Questions C3 (1200 questions)

4. (i) Prove, by counter-example, that the statement $$\text { " } \sec ( A + B ) \equiv \sec A + \sec B , \text { for all } A \text { and } B \text { " }$$ is false.
(ii) Prove that $$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$

1. The function f is given by

$$\mathrm { f } : x \mapsto \frac { x } { x ^ { 2 } - 1 } - \frac { 1 } { x + 1 } , x > 1 .$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { ( x - 1 ) ( x + 1 ) }\).
2. Find the range of f. The function g is given by $$\mathrm { g } : x \mapsto \frac { 2 } { x } , \quad x > 0 .$$
3. Solve \(\operatorname { gf } ( x ) = 70\).

6. (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the values of \(R\) and \(\alpha\) to 3 significant figures.
(b) Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24 ,$$ where \(t\) hours is the number of hours after 1200 .
(c) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(d) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).

7. The function \(f\) is defined by $$f : x \wp \rightarrow | 2 x - a | , x \in \mathbb { R }$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts the axes.
2. On a separate diagram, sketch the graph of \(y = \mathrm { f } ( 2 x )\), showing the coordinates of the points where the graph cuts the axes.
3. Given that a solution of the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\) is \(x = 4\), find the two possible values of \(a\).

8. (a) Prove that $$\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta , \theta \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Solve, giving exact answers in terms of \(\pi\), $$2 ( 1 - \cos 2 \theta ) = \tan \theta , \quad 0 < \theta < \pi$$ [P2 January 2002 Question 6]

9. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 0.5 \mathrm { e } ^ { x } - x ^ { 2 } .$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).

1. Find an equation of the tangent to \(C\) at \(A\). The \(x\)-coordinate of \(B\) is approximately 2.15 . A more exact estimate is to be made of this coordinate using iterations \(x _ { n + 1 } = \ln \mathrm { g } \left( x _ { n } \right)\).
2. Show that a possible form for \(\mathrm { g } ( x )\) is \(\mathrm { g } ( x ) = 4 x\).
3. Using \(x _ { n + 1 } = \ln 4 x _ { n }\), with \(x _ { 0 } = 2.15\), calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give the value of \(x _ { 3 }\) to 4 decimal places.

10. $$\mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1$$

1. Prove that \(\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }\).
2. Find the range of f.
3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
4. Find the range of \(\mathrm { f } ^ { - 1 } ( x )\).

11. Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec ^ { 2 } x\).

12. Express \(\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }\) as a single fraction in its simplest form.
[0pt] [P2 June 2002 Question 2]

13. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{d0c23635-3b9b-4666-9cb4-21b931fb3719-06_626_759_313_537} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3$$ Given that \(\mathrm { f } ( k ) = 0\),

1. show, by calculation, that \(3.1 < k < 3.2\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
3. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\).

14. $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x - 3 , x \in \mathbb { R } , x \geq 1 .$$

1. Find the range of f .
2. Write down the domain and range of \(\mathrm { f } ^ { - 1 }\).
3. Sketch the graph of \(\mathrm { f } ^ { - 1 }\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. Given that \(\mathrm { g } ( x ) = | x - 4 | , x \in \mathbb { R }\),
4. find an expression for \(\operatorname { gf } ( x )\).
5. Solve \(\operatorname { gf } ( x ) = 8\).

15. Express \(\frac { y + 3 } { ( y + 1 ) ( y + 2 ) } - \frac { y + 1 } { ( y + 2 ) ( y + 3 ) }\) as a single fraction in its simplest form.

16. (a) Express \(1.5 \sin 2 x + 2 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate.
(b) Express \(3 \sin x \cos x + 4 \cos ^ { 2 } x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\) and \(c\) are constants to be found.
(c) Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos ^ { 2 } x\).

17. The curve \(C\) with equation \(y = p + q \mathrm { e } ^ { x }\), where \(p\) and \(q\) are constants, passes through the point \(( 0,2 )\). At the point \(P ( \ln 2 , p + 2 q )\) on \(C\), the gradient is 5 .

