Edexcel P4 (Pure Mathematics 4) 2022 October

1. A curve \(C\) has parametric equations

$$x = \frac { t } { t - 3 } \quad y = \frac { 1 } { t } + 2 \quad t \in \mathbb { R } \quad t > 3$$ Show that all points on \(C\) lie on the curve with Cartesian equation $$y = \frac { a x - 1 } { b x }$$ where \(a\) and \(b\) are constants to be found.

1. (a) Express \(\frac { 3 x } { ( 2 x - 1 ) ( x - 2 ) }\) in partial fraction form.
   (b) Hence show that

$$\int _ { 5 } ^ { 25 } \frac { 3 x } { ( 2 x - 1 ) ( x - 2 ) } d x = \ln k$$ where \(k\) is a fully simplified fraction to be found.
(Solutions relying entirely on calculator technology are not acceptable.)

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of triangle \(P Q R\).
Given that

• \(\overrightarrow { P Q } = 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }\)
• \(\overrightarrow { P R } = 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k }\)
  1. Find \(\overrightarrow { R Q }\)
  2. Find the size of angle \(P Q R\), in degrees, to three significant figures.

4. $$g ( x ) = \frac { 1 } { \sqrt { 4 - x ^ { 2 } } }$$

1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(\mathrm { g } ( x )\). Give each coefficient in simplest form.
2. State the range of values of \(x\) for which this expansion is valid.
3. Use the expansion from part (a) to find a fully simplified rational approximation for \(\sqrt { 3 }\) Show your working and make your method clear.

1. 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 2 shows a sketch of part of the curve with equation $$y = \frac { 12 \sqrt { x } } { \left( 2 x ^ { 2 } + 3 \right) ^ { 1.5 } }$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac { 1 } { \sqrt { 2 } }\), the \(x\)-axis and the line with equation \(x = k\). This region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. Given that the volume of this solid is \(\frac { 713 } { 648 } \pi\), use algebraic integration to find the exact value of the constant \(k\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + 3 \tan t \quad y = 2 \cos 2 t \quad - \frac { \pi } { 6 } \leqslant t \leqslant \frac { \pi } { 3 }$$ The curve crosses the \(x\)-axis at point \(P\), as shown in Figure 3.

1. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where f is a function with domain \([ k , 1 + 3 \sqrt { 3 } ]\)
2. Find the exact value of the constant \(k\).
3. Find the range of f.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Use the substitution \(u = \mathrm { e } ^ { x } - 3\) to show that $$\int _ { \ln 5 } ^ { \ln 7 } \frac { 4 \mathrm { e } ^ { 3 x } } { \mathrm { e } ^ { x } - 3 } \mathrm {~d} x = a + b \ln 2$$ where \(a\) and \(b\) are constants to be found.
2. Show, by integration, that $$\int 3 \mathrm { e } ^ { x } \cos 2 x \mathrm {~d} x = p \mathrm { e } ^ { x } \sin 2 x + q \mathrm { e } ^ { x } \cos 2 x + c$$ where \(p\) and \(q\) are constants to be found and \(c\) is an arbitrary constant.

1. A student was asked to prove by contradiction that
   "there are no positive integers \(x\) and \(y\) such that \(3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25\) "
   The start of the student's proof is shown in the box below.

Assume that integers \(x\) and \(y\) exist such that \(3 x ^ { 2 } + 2 x y - y ^ { 2 } = 25\) $$\Rightarrow ( 3 x - y ) ( x + y ) = 25$$ $$\begin{aligned} & \text { If } \quad ( 3 x - y ) = 1 \quad \text { and } ( x + y ) = 25
& \left. \begin{array} { l } 3 x - y = 1
x + y = 25 \end{array} \right\} \Rightarrow 4 x = 26 \Rightarrow x = 6.5 , y = 18.5 \quad \text { Not integers } \end{aligned}$$ Show the calculations and statements that are needed to complete the proof.

1. With respect to a fixed origin \(O\), the equations of lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2
8

10 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 4
- 1
2 \end{array} \right) + \mu \left( \begin{array} { l } 5
4
8 \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
Prove that lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.

1. A spherical ball of ice of radius 12 cm is placed in a bucket of water.

In a model of the situation,

• the ball remains spherical as it melts
• \(t\) minutes after the ball of ice is placed in the bucket, its radius is \(r \mathrm {~cm}\)
• the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius
• the radius of the ball of ice is 6 cm after 15 minutes

Using the model and the information given,

1. find an equation linking \(r\) and \(t\),
2. find the time taken for the ball of ice to melt completely.
3. On Diagram 1 on page 27, sketch a graph of \(r\) against \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-27_662_728_1959_671} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the closed curve with equation $$( x + y ) ^ { 3 } + 10 y ^ { 2 } = 108 x$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 108 - 3 ( x + y ) ^ { 2 } } { 20 y + 3 ( x + y ) ^ { 2 } }$$ The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
   The points \(P\) and \(Q\) represent points that are furthest north and furthest south of the origin \(O\), as shown in Figure 4. Using the result given in part (a),
2. find how far the point \(Q\) is south of \(O\). Give your answer to the nearest 100 m .