( 3 x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your answers to 3 significant figures. $$f ( x ) = e ^ { 2 x } \cos 3 x$$
  62. Show that \(\mathrm { f } ^ { \prime } ( x )\) can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = \operatorname { Re } ^ { 2 x } \cos ( 3 x + \alpha ) ,$$ where \(R\) and \(\alpha\) are the constants found in part (a).
  63. Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm { f } ( x )\) has a turning point. \section*{Advanced Level} \section*{Monday 23 January 2012 - Morning} Time: 1 hour 30 minutes \section*{Materials required for examination
    Mathematical Formulae (Pink)} Items included with question papers
    Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation or integration, or have retrievable mathematical formulae stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit.
    1. Differentiate with respect to \(x\), giving your answer in its simplest form,
    2. \(x ^ { 2 } \ln ( 3 x )\),
    3. \(\frac { \sin 4 x } { x ^ { 3 } }\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-36_464_751_603_335} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the graph of equation \(y = \mathrm { f } ( x )\).
    The points \(P ( - 3,0 )\) and \(Q ( 2 , - 4 )\) are stationary points on the graph.
    Sketch, on separate diagrams, the graphs of
  64. \(y = 3 \mathrm { f } ( x + 2 )\),
  65. \(y = | f ( x ) |\).
    3. The area, \(A \mathrm {~mm} ^ { 2 }\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geq 0$$
  66. Write down the area of the culture at midday.
  67. Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.
    4. The point \(P\) is the point on the curve \(x = 2 \tan \left( y + \frac { \pi } { 12 } \right)\) with \(y\)-coordinate \(\frac { \pi } { 4 }\). Find an equation of the normal to the curve at \(P\).
    5. Solve, for \(0 \leq \theta \leq 180 ^ { \circ }\), $$2 \cot ^ { 2 } 3 \theta = 7 \operatorname { cosec } 3 \theta - 5$$ Give your answers in degrees to 1 decimal place.
    6. $$f ( x ) = x ^ { 2 } - 3 x + 2 \cos \left( \frac { 1 } { 2 } x \right) , \quad 0 \leq x \leq \pi$$
  68. Show that the equation \(\mathrm { f } ( x ) = 0\) has a solution in the interval \(0.8 < x < 0.9\). The curve with equation \(y = \mathrm { f } ( x )\) has a minimum point \(P\).
  69. Show that the \(x\)-coordinate of \(P\) is the solution of the equation $$x = \frac { 3 + \sin \left( \frac { 1 } { 2 } x \right) } { 2 }$$
  70. Using the iteration formula $$x _ { n + 1 } = \frac { 3 + \sin \left( \frac { 1 } { 2 } x _ { n } \right) } { 2 } , \quad x _ { 0 } = 2$$ find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  71. By choosing a suitable interval, show that the \(x\)-coordinate of \(P\) is 1.9078 correct to 4 decimal places.
    7. The function \(f\) is defined by $$f : x \mapsto \frac { 3 ( x + 1 ) } { 2 x ^ { 2 } + 7 x - 4 } - \frac { 1 } { x + 4 } , \quad x \in \mathbb { R } , x > \frac { 1 } { 2 }$$
  72. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 x - 1 }\).
  73. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  74. Find the domain of \(\mathrm { f } ^ { - 1 }\). $$g ( x ) = \ln ( x + 1 )$$
  75. Find the solution of \(\mathrm { fg } ( x ) = \frac { 1 } { 7 }\), giving your answer in terms of e .
    8. (a) Starting from the formulae for \(\sin ( A + B )\) and \(\cos ( A + B )\), prove that $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$
  76. Deduce that $$\tan \left( \theta + \frac { \pi } { 6 } \right) = \frac { 1 + \sqrt { } 3 \tan \theta } { \sqrt { } 3 - \tan \theta }$$
  77. Hence, or otherwise, solve, for \(0 \leq \theta \leq \pi\), $$1 + \sqrt { 3 } \tan \theta = ( \sqrt { 3 } - \tan \theta ) \tan ( \pi - \theta )$$ Give your answers as multiples of \(\pi\). \section*{END}
    1. Express
    $$\frac { 2 ( 3 x + 2 ) } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ Paper Reference(s)
    6665/01 Mathematical Formulae (Pink) \section*{Advanced Level} \section*{Thursday 14 June 2012 - Morning} \section*{Nil} Time: 1 hour 30 minutes Candidates may use any calculator allowed by the regulations of the Joint
    Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit. \section*{as a single fraction in its simplest form.} $$\mathrm { f } ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$
  78. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) } , \quad x \neq - 3$$ The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2 .
