An engineer is studying two irrational vibration frequencies, f1f_1f1 and f2f_2f2 (measured in kHz). It is known that the difference f1−f2f_1 - f_2f1−f2 is a rational number and the product f1f2f_1 f_2f1f2 is also a rational number.
The engineer proposes the following proof to show that f12+f22f_1^2 + f_2^2f12+f22 is rational:
Step 1: f12+f22=(f1−f2)2f_1^2 + f_2^2 = (f_1 - f_2)^2f12+f22=(f1−f2)2
Step 2: Since f1−f2f_1 - f_2f1−f2 is rational, (f1−f2)2(f_1 - f_2)^2(f1−f2)2 is rational.
Step 3: Therefore, f12+f22f_1^2 + f_2^2f12+f22 is rational.
(i) Identify the error in the engineer's proof.
(ii) Write down a correct proof that f12+f22f_1^2 + f_2^2f12+f22 is rational.
Prove by contradiction that the difference x−yx - yx−y is irrational, where xxx is any irrational number and yyy is any rational number.
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.