1. Find the value of \(p\) and the value of \(q\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
2. Show that the area of \(\triangle O L M\), where \(O\) is the origin, is approximately 53.8. \section*{18.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-08_487_695_259_683}
   \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { - x } - 1\).
3. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac { 1 } { 2 } | x - 1 |\). Show the coordinates of the points where the graph meets the axes. The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).
4. Show that \(x = \alpha\) is a root of the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
5. Show that \(- 1 < \alpha < 0\). The iterative formula \(x _ { \mathrm { n } + 1 } = - \ln \left[ \frac { 1 } { 2 } \left( 3 - x _ { n } \right) \right]\) is used to solve the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
6. Starting with \(x _ { 0 } = - 1\), find the values of \(x _ { 1 }\) and \(x _ { 2 }\).
7. Show that, to 2 decimal places, \(\alpha = - 0.58\).

19. The function f is defined by \(\mathrm { f } : x \mapsto \frac { 3 x - 1 } { x - 3 } , x \in \mathbb { R } , x \neq 3\).

1. Prove that \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) for all \(x \in \mathbb { R } , x \neq 3\).
2. Hence find, in terms of \(k , \operatorname { ff } ( k )\), where \(x \neq 3\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-09_817_1139_623_406}
   \end{figure} Figure 3 shows a sketch of the one-one function g , defined over the domain \(- 2 \leq x \leq 2\).
3. Find the value of \(\mathrm { fg } ( - 2 )\).
4. Sketch the graph of the inverse function \(\mathrm { g } ^ { - 1 }\) and state its domain. The function h is defined by \(\mathrm { h } : x \mapsto 2 \mathrm {~g} ( x - 1 )\).
5. Sketch the graph of the function h and state its range.

20. Express \(\frac { x } { ( x + 1 ) ( x + 3 ) } + \frac { x + 12 } { x ^ { 2 } - 9 }\) as a single fraction in its simplest form.
21. (a) Sketch the graph of \(y = | 2 x + a | , a > 0\), showing the coordinates of the points where the graph meets the coordinate axes.
(b) On the same axes, sketch the graph of \(y = \frac { 1 } { x }\).
(c) Explain how your graphs show that there is only one solution of the equation $$x | 2 x + a | - 1 = 0$$ (d) Find, using algebra, the value of \(x\) for which \(x | 2 x + 1 | - 1 = 0\).
22. The curve with equation \(y = \ln 3 x\) crosses the \(x\)-axis at the point \(P ( p , 0 )\).
(a) Sketch the graph of \(y = \ln 3 x\), showing the exact value of \(p\). The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
(b) Show that \(x = q\) is a solution of the equation \(x ^ { 2 } + \ln 3 x = 0\).
(c) Show that the equation in part (b) can be rearranged in the form \(x = \frac { 1 } { 3 } \mathrm { e } ^ { - x ^ { 2 } }\).
(d) Use the iteration formula \(x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } ^ { 2 } }\), with \(x _ { 0 } = \frac { 1 } { 3 }\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Hence write down, to 3 decimal places, an approximation for \(q\).
23. (a) Express \(\sin x + \sqrt { 3 } \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Show that the equation \(\sec x + \sqrt { 3 } \operatorname { cosec } x = 4\) can be written in the form $$\sin x + \sqrt { 3 } \cos x = 2 \sin 2 x$$ (c) Deduce from parts (a) and (b) that sec \(x + \sqrt { 3 } \operatorname { cosec } x = 4\) can be written in the form $$\sin 2 x - \sin \left( x + 60 ^ { \circ } \right) = 0$$ 24. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \geq 0\). The curve meets the coordinate axes at the points \(( 0 , c )\) and \(( d , 0 )\). In separate diagrams sketch the curve with equation