  79. Use the iteration formula $$x _ { n + 1 } = \sqrt { \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) } , \quad n \geq 0$$ with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
  80. By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-39_314_561_278_470} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) which has equation $$y = \mathrm { e } ^ { x \sqrt { 3 } } \sin 3 x , - \frac { \pi } { 3 } \leq x \leq \frac { \pi } { 3 } .$$
  81. Find the \(x\)-coordinate of the turning point \(P\) on \(C\), for which \(x > 0\). Give your answer as a multiple of \(\pi\).
  82. Find an equation of the normal to \(C\) at the point where \(x = 0\).
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-39_312_666_278_1836} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\).
    The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
    On separate diagrams, sketch the curve with equation
  83. \(y = | \mathrm { f } ( x ) |\)
  84. \(y = \mathrm { f } ( | x | )\)
  85. \(y = 2 \mathrm { f } ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
    5. (a) Express \(4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
  86. Hence show that $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta .$$
  87. Hence or otherwise solve, for \(0 < \theta < \pi\), $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of \(\pi\).
    6. The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , & x \in \mathbb { R } ,
    \mathrm {~g} : x \mapsto \ln x , & x > 0 . \end{array}$$
  88. State the range of f.
  89. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  90. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\).
  91. Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
  92. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.
    7. (a) Differentiate with respect to \(x\),
    1. \(x ^ { \frac { 1 } { 2 } } \ln ( 3 x )\),
    2. \(\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }\), giving your answer in its simplest form.
  93. Given that \(x = 3 \tan 2 y\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
    8. \(\mathrm { f } ( x ) = 7 \cos 2 x - 24 \sin 2 x\). Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
  94. find the value of \(R\) and the value of \(\alpha\).
  95. Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for \(0 \leq x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
  96. Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a\), \(b\), and \(c\) are constants to be found.
  97. Hence, using your answers to parts (a) and (c), deduce the maximum value of \(14 \cos ^ { 2 } x - 48 \sin x \cos x\). \section*{Advanced Level} Materials required for examination
    Mathematical Formulae (Pink) Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation or integration, or have retrievable mathematical formulae stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 . Items included with question papers Nil You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit. \section*{Friday 25 January 2013 - Afternoon} Time: 1 hour 30 minutes
    1. The curve \(C\) has equation
    $$y = ( 2 x - 3 ) ^ { 5 }$$ The point \(P\) lies on \(C\) and has coordinates \(( w , - 32 )\).
    Find
  98. the value of \(w\),
  99. the equation of the tangent to \(C\) at the point \(P\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    2. \(\mathrm { g } ( x ) = \mathrm { e } ^ { x - 1 } + x - 6\)
  100. Show that the equation \(\mathrm { g } ( x ) = 0\) can be written as $$x = \ln ( 6 - x ) + 1 , \quad x < 6$$ The root of \(\mathrm { g } ( x ) = 0\) is \(\alpha\).
    The iterative formula $$x _ { n + 1 } = \ln \left( 6 - x _ { n } \right) + 1 , \quad x _ { 0 } = 2$$ is used to find an approximate value for \(\alpha\).
  101. Calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
  102. By choosing a suitable interval, show that \(\alpha = 2.307\) correct to 3 decimal places.
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-42_415_734_296_386} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
    The curve passes through the points \(Q ( 0,2 )\) and \(P ( - 3,0 )\) as shown.
  103. Find the value of ff (-3). On separate diagrams, sketch the curve with equation
  104. \(y = \mathrm { f } ^ { - 1 } ( x )\),
  105. \(y = \mathrm { f } ( | x | ) - 2\),
  106. \(y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
    4. (a) Express \(6 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
  107. $$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leq \theta \leq 2 \pi$$ Calculate
    1. the maximum value of \(\mathrm { p } ( \theta )\),
    2. the value of \(\theta\) at which the maximum occurs.
