Edexcel P3 (Pure Mathematics 3) 2024 June

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 | x - 5 | + 10$$ The point \(P\), shown in Figure 1, is the vertex of the graph.

1. State the coordinates of \(P\)
2. Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
3. Find the point to which \(P\) is mapped, when the graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = 3 \mathrm { f } ( x - 2 )\)

2. $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$

1. Write \(g ( x )\) in the form $$A x + B + \frac { C } { x - 2 }$$ where \(A , B\) and \(C\) are integers to be found.
2. Hence use algebraic integration to show that $$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$ where \(\alpha\) and \(\beta\) are integers to be found.

1. (i) The variables \(x\) and \(y\) are connected by the equation

$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$ Sketch the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\)
Show on your sketch the coordinates of the points of intersection of the graph with the axes.
(ii) \begin{figure}[h] \end{figure} Figure 2 shows the linear relationship between \(\log _ { 3 } N\) and \(t\).
Show that \(N = a b ^ { t }\) where \(a\) and \(b\) are constants to be found.

4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$

1. Write \(\mathrm { f } ( x )\) in the form $$a \sin 2 x + b \cos 2 x + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
2. Use the answer to part (a) to write \(\mathrm { f } ( x )\) in the form $$R \sin ( 2 x + \alpha ) + c$$ where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
   Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 significant figures.
3. Hence, or otherwise,
   1. state the maximum value of \(\mathrm { f } ( x )\)
   2. find the second smallest positive value of \(x\) at which a maximum value of \(\mathrm { f } ( x )\) occurs. Give your answer to 3 significant figures.

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

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0
& \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
2. Use differentiation to prove that g is an increasing function.
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the range of fg

6. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve with equation $$y = \sqrt { 4 x - 7 }$$ The line \(l\), shown in Figure 3, is the normal to the curve at the point \(P ( 8,5 )\)

1. Use calculus to show that an equation of \(l\) is $$5 x + 2 y - 50 = 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and \(l\).
2. Use algebraic integration to find the exact area of \(R\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
2. Hence or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$

8. \begin{figure}[h] \end{figure} Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground. The vertical height, \(h\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit. The ball travels a horizontal distance of \(d\) metres before it first hits the ground.
The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
The path of the ball is modelled by the equation $$h = 1.5 x - 0.5 x \mathrm { e } ^ { 0.02 x } \quad 0 \leqslant x \leqslant d$$ \section*{Use the model to answer parts (a), (b) and (c).}

1. Find the value of \(d\), giving your answer to 2 decimal places.
   (Solutions relying entirely on calculator technology are not acceptable.)
2. Show that the maximum value of \(h\) occurs when $$x = 50 \ln \left( \frac { 150 } { x + 50 } \right)$$ Using the iteration formula $$x _ { n + 1 } = 50 \ln \left( \frac { 150 } { x _ { n } + 50 } \right) \quad \text { with } x _ { 1 } = 30$$
3. 1. find the value of \(x _ { 2 }\) to 2 decimal places,
   2. find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places.
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9. \begin{figure}[h] \end{figure} The curve shown in Figure 5 has equation $$x = 4 \sin ^ { 2 } y - 1 \quad 0 \leqslant y \leqslant \frac { \pi } { 2 }$$ The point \(P \left( k , \frac { \pi } { 3 } \right)\) lies on the curve.

1. Verify that \(k = 2\)
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\)
   2. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 \sqrt { x + 1 } \sqrt { 3 - x } }\) The normal to the curve at \(P\) cuts the \(x\)-axis at the point \(N\).
3. Find the exact area of triangle \(O P N\), where \(O\) is the origin. Give your answer in the form \(a \pi + b \pi ^ { 2 }\) where \(a\) and \(b\) are constants.