(a) \(y = \mathrm { f } ^ { - 1 } ( x )\),
(b) \(y = 3 \mathrm { f } ( 2 x )\). Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\mathrm { f } : x \mapsto 3 \left( 2 ^ { - x } \right) - 1 , x \in \mathbb { R } , x \geq 0 ,$$ (c) state
(i) the value of \(c\),
(ii) the range of \(f\).
(d) Find the value of \(d\), giving your answer to 3 decimal places. The function g is defined by $$\mathrm { g } : x \rightarrow \log _ { 2 } x , x \in \mathbb { R } , x \geq 1 .$$ (e) Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
25. (a) Simplify \(\frac { x ^ { 2 } + 4 x + 3 } { x ^ { 2 } + x }\).
(b) Find the value of \(x\) for which \(\log _ { 2 } \left( x ^ { 2 } + 4 x + 3 \right) - \log _ { 2 } \left( x ^ { 2 } + x \right) = 4\).
26. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } - 2 x + 3 , x \in \mathbb { R } , 0 \leq x \leq 4
& \mathrm {~g} : x \mapsto \lambda x ^ { 2 } + 1 , \text { where } \lambda \text { is a constant, } x \in \mathbb { R } . \end{aligned}$$ (a) Find the range of f .
(b) Given that \(\operatorname { gf } ( 2 ) = 16\), find the value of \(\lambda\).
27. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , - 1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A ( 2,0 )\) and has a maximum at the point \(B \left( \frac { 4 } { 3 } , 1 \right)\). In separate diagrams, show a sketch of the curve with equation
(a) \(y = \mathrm { f } ( x + 1 )\),
(b) \(y = | \mathrm { f } ( x ) |\),
(c) \(y = \mathrm { f } ( | x | )\),
marking on each sketch the coordinates of points at which the curve
(i) has a turning point,
(ii) meets the \(x\)-axis.
28. (a) Sketch, on the same set of axes, the graphs of $$y = 2 - \mathrm { e } ^ { - x } \text { and } y = \sqrt { } x$$ [It is not necessary to find the coordinates of any points of intersection with the axes.]
Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } + \sqrt { } x - 2 , x \geq 0\),
(b) explain how your graphs show that the equation \(\mathrm { f } ( x ) = 0\) has only one solution,
(c) show that the solution of \(\mathrm { f } ( x ) = 0\) lies between \(x = 3\) and \(x = 4\). The iterative formula \(x _ { n + 1 } = \left( 2 - \mathrm { e } ^ { - x _ { n } } \right) ^ { 2 }\) is used to solve the equation \(\mathrm { f } ( x ) = 0\).
(d) Taking \(x _ { 0 } = 4\), write down the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), and hence find an approximation to the solution of \(\mathrm { f } ( x ) = 0\), giving your answer to 3 decimal places.
[0pt] [P2 June 2003 Question 5] 28a. (i) Given that \(\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }\), prove that \(\tan x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }\).
(ii) (a) Prove that \(\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta\).
(b) Verify that \(\theta = 180 ^ { \circ }\) is a solution of the equation \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
(c) Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360 ^ { \circ }\), of the equation using \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
[0pt] [P2 June 2003 Question 8]
29. (a) Express as a fraction in its simplest form $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 }$$ (b) Hence solve $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 } = 1$$ [P2 November 2003 Question 1]
30. Prove that $$\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } \equiv \cos 2 \theta$$

