1. Let \(C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos ( r \theta )\). Show that \(C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos ( 10 \theta )\).
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2. During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by

$$\frac { d y } { d t } = 0.3 x - 0.2 y \quad \text { and } \quad \frac { d z } { d t } = 0.2 y + 0.1 x$$ where \(x , y\) and \(z\) are the amounts in kg of \(X , Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.

1. Show that \(x = A e ^ { - 0.4 t }\), stating the value of \(A\).
2. Show that \(\frac { d x } { d t } + \frac { d y } { d t } + \frac { d z } { d t } = 0\). Comment on this result in the context of the industrial process.
3. Express \(y\) in terms of \(t\).
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