Edexcel P3 (Pure Mathematics 3) 2023 June

1. $$g ( x ) = x ^ { 6 } + 2 x - 1000$$

1. Show that \(\mathrm { g } ( x ) = 0\) has a root \(\alpha\) in the interval [3,4] Using the iteration formula $$x _ { n + 1 } = \sqrt [ 6 ] { 1000 - 2 x _ { n } } \quad \text { with } x _ { 1 } = 3$$
2. 1. find, to 4 decimal places, the value of \(x _ { 2 }\)
   2. find, by repeated iteration, the value of \(\alpha\). Give your answer to 4 decimal places.

2. \begin{figure}[h] \end{figure} Figure 1 shows the linear relationship between \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
The line passes through the points \(( 0,4 )\) and \(( 2,0 )\) as shown.

1. 1. Find an equation linking \(\log _ { 6 } T\) and \(\log _ { 6 } x\)
   2. Hence find the exact value of \(T\) when \(x = 216\)
2. Find an equation, not involving logs, linking \(T\) with \(x\)

1. (i) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \ln \left( \sin ^ { 2 } 3 x \right)\) writing your answer in simplest form.
   (ii) (a) Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( 3 x ^ { 2 } - 4 \right) ^ { 6 }\)
   (b) Hence show that

$$\int _ { 0 } ^ { \sqrt { 2 } } x \left( 3 x ^ { 2 } - 4 \right) ^ { 5 } \mathrm {~d} x = R$$ where \(R\) is an integer to be found.
(Solutions relying on calculator technology are not acceptable.)

1. The function f is defined by

$$\mathrm { f } ( x ) = 2 x ^ { 2 } - 5 \quad x \geqslant 0 \quad x \in \mathbb { R }$$

1. State the range of f On the following page there is a diagram, labelled Diagram 1, which shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
2. On Diagram 1, sketch the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) meets the curve with equation \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(P\) Using algebra and showing your working,
3. find the exact \(x\) coordinate of \(P\)
   \includegraphics[max width=\textwidth, alt={}]{bef290fb-fbac-4c9c-981e-5e323ac7182e-09_607_610_248_731}
   \section*{Diagram 1}

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Solve, for \(0 < x < \pi\)
   $$( x - 2 ) ( \sqrt { 3 } \sec x + 2 ) = 0$$
2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$10 \sin \theta = 3 \cos 2 \theta$$

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 | x - 2 | - 10$$ The vertex of the graph is at point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\)
2. Find \(\mathrm { ff } ( 0 )\)
3. Solve the inequality $$3 | x - 2 | - 10 < 5 x + 10$$
4. Solve the equation $$\mathrm { f } ( | x | ) = 0$$

1. A scientist is studying two different populations of bacteria.

The number of bacteria \(N\) in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that

• there were 2500 bacteria in this population at the start of the study
• there were 10000 bacteria 8 hours later
  1. find the exact value of \(A\) and the value of \(k\) to 4 significant figures.

The number of bacteria \(N\) in the second population is modelled by the equation $$N = 60000 \mathrm { e } ^ { - 0.6 t } \quad t \geqslant 0$$ where \(t\) is the time in hours from the start of the study.

Find the rate of decrease of bacteria in this population exactly 5 hours from the start of the study. Give your answer to 3 significant figures. When \(t = T\), the number of bacteria in the two different populations was the same.

Find the value of \(T\), giving your answer to 3 significant figures.
(Solutions relying entirely on calculator technology are not acceptable.)

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

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = A ( 2 x + 1 ) ^ { 2 } ( 1 - 4 x ) \mathrm { e } ^ { - 4 x }$$ where \(A\) is a constant to be found.
2. Hence find the exact coordinates of the two stationary points on \(C\). The function g is defined by $$g ( x ) = 8 f ( x - 2 )$$
3. Find the coordinates of the maximum stationary point on the curve with equation \(y = g ( x )\).

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$x = \frac { 2 y ^ { 2 } + 6 } { 3 y - 3 }$$

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) giving your answer as a fully simplified fraction. The tangents at points \(P\) and \(Q\) on the curve are parallel to the \(y\)-axis, as shown in Figure 4.
2. Use the answer to part (a) to find the equations of these two tangents.