      5. (i) Differentiate with respect to \(x\)
  108. \(y = x ^ { 3 } \ln 2 x\),
  109. \(y = ( x + \sin 2 x ) ^ { 3 }\). Given that \(x = \cot y\),
    (ii) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }\).
    6. (i) Without using a calculator, find the exact value of $$\left( \sin 22.5 ^ { \circ } + \cos 22.5 ^ { \circ } \right) ^ { 2 }$$ You must show each stage of your working.
    (ii) (a) Show that \(\cos 2 \theta + \sin \theta = 1\) may be written in the form $$k \sin ^ { 2 } \theta - \sin \theta = 0 , \text { stating the value of } k .$$
  110. Hence solve, for \(0 \leq \theta < 360 ^ { \circ }\), the equation $$\cos 2 \theta + \sin \theta = 1$$
    1. \(\quad \mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geq 0\).
    2. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\).
    3. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-43_472_808_559_351} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
  111. Calculate the range of \(\mathrm { h } ( x )\).
    8. The value of Bob's car can be calculated from the formula $$V = 17000 \mathrm { e } ^ { - 0.25 t } + 2000 \mathrm { e } ^ { - 0.5 t } + 500 .$$ where \(V\) is the value of the car in pounds ( \(\pounds\) ) and \(t\) is the age in years.
  112. Find the value of the car when \(t = 0\).
  113. Calculate the exact value of \(t\) when \(V = 9500\).
  114. Find the rate at which the value of the car is decreasing at the instant when \(t = 8\). Give your answer in pounds per year to the nearest pound. Mathematical Formulae (Pink) Nil \section*{6665/01R Edexcel GCE} \section*{Core Mathematics C3 (R)} Advanced Subsidiary \section*{Thursday 13 June 2013 - Morning} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. This paper is strictly for students outside the UK. In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
    Answer ALL the questions.
    You must write your answer for each question in the space following the question.
    When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 .
    There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.
    1. Express
    $$\frac { 3 x + 5 } { x ^ { 2 } + x - 12 } - \frac { 2 } { x - 3 }$$ as a single fraction in its simplest form.
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-44_323_723_571_388} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x > 0\), where f is an increasing function of \(x\). The curve crosses the \(x\)-axis at the point \(( 1,0 )\) and the line \(x = 0\) is an asymptote to the curve. On separate diagrams, sketch the curve with equation
  115. \(y = \mathrm { f } ( 2 x ) , x > 0\)
  116. \(y = | \mathrm { f } ( x ) | , \quad x > 0\) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the \(x\)-axis.
    3. $$\mathrm { f } ( x ) = 7 \cos x + \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - a )\), where \(\mathrm { R } > 0\) and \(0 < a < 90 ^ { \circ }\),
  117. find the exact value of \(R\) and the value of \(a\) to one decimal place.
  118. Hence solve the equation $$7 \cos x + \sin x = 5$$ for \(0 \leq x < 360 ^ { \circ }\), giving your answers to one decimal place.
  119. State the values of \(k\) for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval \(0 \leq x < 360 ^ { \circ }\).
    4. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 | x | + 3 , & x \in
    \mathrm {~g} : x \mapsto 3 - 4 x , & x \in \end{array}$$
  120. State the range of f.
  121. Find \(\mathrm { fg } ( 1 )\).
  122. Find \(\mathrm { g } ^ { - 1 }\), the inverse function of g .
  123. Solve the equation $$\operatorname { gg } ( x ) + [ \operatorname { g } ( x ) ] ^ { 2 } = 0$$
    1. (a) Differentiate
    $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
  124. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
  125. Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.
    6. (i) Use an appropriate double angle formula to show that $$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
    (ii) Solve, for \(0 \leq \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-45_319_574_264_1882} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) e ^ { x ^ { 2 } }$$ The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2.
  126. Calculate the \(x\)-coordinate of \(A\) and the \(x\)-coordinate of \(B\), giving your answers to 3 decimal places.
  127. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve has a minimum turning point \(P\) as shown in Figure 2.
  128. Show that the \(x\)-coordinate of \(P\) is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
  129. Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places. The \(x\)-coordinate of \(P\) is \(\alpha\).
  130. By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-46_405_807_196_347} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 e ^ { - k t } } , \quad \mathrm { t } \geq 0$$ where \(k\) is a positive constant.
