Edexcel P3 (Pure Mathematics 3) 2022 October

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$f ( x ) = \frac { 2 x ^ { 3 } - 4 x - 15 } { x ^ { 2 } + 3 x + 4 }$$

1. Show that $$f ( x ) \equiv A x + B + \frac { C ( 2 x + 3 ) } { x ^ { 2 } + 3 x + 4 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence, find $$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

2. The functions f and g are defined by $$\begin{array} { l l } f ( x ) = 5 - \frac { 4 } { 3 x + 2 } & x \geqslant 0
g ( x ) = \left| 4 \sin \left( \frac { x } { 3 } + \frac { \pi } { 6 } \right) \right| & x \in \mathbb { R } \end{array}$$

1. Find the range of f
2. 1. Find \(\mathrm { f } ^ { - 1 } ( x )\)
   2. Write down the domain of \(\mathrm { f } ^ { - 1 }\)
3. Find \(\mathrm { fg } ( - \pi )\)

3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } \mathrm { e } ^ { 3 x } \quad x \in \mathbb { R }$$ The curve has a maximum turning point at \(A\) and a minimum turning point at \(( 2,0 )\)

1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has at least two distinct roots,
2. state the range of possible values for \(k\).

4. $$y = \log _ { 10 } ( 2 x + 1 )$$

1. Express \(x\) in terms of \(y\).
2. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   4.
3. Express \(x\) in terms of \(y\).

   	\(\begin{array} { c } \text { Leave }
   \text { blank } \end{array}\)

   (2)
4. Hence, giving your answer in terms of \(x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-10_2662_111_107_1950}

5. \begin{figure}[h] \end{figure} The profit made by a company, \(\pounds P\) million, \(t\) years after the company started trading, is modelled by the equation $$P = \frac { 4 t - 1 } { 10 } + \frac { 3 } { 4 } \ln \left[ \frac { t + 1 } { ( 2 t + 1 ) ^ { 2 } } \right]$$ The graph of \(P\) against \(t\) is shown in Figure 2. According to the model,

1. show that exactly one year after it started trading, the company had made a loss of approximately £ 830000 A manager of the company wants to know the value of \(t\) for which \(P = 0\)
2. Show that this value of \(t\) occurs in the interval [6,7]
3. Show that the equation \(P = 0\) can be expressed in the form $$t = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { ( 2 t + 1 ) ^ { 2 } } { t + 1 } \right]$$
4. Using the iteration formula $$t _ { n + 1 } = \frac { 1 } { 4 } + \frac { 15 } { 8 } \ln \left[ \frac { \left( 2 t _ { n } + 1 \right) ^ { 2 } } { t _ { n } + 1 } \right] \text { with } t _ { 1 } = 6$$ find the value of \(t _ { 2 }\) and the value of \(t _ { 6 }\), giving your answers to 3 decimal places.
5. Hence find, according to the model, how many months it takes in total, from when the company started trading, for it to make a profit.
   (2)

6. $$y = \frac { 2 + 3 \sin x } { \cos x + \sin x }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a \tan x + b \sec x + c } { \sec x + 2 \sin x }$$ where \(a , b\) and \(c\) are integers to be found.

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(C _ { 1 }\) with equation $$y = 5 - | 3 x - 22 |$$

1. Write down the coordinates of
   1. the vertex of \(C _ { 1 }\)
   2. the intersection of \(C _ { 1 }\) with the \(y\)-axis.
2. Find the \(x\) coordinates of the intersections of \(C _ { 1 }\) with the \(x\)-axis. Diagram 1, shown on page 21, is a copy of Figure 3.
3. On Diagram 1, sketch the curve \(C _ { 2 }\) with equation $$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$ Identify clearly the coordinates of any points of intersection of \(C _ { 2 }\) with the coordinate axes.
4. Find the coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573} \section*{Diagram 1} Solutions relying entirely on calculator technology are not acceptable.

1. Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
2. Find
   1. the minimum value of \(\mathrm { f } ( x )\)
   2. the smallest value of \(x\) at which this minimum value occurs.
3. State the \(y\) coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
4. State the smallest value of \(x\) at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
   In this question you must show all stages of your working.}

9. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that \(\cos 2 \theta - \sin 3 \theta \neq 0\)

1. prove that $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } \equiv \frac { 1 + \sin \theta } { 1 - 2 \sin \theta - 4 \sin ^ { 2 } \theta }$$
2. Hence solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } = 2 \operatorname { cosec } \theta$$ Give your answers to one decimal place.
   \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-28_2257_52_309_1983}