13 The equation \(z ^ { 3 } + k z ^ { 2 } + 9 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
13
- Show that
$$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$
13
- (ii) Show that
$$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$
13
- The equation \(9 z ^ { 3 } - 40 z ^ { 2 } + r z + s = 0\) has roots \(\alpha \beta + \gamma , \beta \gamma + \alpha\) and \(\gamma \alpha + \beta\).
13
- Show that
$$k = - \frac { 40 } { 9 }$$
Question 13 continues on the next page
13
- (ii) Without calculating the values of \(\alpha , \beta\) and \(\gamma\), find the value of \(s\).
Show working to justify your answer.
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A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram.
The tube is filled with oil.
When the compression of the spring is \(\varepsilon\) metres, the thrust in the spring is \(9 m \varepsilon\) newtons.
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-24_506_250_721_895}
The mass is held at rest in a position where the compression of the spring is \(\frac { 20 } { 9 }\) metres.
The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6 m v\) newtons to act on the mass, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the mass.
At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.