Edexcel P3 (Pure Mathematics 3) 2021 October

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }\)
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.
      (3)

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = | 3 x - 13 | + 5 \quad x \in \mathbb { R }$$ The vertex of the graph is at point \(P\), as shown in Figure 1.

1. State the coordinates of \(P\).
2. 1. State the range of f .
   2. Find the value of ff(4)
3. Solve, using algebra and showing your working, $$16 - 2 x > | 3 x - 13 | + 5$$ The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x + b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x + b )\) is \(( 4,20 )\) Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\).

3. \begin{figure}[h] \end{figure} The total mass of gold, \(G\) tonnes, extracted from a mine is modelled by the equation $$G = 40 - 30 \mathrm { e } ^ { 1 - 0.05 t } \quad t \geqslant k \quad G \geqslant 0$$ where \(t\) is the number of years after 1st January 1800.
Figure 2 shows a sketch of \(G\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).

1. 1. Find the value of \(k\).
   2. Hence find the year and month in which gold started being extracted from the mine.
2. Find the total mass of gold extracted from the mine up to 1st January 1870. There is a limit to the mass of gold that can be extracted from the mine.
3. State the value of this limit.
   M

4. In this question you should show detailed reasoning. \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$ can be written in the form $$\tan \theta = 2 \sqrt { 3 }$$
2. Hence, or otherwise, solve for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$ giving your answers to one decimal place.

5. (i) Find, by algebraic integration, the exact value of $$\int _ { 2 } ^ { 4 } \frac { 8 } { ( 2 x - 3 ) ^ { 3 } } d x$$ (ii) Find, in simplest form, $$\int x \left( x ^ { 2 } + 3 \right) ^ { 7 } d x$$

6. (i) The curve \(C _ { 1 }\) has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that \(C _ { 1 }\) has a stationary point at \(x = \frac { \sqrt { p } } { 2 }\) where \(p\) is a constant to be found.
(ii) A different curve \(C _ { 2 }\) has equation $$y = 4 x - 12 \sin ^ { 2 } x$$

1. Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where \(A\) and \(B\) are constants to be found.
2. Hence, state the maximum gradient of this curve.

7 The mass, \(M \mathrm {~kg}\), of a species of tree can be modelled by the equation $$\log _ { 10 } M = 1.93 \log _ { 10 } r + 0.684$$ where \(r \mathrm {~cm}\) is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm .
According to the model,

1. find the mass of this tree, giving your answer to 2 significant figures.
2. Show that the equation of the model can be written in the form $$M = p r ^ { q }$$ giving the values of the constants \(p\) and \(q\) to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(p\). Q

8. A curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \arcsin \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2 \quad - \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }$$

1. Sketch \(C\).
2. Given \(x = 2 \sin y\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { A - x ^ { 2 } } }$$ where \(A\) is a constant to be found. The point \(P\) lies on \(C\) and has \(y\) coordinate \(\frac { \pi } { 4 }\)
3. Find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
   (3)

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$

1. Find \(f ^ { \prime } ( x )\). The line \(l\) is the normal to the curve at \(O\) and meets the curve again at the point \(P\). The point \(P\) lies in the 3rd quadrant, as shown in Figure 3.
2. Show that the \(x\) coordinate of \(P\) is a solution of the equation $$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
3. Using the iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$ find, to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the \(x\) coordinate of \(P\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$

1. Show that $$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$ where \(p , q\) and \(r\) are constants to be found. The curve touches the positive \(x\)-axis for the second time when \(x = a\), as shown in Figure 4. The regions bounded by the curve, the \(y\)-axis and the \(x\)-axis up to \(x = a\) are shown shaded in Figure 4.
2. Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}