Five numbers from their mean, median, mode and range

Five positive integers have mean \(6\), median \(5\), a single mode of \(3\) and range \(9\).

1. Find the five integers, explaining why there is only one possible set.[3]
2. A sixth integer \(k\) is added so that the mean becomes \(7\). Find \(k\) and the new median.[2]
3. Show that the standard deviation increases when \(k\) is added.[2]