SPS SPS FM (SPS FM) 2020 May

1. Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)
[0pt] [4 marks]

2. A plane has equation \(\mathbf { r } \cdot \left[ \begin{array} { l } 1
1
1 \end{array} \right] = 7\)
A line has equation \(\mathbf { r } = \left[ \begin{array} { l } 2
0
1 \end{array} \right] + \mu \left[ \begin{array} { l } 1
0
1 \end{array} \right]\)
Calculate the acute angle between the line and the plane.
Give your answer to the nearest \(0.1 ^ { \circ }\)
[0pt] [3 marks]

3. Show that $$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$ where \(m\) and \(n\) are integers.
[0pt] [3 marks]

4.

1. If \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ [3 marks]
2. Express \(\sin ^ { 5 } \theta\) in terms of \(\sin 5 \theta , \sin 3 \theta\) and \(\sin \theta\)
   [0pt] [4 marks]
3. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }$$

5. The equation \(z ^ { 3 } + k z ^ { 2 } + 9 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. 1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$
2. (ii) Show that $$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$

6. The points \(A , B\) and \(C\) have coordinates \(A ( 4,5,2 ) , B ( - 3,2 , - 4 )\) and \(C ( 2,6,1 )\)
Use a vector product to show that the area of triangle \(A B C\) is \(\frac { 5 \sqrt { 11 } } { 2 }\)
[0pt] [4 marks]

7. Prove by induction that \(\mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n\) is divisible by 6 for all integers \(n \geq 1\)

8. Let $$S _ { n } = \sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }$$ where \(n \geq 1\) Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where \(a\), \(b\) and \(c\) are integers.
[0pt] [6 marks]

9.
\includegraphics[max width=\textwidth, alt={}, center]{ab2949b2-11f2-4682-ab0c-25ecee2d665a-4_268_648_1169_623} Two tanks, \(A\) and \(B\), each have a capacity of 800 litres. At time \(t = 0\) both tanks are full of pure water. When \(t > 0\), water flows in the following ways:

• Water with a salt concentration of \(\mu\) grams per litre flows into tank \(A\) at a constant rate
• Water flows from tank \(A\) to tank \(B\) at a rate of 16 litres per minute
• Water flows from tank \(B\) to tank \(A\) at a rate of \(r\) litres per minute
• Water flows out of tank \(B\) through a waste pipe
• The amount of water in each tank remains at 800 litres.

This system is represented by the coupled differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 36 - 0.02 x + 0.005 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.02 x - 0.02 y \end{aligned}$$ Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\).

10. \begin{figure}[h] \end{figure} A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
[0pt] [The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

1. Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\).

11. A particle, \(P\), of mass 0.4 kg is moving along the positive \(x\)-axis, in the positive \(x\) direction under the action of a single force. At time \(t\) seconds, \(t > 0 , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force is acting in the direction of \(x\) increasing and has magnitude \(\frac { k } { v }\) newtons, where \(k\) is a constant. At \(x = 3 , v = 2\) and at \(x = 6 , v = 2.5\)

1. Show that \(v ^ { 3 } = \frac { 61 x + 9 } { 24 }\) The time taken for the speed of \(P\) to increase from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
2. Use algebraic integration to show that \(T = \frac { 81 } { 61 }\)

12.
[0pt] [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(A\) has mass 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\) The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\)

1. Find the velocity of \(A\) immediately after the collision.
2. Find the magnitude of the impulse received by \(A\) in the collision.
3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.

13. Six women and five men stand in a line for a photo.

1. In how many arrangements will all the men stand next to each other and all the women stand next to each other?
2. In how many arrangements will all the men be apart?

14. Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.

1. Calculate the value of Spearman's rank correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to examine whether there is any association between time taken for the run and number of push-ups achieved. [4]
3. Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient. State, with a reason, which test is more appropriate for these data.

15. A researcher at a large company thinks that there may be some relationship between the numbers of working days lost due to illness per year and the ages of the workers in the company. The researcher selects a random sample of 190 workers. The ages of the workers and numbers of days lost for a period of 1 year are summarised below.

1. Carry out a test at the \(1 \%\) significance level to investigate whether the researcher's belief appears to be true. Your working should include a table showing the contributions of each cell to the test statistic.
2. For the 'Over 50 ' age group, comment briefly on how the working days lost compare with what would be expected if there were no association.