    The graph of \(P\) against \(t\) is shown in Figure 3.
    Use the given equation to
  131. find the population at the start of the study,
  132. find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,
  133. calculate the value of \(k\) to 3 decimal places. Using this value for \(k\),
  134. find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
  135. Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study. Mathematical Formulae (Pink) Items included with question papers Nil \section*{Thursday 13 June 2013 - Morning} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
    Check that you have the correct question paper.
    Answer ALL the questions.
    You must write your answer for each question in the space following the question.
    When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 8 questions in this question paper. The total mark for this paper is 75 .
    There are 32 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.
    1. Given that
    $$\frac { 3 x ^ { 4 } - 2 x ^ { 3 } - 5 x ^ { 2 } - 4 } { x ^ { 2 } - 4 } \equiv a x ^ { 2 } + b x + c + \frac { d x + e } { x ^ { 2 } - 4 } , \quad x \neq \pm 2$$ find the values of the constants \(a , b , c , d\) and \(e\).
    2. Given that $$\mathrm { f } ( x ) = \ln x , \quad x > 0$$ sketch on separate axes the graphs of
    (i) \(y = \mathrm { f } ( x )\),
    (ii) \(y = | \mathrm { f } ( x ) |\),
    (iii) \(y = - \mathrm { f } ( x - 4 )\). Show, on each diagram, the point where the graph meets or crosses the \(x\)-axis.
    In each case, state the equation of the asymptote.
    3. Given that $$2 \cos ( x + 50 ) ^ { \circ } = \sin ( x + 40 ) ^ { \circ }$$
  136. Show, without using a calculator, that $$\tan x ^ { \circ } = \frac { 1 } { 3 } \tan 40 ^ { \circ }$$
  137. Hence solve, for \(0 \leq \theta < 360\), $$2 \cos ( 2 \theta + 50 ) ^ { \circ } = \sin ( 2 \theta + 40 ) ^ { \circ }$$ giving your answers to 1 decimal place. $$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$
  138. Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
  139. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
  140. Starting with \(x _ { 0 } = 0.5\), use the iteration formula $$x _ { n + 1 } = \frac { 4 } { 5 } e ^ { - x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  141. Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.
    5. Given that $$x = \sec ^ { 2 } 3 y , \quad 0 < y < \frac { \pi } { 6 }$$
  142. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  143. Hence show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 x ( x - 1 ) ^ { \frac { 1 } { 2 } } }$$
  144. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\). Give your answer in its simplest form.
    6. Find algebraically the exact solutions to the equations
  145. \(\ln ( 4 - 2 x ) + \ln ( 9 - 3 x ) = 2 \ln ( x + 1 ) , \quad - 1 < x < 2\),
  146. \(2 ^ { x } \mathrm { e } ^ { 3 x + 1 } = 10\). Give your answer to (b) in the form \(\frac { a + \ln b } { c + \ln d }\) where \(a , b , c\) and \(d\) are integers.
    7. The function f has domain \(- 2 \leq x \leq 6\) and is linear from \(( - 2,10 )\) to \(( 2,0 )\) and from \(( 2,0 )\) to (6, 4). A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-48_604_645_317_1845} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  147. Write down the range of f .
  148. Find \(\mathrm { ff } ( 0 )\). The function g is defined by $$\mathrm { g } : x \rightarrow \frac { 4 + 3 x } { 5 - x } , \quad x \in \mathbb { R } , \quad x \neq 5$$
  149. Find \(\mathrm { g } ^ { - 1 } ( x )\).
  150. Solve the equation \(\operatorname { gf } ( x ) = 16\).
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-49_302_809_251_347} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Kate crosses a road, of constant width 7 m , in order to take a photograph of a marathon runner, John, approaching at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Kate is 24 m ahead of John when she starts to cross the road from the fixed point \(A\).
