Edexcel P4 (Pure Mathematics 4) 2021 January

1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( \frac { 1 } { 4 } - 5 x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 20 }$$ giving each coefficient in its simplest form. By substituting \(x = \frac { 1 } { 100 }\) into the answer for (a),
(b) find an approximation for \(\sqrt { 5 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of parallelogram \(A B C D\).
Given that \(\overrightarrow { A B } = 6 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { B C } = 2 \mathbf { i } + 5 \mathbf { j } + 8 \mathbf { k }\)

1. find the size of angle \(A B C\), giving your answer in degrees, to 2 decimal places.
2. Find the area of parallelogram \(A B C D\), giving your answer to one decimal place.

3. Prove by contradiction that there is no greatest odd integer.

4. The curve \(C\) is defined by the parametric equations $$x = \frac { 1 } { t } + 2 \quad y = \frac { 1 - 2 t } { 3 + t } \quad t > 0$$

1. Show that the equation of \(C\) can be written in the form \(y = \mathrm { g } ( x )\) where g is the function $$\mathrm { g } ( x ) = \frac { a x + b } { c x + d } \quad x > k$$ where \(a , b , c , d\) and \(k\) are integers to be found.
2. Hence, or otherwise, state the range of g .
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5. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.
Using the substitution \(u = 3 + \sqrt { 2 x - 1 }\) find the exact value of $$\int _ { 1 } ^ { 13 } \frac { 4 } { 3 + \sqrt { 2 x - 1 } } d x$$ giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found.

6. A curve has equation $$4 y ^ { 2 } + 3 x = 6 y \mathrm { e } ^ { - 2 x }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve crosses the \(y\)-axis at the origin and at the point \(P\).
2. Find the equation of the normal to the curve at \(P\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.

7. \begin{figure}[h] \end{figure}

1. Find \(\int \mathrm { e } ^ { 2 x } \sin x \mathrm {~d} x\) Figure 2 shows a sketch of part of the curve with equation $$y = \mathrm { e } ^ { 2 x } \sin x \quad x \geqslant 0$$ The finite region \(R\) is bounded by the curve and the \(x\)-axis and is shown shaded in Figure 2.
2. Show that the exact area of \(R\) is \(\frac { \mathrm { e } ^ { 2 \pi } + 1 } { 5 }\)
   (Solutions relying on calculator technology are not acceptable.)
   Question 7 continue

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 1
5
4 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 1
5 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2
- 2
- 5 \end{array} \right) + \mu \left( \begin{array} { r } 4
- 3
b \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Prove that for all values of \(b \neq 7\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$ The finite region, shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \sqrt { 3 }\) The region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the exact volume of this solid of revolution is given by $$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$ where \(p\) and \(k\) are constants to be found.
2. Hence find, by algebraic integration, the exact volume of this solid of revolution.

10. (a) Write \(\frac { 1 } { ( H - 5 ) ( H + 3 ) }\) in partial fraction form. The depth of water in a storage tank is being monitored.
The depth of water in the tank, \(H\) metres, is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - \frac { ( H - 5 ) ( H + 3 ) } { 40 }$$ where \(t\) is the time, in days, from when monitoring began.
Given that the initial depth of water in the tank was 13 m ,
(b) solve the differential equation to show that $$H = \frac { 10 + 3 \mathrm { e } ^ { - 0.2 t } } { 2 - \mathrm { e } ^ { - 0.2 t } }$$ (c) Hence find the time taken for the depth of water in the tank to fall to 8 m .
(Solutions relying entirely on calculator technology are not acceptable.) According to the model, the depth of water in the tank will eventually fall to \(k\) metres.
(d) State the value of the constant \(k\).