Expand and simplify (p+3q)2(p + 3q)^2(p+3q)2.
An electronics engineer claims that for any non-zero voltage ratio vvv, the expression v+9vv + \frac{9}{v}v+v9 is always greater than or equal to 6.
The engineer tests two values: When v=9v = 9v=9, 9+99=10≥69 + \frac{9}{9} = 10 \ge 69+99=10≥6. When v=3v = 3v=3, 3+93=6≥63 + \frac{9}{3} = 6 \ge 63+39=6≥6.
Provide a counter-example to show that the engineer's claim is incorrect.
Given that xxx and yyy are positive real numbers such that x≠3yx \neq 3yx=3y, use proof by contradiction to prove that:
xy+9yx>6 \frac{x}{y} + \frac{9y}{x} > 6 yx+x9y>6368 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.