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \mapsto | x - a | + a , x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto 4 x + a , \quad x \in \mathbb { R } . \end{aligned}$$ where \(a\) is a positive constant.
(a) On the same diagram, sketch the graphs of f and g , showing clearly the coordinates of any points at which your graphs meet the axes.
(b) Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect.
(c) Find an expression for \(\mathrm { fg } ( x )\).
(d) Solve, for \(x\) in terms of \(a\), the equation $$\mathrm { fg } ( x ) = 3 a$$

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where

$$\mathrm { f } ( x ) = 3 \ln x + \frac { 1 } { x } , \quad x > 0$$ The point \(P\) is a stationary point on \(C\).
(a) Calculate the \(x\)-coordinate of \(P\).
(b) Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. The point \(Q\) on \(C\) has \(x\)-coordinate 1 .
(c) Find an equation for the normal to \(C\) at \(Q\). The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
(d) Show that the \(x\)-coordinate of \(R\)
(i) satisfies the equation \(6 \ln x + x + \frac { 2 } { x } - 3 = 0\),
(ii) lies between 0.13 and 0.14 .
33. The function f is given by \(\mathrm { f } : x \mapsto 2 + \frac { 3 } { x + 2 } , x \in \mathbb { R } , x \neq - 2\).
(a) Express \(2 + \frac { 3 } { x + 2 }\) as a single fraction.
(b) Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
(c) Write down the domain of \(\mathrm { f } ^ { - 1 }\).
34. The function f is even and has domain \(\mathbb { R }\). For \(x \geq 0 , \mathrm { f } ( x ) = x ^ { 2 } - 4 a x\), where \(a\) is a positive constant.
(a) In the space below, sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of all the points at which the curve meets the axes.
(b) Find, in terms of \(a\), the value of \(\mathrm { f } ( 2 a )\) and the value of \(\mathrm { f } ( - 2 a )\). Given that \(a = 3\),
(c) use algebra to find the values of \(x\) for which \(\mathrm { f } ( x ) = 45\).
35. Given that \(y = \log _ { a } x , x > 0\), where \(a\) is a positive constant,
(a) (i) express \(x\) in terms of \(a\) and \(y\),
(ii) deduce that \(\ln x = y \ln a\).
(b) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln a }\). The curve \(C\) has equation \(y = \log _ { 10 } x , x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10 . Using the result in part (b),
(c) find an equation for the tangent to \(C\) at \(A\). The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
(d) Find the exact \(x\)-coordinate of \(B\).
36. (i) (a) Express ( \(12 \cos \theta - 5 \sin \theta\) ) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.
(ii) Solve $$8 \cot \theta - 3 \tan \theta = 2 ,$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.
37. Express as a single fraction in its simplest form $$\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 } \times \frac { 2 x ^ { 2 } + 6 x } { ( x - 5 ) ^ { 2 } }$$ [P2 June 2004 Question 1]
38. (i) Given that \(\sin x = \frac { 3 } { 5 }\), use an appropriate double angle formula to find the exact value of \(\sec 2 x\).
(ii) Prove that $$\cot 2 x + \operatorname { cosec } 2 x \equiv \cot x , \quad \left( x \neq \frac { n \pi } { 2 } , n \in \mathrm { Z } \right)$$ [P2 June 2004 Question 2]
39. $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 1$$ The equation \(\mathrm { f } ( x ) = 0\) has only one positive root, \(\alpha\).
(a) Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as $$x = \sqrt { \left( \frac { 4 x + 1 } { x + 1 } \right) } , x \neq - 1$$ The iterative formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) is used to find an approximation to \(\alpha\).
(b) Taking \(x _ { 1 } = 1\), find, to 2 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
(c) By choosing values of \(x\) in a suitable interval, prove that \(\alpha = 1.70\), correct to 2 decimal places.
(d) Write down a value of \(x _ { 1 }\) for which the iteration formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) does not produce a valid value for \(x _ { 2 }\). Justify your answer.
40. $$\mathrm { f } ( x ) = x + \frac { \mathrm { e } ^ { x } } { 5 } , \quad x \in \mathbb { R }$$ (a) Find \(\mathrm { f } ^ { \prime } ( x )\). The curve \(C\), with equation \(y = \mathrm { f } ( x )\), crosses the \(y\)-axis at the point \(A\).
(b) Find an equation for the tangent to \(C\) at \(A\).
(c) Complete the table, giving the values of \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\) to 2 decimal places.

1. The function f is given by

$$f : x \mapsto \ln ( 3 x - 6 ) , \quad x \in \mathbb { R } , \quad x > 2 .$$ (a) Find \(\mathrm { f } ^ { - 1 } ( x )\).
(b) Write down the domain of \(\mathrm { f } ^ { - 1 }\) and the range of \(\mathrm { f } ^ { - 1 }\).
(c) Find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 3\). The function g is given by $$\mathrm { g } : x \mapsto \ln | 3 x - 6 | , \quad x \in \mathbb { R } , \quad x \neq 2 .$$ (d) Sketch the graph of \(y = \mathrm { g } ( x )\).
(e) Find the exact coordinates of all the points at which the graph of \(y = \mathrm { g } ( x )\) meets the coordinate axes.

2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.

5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at ( \(0 , a\) ) and ( \(b , 0\) ).
\includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}

1. State the value of \(a\), and show that \(b = \ln 3\).
2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).

6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1.0.
2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
   (1 mark)
3. 1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
   2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      (3 marks)

7

1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\).
   \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-006_819_824_456_591} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , \quad x \neq - 2 \end{array}$$

1. State the range of f.
2. 1. Find fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
3. 1. Explain why the function f does not have an inverse.
   2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

9

1. Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
2. Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
3. The sketch shows the graph of \(y = x ^ { - 2 } \ln x\).
   \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-007_593_1034_696_543}
   1. Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
   2. The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is $$\frac { 1 } { 5 } ( 4 - \ln 5 )$$

10

1. 1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
   2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \tan 3 x\).
   (2 marks)
2. Given that \(y = \frac { 3 x + 1 } { 2 x + 1 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }\).
   (3 marks)

2 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.