    John passes her as she reaches the other side of the road at a variable point \(B\), as shown in Figure 2. Kate's speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she moves in a straight line, which makes an angle \(\theta\), \(0 < \theta < 150 ^ { \circ }\), with the edge of the road, as shown in Figure 2. You may assume that \(V\) is given by the formula $$V = \frac { 21 } { 24 \sin \theta + 7 \cos \theta } , \quad 0 < \theta < 150 ^ { \circ }$$
  151. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants and where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places. Given that \(\theta\) varies,
  152. find the minimum value of \(V\). Given that Kate's speed has the value found in part (b),
  153. find the distance \(A B\). Given instead that Kate's speed is \(1.68 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  154. find the two possible values of the angle \(\theta\), given that \(0 < \theta < 150 ^ { \circ }\). Mathematical Formulae (Pink) Nil Paper Reference(s)
    6665A/01 \section*{Pearson Edexcel International Advanced Level} \section*{Advanced} \section*{Monday 27 January 2014 - Morning} Time: 1 hour 30 minutes Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
    • Use black ink or ball-point pen.
    • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used.
    • Fill in the boxes at the top of this page with your name, centre number and candidate number.
    • Answer all questions and ensure that your answers to parts of questions are clearly labelled.
    • Answer the questions in the spaces provided
    • there may be more space than you need.
    • You should show sufficient working to make your methods clear. Answers without working may not gain full credit.
    • When a calculator is used, the answer should be given to an appropriate degree of accuracy.
    \section*{Information}
    • The total mark for this paper is 75 .
    • The marks for each question are shown in brackets
    • use this as a guide as to how much time to spend on each question.
    \section*{Advice}
    • Read each question carefully before you start to answer it.
    • Try to answer every question.
    • Check your answers if you have time at the end.
    1. $$\mathrm { f } ( x ) = \sec x + 3 x - 2 , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
  155. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ 0.2,0.4 ]\).
  156. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x }$$ The solution of \(\mathrm { f } ( x ) = 0\) is \(\alpha\), where \(\alpha = 0.3\) to 1 decimal place.
  157. Starting with \(x _ { 0 } = 0.3\), use the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  158. State the value of \(\alpha\) correct to 3 decimal places.
    2. $$f ( x ) = \frac { 15 } { 3 x + 4 } - \frac { 2 x } { x - 1 } + \frac { 14 } { ( 3 x + 4 ) ( x - 1 ) } , \quad x > 1$$
  159. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
  160. Hence, or otherwise, find \(\mathrm { f } ^ { \prime } ( x )\), giving your answer as a single fraction in its simplest form.
    3. (a) By writing \(\operatorname { cosec } x\) as \(\frac { 1 } { \sin x }\), show that $$\frac { \mathrm { d } ( \operatorname { cosec } x ) } { \mathrm { d } x } = - \operatorname { cosec } x \cot x$$ Given that \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , \quad 0 < x < \frac { \pi } { 2 }\),
  161. find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , \quad 0 < x < \frac { \pi } { 2 }\), has a single turning point.
  162. Show that the \(x\)-coordinate of this turning point is at \(x = \frac { 1 } { 2 } \arctan k\) where the value of the constant \(k\) should be found.
    4. A pot of coffee is delivered to a meeting room at 11 am . At a time \(t\) minutes after 11 am the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the coffee in the pot is given by the equation $$\theta = A + 60 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants.
    Given also that the temperature of the coffee at 11 am is \(85 ^ { \circ } \mathrm { C }\) and that 15 minutes later it is \(58 ^ { \circ } \mathrm { C }\),
  163. find the value of A .
  164. Show that \(k = \frac { 1 } { 15 } \ln \left( \frac { 20 } { 11 } \right)\).
  165. Find, to the nearest minute, the time at which the temperature of the coffee reaches \(50 ^ { \circ } \mathrm { C }\).
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-51_426_679_221_411} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The curve shown in Figure 1 has equation $$x = 3 \sin y + 3 \cos y , \quad - \frac { \pi } { 4 } < y < \frac { \pi } { 4 }$$
  166. Express the equation of the curve in the form $$x = R \sin ( y + \alpha ) , \text { where } R \text { and } \alpha \text { are constants, } R > 0 \text { and } 0 < \alpha < \frac { \pi } { 2 }$$
  167. Find the coordinates of the point on the curve where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(\frac { 1 } { 2 }\). Give your answers to 3 decimal places.
    6. Given that \(a\) and \(b\) are constants and that \(0 < a < b\),
  168. on separate diagrams, sketch the graph with equation
    1. \(y = | 2 x + a |\),
    2. \(y = | 2 x + a | - b\). Show on each sketch the coordinates of each point at which the graph crosses or meets the axes.
