N2009 P2 Q7

A company buys \(p\%\) of its electronic components from supplier \(A\) and the remaining \((100 - p)\%\) from supplier \(B\). The probability that a randomly chosen component supplied by \(A\) is faulty is \(0.05\). The probability that a randomly chosen component supplied by \(B\) is faulty is \(0.03\).

1. Given that \(p = 25\), find the probability that a randomly chosen component is faulty.[2]
2. For a general value of \(p\), the probability that a randomly chosen component that is faulty was supplied by \(A\) is denoted by \(\mathrm{f}(p)\). Show that \(\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}\). Prove by differentiation that \(\mathrm{f}\) is an increasing function for \(0 \leq p \leq 100\), and explain what this statement means in the context of the question.[6]