Year 10 extension

1. How many digits does the number 4¹⁰ × 5¹⁸ have when it is written out in full?
2. A shop in Nelson raises the price of a kayak by 20%. A week later it wants to bring the price back to exactly what it was before. By what percentage must it reduce the new price?
3. Aroha leaves her tent and walks 7 km north, then 4 km east, then 1 km south, then another 4 km east. How far is she now from her tent, in a straight line?
4. Without a calculator, work out 2027² − 2023².
5. A straight line passes through the points (1, 4) and (3, 10). Where does it cross the x-axis?
6. Mere has a bag of 5 lollies: 3 red and 2 blue. She takes out two lollies at random, one after the other, without putting the first one back. What is the probability that both lollies are the same colour?
7. Two water tanks on a Waikato farm are exactly similar in shape. The smaller tank is 1.2 m tall and holds 400 litres. The larger tank is 1.8 m tall. How much does the larger tank hold?
8. A number x satisfies x + 1/x = 5. What is the value of x² + 1/x²?
9. A school running track has two straight sections joined by two semicircular ends. Ana runs on the inside line. Kahu runs in the next lane, always exactly 1.2 m further out than Ana. How much further than Ana does Kahu run in one full lap?
10. Five positive whole numbers have a mean of 7 and a median of 8. Their mode is 2, and no other value appears as often as 2 does. What is the largest possible value of the biggest number?