  169. Solve, for \(x\), the equation $$| 2 x + a | - b = \frac { 1 } { 3 } x$$ giving any answers in terms of \(a\) and \(b\).
    7. (i) (a) Prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ (You may use the double angle formulae and the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B )\)
  170. Hence solve the equation $$2 \cos 3 \theta + \cos 2 \theta + 1 = 0$$ giving answers in the interval \(0 < \theta < \pi\).
    Solutions based entirely on graphical or numerical methods are not acceptable.
    (ii) Given that \(\theta = \arcsin x\) and that \(0 < \theta < \frac { \pi } { 2 }\), show that $$\cot \theta = \frac { \sqrt { \left( 1 - x ^ { 2 } \right) } } { x } , \quad 0 < x < 1$$
    1. The function f is defined by
    $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in$$
  171. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\), and give its domain.
  172. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
  173. Find the value of \(k\). \section*{Materials required for examination
    Mathematical Formulae (Pink)} Nil \section*{6665/01R
    Edexcel GCE} \section*{Core Mathematics C3 (R)} \section*{Thursday 13 June 2014 - Morning} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. This paper is strictly for students outside the UK. In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.
    Answer ALL the questions.
    You must write your answer for each question in the space following the question.
    When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 7 questions in this question paper. The total mark for this paper is 75 .
    There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit.
    1. Express
    $$\frac { 3 } { 2 x + 3 } - \frac { 1 } { 2 x - 3 } + \frac { 6 } { 4 x ^ { 2 } - 9 }$$ as a single fraction in its simplest form.
    2. A curve \(C\) has equation \(y = \mathrm { e } ^ { 4 x } + x ^ { 4 } + 8 x + 5\).
  174. Show that the \(x\) coordinate of any turning point of \(C\) satisfies the equation $$x ^ { 3 } = - 2 - \mathrm { e } ^ { 4 x }$$
  175. On a pair of axes, sketch, on a single diagram, the curves with equations
    1. \(y = x ^ { 3 }\),
    2. \(y = - 2 - e ^ { 4 x }\) On your diagram give the coordinates of the points where each curve crosses the \(y\)-axis and state the equation of any asymptotes.
  176. Explain how your diagram illustrates that the equation \(x ^ { 3 } = - 2 - e ^ { 4 x }\) has only one root. The iteration formula $$x _ { n + 1 } = \left( - 2 - e ^ { 4 x _ { n } } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = - 1$$ can be used to find an approximate value for this root.
  177. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 5 decimal places.
  178. Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve \(C\).
    3. (i) (a) Show that \(2 \tan x - \cot x = 5 \operatorname { cosec } x\) may be written in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ stating the values of the constants \(a , b\) and \(c\).
  179. Hence solve, for \(0 \leq x < 2 \pi\), the equation $$2 \tan x - \cot x = 5 \operatorname { cosec } x$$ giving your answers to 3 significant figures.
    (ii) Show that $$\tan \theta + \cot \theta \equiv \lambda \operatorname { cosec } 2 \theta , \quad \theta = \frac { n \pi } { 2 } , \quad n \in$$ stating the value of the constant \(\lambda\).
    4. (i) Given that $$x = \sec ^ { 2 } 2 y , \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 x \sqrt { ( x - 1 ) } }$$ (ii) Given that
    find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form.
    (iii) Given that $$f ( x ) = \frac { 3 \cos x } { ( x + 1 ) ^ { \frac { 1 } { 3 } } } , \quad x \neq - 1$$ show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x ) } { ( x + 1 ) ^ { \frac { 4 } { 3 } } } , \quad x \neq - 1$$ where \(\mathrm { g } ( x )\) is an expression to be found.
