Question 1.1 In the first lecture, we looked at how Diophantus of Alexandria (c. 300 AD) parameterised all rational points R on the unit circle. Q R O P X Y

α θ

(a) Explain how the coordinates of any point on the unit circle can be parameterised as (x, y) = (cos θ, sin θ) , for some θ ∈ [0, 2π). (b) By setting t = tan α, give all the necessary details to show that the coordinates

of R(x, y) have parameterisation (x, y) = 1 1 + − t t2 2 , 1 + 2tt2 , t ∈ Q. (c) Express the angle α in terms of the angle θ, clearly stating any theorem(s) you may use.

(d) Using parts (a), (b) and (c), deduce a formula for tan θ in terms of tan (θ/2).

[50 marks] Please turn over Question 1.2 Find all integer solutions to 341x + 98y = 15

and determine the solution (x, y) with smallest positive x. [30 marks] Question 1.3 Prove that if r is a rational number, then r2 is also rational.

Is it true that if s is irrational, then s2 is irrational? Justify your answer.