N2020 P2 Q5

Shania and Tina are playing a game. Shania has a bag containing one green disc, \(r\)red discs and \(2r\) blue discs, where \(r>1\). A red disc is worth \(5\) points, a blue disc is worth \(2\) points and the green disc is worth \(0\) points. Tina takes two discs from the bag at random. Tina’s score is found by multiplying together the number of points for each of the two discs she takes.

1. State Tina’s possible scores.[1]
2. Show that the expectation of Tina’s score is \(\frac{27r-11}{3r+1}\), and find an expression for the variance.[7]
3. Given that the variance of Tina’s score is \(38\), find the value of \(r\).[2]