    (ii) Given that $$y = \left( x ^ { 2 } + x ^ { 3 } \right) \ln 2 x$$ \(x = \frac { \mathrm { e } } { 2 }\), giving your answer in its simplest form. \begin{displayquote} (iii \end{displayquote} Given that
    show that $$\begin{aligned} & \mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { g } ( x } { ( x + 1 }
    & \text { on to be found. } \end{aligned}$$ 5. (a) Sketch the graph with equation $$y = | 4 x - 3 |$$ stating the coordinates of any points where the graph cuts or meets the axes. Find the complete set of values of \(x\) for which
  180. $$| 4 x - 3 | > 2 - 2 x$$
  181. $$| 4 x - 3 | > \frac { 3 } { 2 } - 2 x$$
    1. The function f is defined by
    $$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in , \quad k \text { is a positive constant. }$$
  182. State the range of f .
  183. Find \(\mathrm { f } ^ { - 1 }\) and state its domain. The function g is defined by $$\mathrm { g } : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
  184. Solve the equation $$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$ giving your answer in its simplest form.
  185. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  186. Find, in terms of the constant \(k\), the solution of the equation $$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$ 7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-55_314_585_255_459} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leq x \leq 2 \pi\).
  187. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
  188. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leq t \leq 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
  189. the maximum and minimum values of \(H\) predicted by the model,
  190. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
    [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] Mathematical Formulae (Pink) Nil \section*{Advanced} \section*{Monday 16 June 2014 - Morning} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
    Check that you have the correct question paper.
    Answer ALL the questions.
    You must write your answer for each question in the space following the question.
    When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    The marks for the parts of questions are shown in round brackets, e.g. (2).
    There are 9 questions in this question paper. The total mark for this paper is 75 .
    There are 32 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.
    1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
    $$\mathrm { f } ( x ) = \frac { 4 x + 1 } { x - 2 } , \quad x > 2$$
  191. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { - 9 } { ( x - 2 ) ^ { 2 } }$$ Given that \(P\) is a point on \(C\) such that \(\mathrm { f } ^ { \prime } ( x ) = - 1\),
  192. find the coordinates of \(P\).
    2. Find the exact solutions, in their simplest form, to the equations
  193. \(2 \ln ( 2 x + 1 ) - 10 = 0\)
  194. \(3 ^ { x } \mathrm { e } ^ { 4 x } = \mathrm { e } ^ { 7 }\)
    3. The curve \(C\) has equation \(x = 8 y \tan 2 y\). The point \(P\) has coordinates \(\left( \pi , \frac { \pi } { 8 } \right)\).
  195. Verify that \(P\) lies on \(C\).
  196. Find the equation of the tangent to \(C\) at \(P\) in the form \(a y = x + b\), where the constants \(a\) and \(b\) are to be found in terms of \(\pi\).
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-56_442_519_255_1905} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in\).
    The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\).
    The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\).
    Sketch, on separate diagrams, the graphs of
  197. \(y = | \mathrm { f } ( x ) |\)
  198. \(y = 2 \mathrm { f } ( - \mathrm { x } ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
    Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
  199. state the value of \(a\) and the value of \(b\).
    5. \(\operatorname { g } ( x ) = \frac { x } { x + 3 } + \frac { 3 ( 2 x + 1 ) } { x ^ { 2 } + x - 6 } , \quad x > 3\)
  200. Show that \(\mathrm { g } ( x ) = \frac { x + 1 } { x - 2 } , x > 3\)
  201. Find the range of g .
  202. Find the exact value of \(a\) for which \(\mathrm { g } ( a ) = \mathrm { g } ^ { - 1 } ( a )\).
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-57_300_773_699_367} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$ The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).
  203. Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2.
  204. Show that the \(x\) coordinate of \(R\) is a solution of the equation $$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
  205. find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.
    7. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x = \cot x , \quad x \neq 90 n ^ { \circ } , \quad n \in$$
  206. Hence, or otherwise, solve, for \(0 \leq \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { } 3$$ You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geq 0 , \quad t \in$$
  207. Calculate the number of primroses at the start of the study.
  208. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
  209. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
  210. Explain why the population of primroses can never be 270 .
    9. (a) Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
    Give the value of \(\alpha\) to 3 decimal places.
    (3) $$\mathrm { H } ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$ Find
    1. the maximum value of \(\mathrm { H } ( \theta )\),
    2. the smallest value of \(\theta\), for \(0 \leq \theta \leq \pi\), at which this maximum value occurs.
      (3) Find
    1. the minimum value of \(\mathrm { H } ( \theta )\),
    2. the largest value of \(\theta\), for \(0 \leq \theta \leq \pi\), at which this minimum value occurs.
      (3)
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