Two positive irrational numbers, xxx and yyy, have a rational sum and a rational product.
Jamie is trying to prove that x2+y2x^2 + y^2x2+y2 is rational.
Here is Jamie's proof:
Step 1: x2+y2=(x+y)2x^2 + y^2 = (x + y)^2x2+y2=(x+y)2
Step 2: x+yx + yx+y is rational, so (x+y)2(x + y)^2(x+y)2 is rational.
Step 3: Therefore, x2+y2x^2 + y^2x2+y2 is rational.
Identify Jamie's mistake.
Write down a correct version of the proof that x2+y2x^2 + y^2x2+y2 is rational.
Prove by contradiction that the sum of any rational number and any irrational number is irrational.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.