Edexcel P3 (Pure Mathematics 3) 2023 October

1. A curve has equation \(y = \mathrm { f } ( x )\) where

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

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [1,2] The iterative formula $$x _ { n + 1 } = \sqrt { 5 x _ { n } - \mathrm { e } ^ { x _ { n } } }$$ with \(x _ { 1 } = 1\) is used to find an approximate value for the root \(\alpha\).
2. 1. Find the value of \(x _ { 2 }\) to 4 decimal places.
   2. Find, by repeated iteration, the value of \(\alpha\), giving your answer to 4 decimal places.

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { x + 3 } { x - 4 } \quad x \in \mathbb { R } , x \neq 4$$

1. Find ff(6)
2. Find \(f ^ { - 1 }\) The function \(g\) is defined by $$g ( x ) = x ^ { 2 } + 5 \quad x \in \mathbb { R } , x > 0$$
3. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 7$$

1. (a) Using the identity for \(\cos ( A + B )\), prove that

$$\cos 2 A \equiv 2 \cos ^ { 2 } A - 1$$ (b) Hence, using algebraic integration, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 8 } } \left( 5 - 4 \cos ^ { 2 } 3 x \right) d x$$

1. A new mobile phone is released for sale.

The total sales \(N\) of this phone, in thousands, is modelled by the equation $$N = 125 - A \mathrm { e } ^ { - 0.109 t } \quad t \geqslant 0$$ where \(A\) is a constant and \(t\) is the time in months after the phone was released for sale.
Given that when \(t = 0 , N = 32\)

1. state the value of \(A\). Given that when \(t = T\) the total sales of the phone was 100000
2. find, according to the model, the value of \(T\). Give your answer to 2 decimal places.
3. Find, according to the model, the rate of increase in total sales when \(t = 7\), giving your answer to 3 significant figures.
   (Solutions relying entirely on calculator technology are not acceptable.) The total sales of the mobile phone is expected to reach 150000
   Using this information,
4. give a reason why the given equation is not suitable for modelling the total sales of the phone.

1. The curve \(C\) has equation

$$y = \frac { \ln \left( x ^ { 2 } + k \right) } { x ^ { 2 } + k } \quad x \in \mathbb { R }$$ where \(k\) is a positive constant.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x \left( B - \ln \left( x ^ { 2 } + k \right) \right) } { \left( x ^ { 2 } + k \right) ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found. Given that \(C\) has exactly three turning points,
2. find the \(x\) coordinate of each of these points. Give your answer in terms of \(k\) where appropriate.
3. find the upper limit to the value for \(k\).

1. An area of sea floor is being monitored.

The area of the sea floor, \(S \mathrm {~km} ^ { 2 }\), covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$

1. find, according to the model, the area of sea floor covered by coral reefs when \(t = 2\)
2. find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(q\)

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

1. Find the range of f
2. Show that $$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$ where \(A\) and \(B\) are constants to be found. Given that the line \(y = k\), where \(k\) is a constant, \(k > 0\), intersects the curve at exactly two distinct points,
3. find the exact range of values of \(k\)

1. (a) Prove that

$$2 \operatorname { cosec } ^ { 2 } 2 \theta ( 1 - \cos 2 \theta ) \equiv 1 + \tan ^ { 2 } \theta$$ (b) Hence solve for \(0 < x < 360 ^ { \circ }\), where \(x \neq ( 90 n ) ^ { \circ } , n \in \mathbb { N }\), the equation $$2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = 4 + 3 \sec x$$ giving your answers to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)

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

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = | 2 - 4 \ln ( x + 1 ) | \quad x > k$$ where \(k\) is a constant.
Given that the curve

• has an asymptote at \(x = k\)
• cuts the \(y\)-axis at point \(A\)
• meets the \(x\)-axis at point \(B\)
  as shown in Figure 2,
  1. state the value of \(k\)
  2. 1. find the \(y\) coordinate of \(A\)
     2. find the exact \(x\) coordinate of \(B\)
  3. Using algebra and showing your working, find the set of values of \(x\) such that

$$| 2 - 4 \ln ( x + 1 ) | > 3$$

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

A curve \(C\) has equation $$x = \sin ^ { 2 } 4 y \quad 0 \leqslant y \leqslant \frac { \pi } { 8 } \quad 0 \leqslant x \leqslant 1$$ The point \(P\) with \(x\) coordinate \(\frac { 1 } { 4 }\) lies on \(C\)

1. Find the exact \(y\) coordinate of \(P\)
2. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\)
3. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { q + r ( x + s ) ^ { 2 } } }$$ where \(q , r\) and \(s\) are constants to be found. Using the answer to part (c),
4. 1. state the \(x\) coordinate of the point where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is a minimum,
   2. state the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at this point.