Edexcel P4 (Pure Mathematics 4) 2022 January

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

$$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting \(x = 1\) into the answer to part (a),
(b) find an approximation for \(\sqrt { 7 }\), giving your answer to 4 decimal places.

2. The curve \(C\) has parametric equations $$x = \frac { t ^ { 4 } } { 2 t + 1 } \quad y = \frac { t ^ { 3 } } { 2 t + 1 } \quad t > 0$$

1. Write down \(\frac { x } { y }\) in terms of \(t\), giving your answer in simplest form.
2. Hence show that all points on \(C\) satisfy the equation $$x ^ { 3 } - 2 x y ^ { 3 } - y ^ { 4 } = 0$$

3. The curve \(C\) has equation $$3 y ^ { 2 } - 11 x ^ { 2 } + 11 x y = 20 y - 36 x + 28$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 4 , k )\), where \(k\) is a constant, lies on \(C\).
   Given that \(k < 0\)
2. find the value of the gradient of \(C\) at \(P\)

4. $$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
3. Find $$\int _ { 3 } ^ { 5 } f ( x ) d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers to be found.

5. 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 } 4
4
- 5 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 3

6 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }

7. Water is flowing into a large container and is leaking from a hole at the base of the container. At time \(t\) seconds after the water starts to flow, the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the container is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 300 - k V$$ where \(k\) is a constant.

1. Solve the differential equation to show that, according to the model, $$V = \frac { 300 } { k } + A \mathrm { e } ^ { - k t }$$ where \(A\) is a constant.
   (5) Given that the container is initially empty and that when \(t = 10\), the volume of water is increasing at a rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
2. find the exact value of \(k\).
3. Hence find, according to the model, the time taken for the volume of water in the container to reach 6 litres. Give your answer to the nearest second.

8. Use proof by contradiction to prove that, for all positive real numbers \(x\) and \(y\), $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$

13
- 1
4 \end{array} \right) + \mu \left( \begin{array} { r } 5
1
- 3 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection \(A\).
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to one decimal place. A circle with centre \(A\) and radius 35 cuts the line \(l _ { 1 }\) at the points \(P\) and \(Q\). Given that the \(x\) coordinate of \(P\) is greater than the \(x\) coordinate of \(Q\),
3. find the coordinates of \(P\) and the coordinates of \(Q\). 6. Use integration by parts to show that $$\int \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x = p \mathrm { e } ^ { 2 x } \sin 3 x + q \mathrm { e } ^ { 2 x } \cos 3 x + k$$ where \(p\) and \(q\) are rational numbers to be found and \(k\) is an arbitrary constant.
   (6)
   7. Water is flowing into a large container and is leaking from a hole at the base of the container. At time \(t\) seconds after the water starts to flow, the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the container is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 300 - k V$$ where \(k\) is a constant.
4. Solve the differential equation to show that, according to the model, $$V = \frac { 300 } { k } + A \mathrm { e } ^ { - k t }$$ where \(A\) is a constant.
   (5) Given that the container is initially empty and that when \(t = 10\), the volume of water is increasing at a rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
5. find the exact value of \(k\).
6. Hence find, according to the model, the time taken for the volume of water in the container to reach 6 litres. Give your answer to the nearest second.
   8. Use proof by contradiction to prove that, for all positive real numbers \(x\) and \(y\), $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$ 9. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-24_632_734_214_607} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of a closed curve with parametric equations $$x = 5 \cos \theta \quad y = 3 \sin \theta - \sin 2 \theta \quad 0 \leqslant \theta < 2 \pi$$ The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis to form a solid of revolution.
7. Show that the volume, \(V\), of the solid of revolution is given by $$V = 5 \pi \int _ { \alpha } ^ { \beta } \sin ^ { 3 } \theta ( 3 - 2 \cos \theta ) ^ { 2 } \mathrm {~d} \theta$$ where \(\alpha\) and \(\beta\) are constants to be found.
8. Use the substitution \(u = \cos \theta\) and algebraic integration to show that \(V = k \pi\) where \(k\) is a rational number to be found. \includegraphics[max width=\textwidth, alt={}, center]{594542dd-ee2d-49b6-9fab-77b2d1a44f8c-28_2649_1889_109_178}