CAIE S2 (Statistics 2) 2020 November

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.
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4
\includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).