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]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-12_526_1052_287_474}
\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]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-13_571_1326_936_438}
\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
$$\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } \equiv \cos 2 \theta$$
- 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
$$\mathrm { fg } ( x ) = 3 a$$
- 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.
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\) | 0.45 | 0.91 | | | |
- 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.