1.1.3 Proof by contradiction (A-level only)

Expand and simplify $p + 3q)^2$ .

[1]

An electronics engineer claims that for any non-zero voltage ratio $v$ , the expression $\frac{9}{v}$ is always greater than or equal to 6.

The engineer tests two values: When $v = 9$ , $\frac{9}{9} = 10 \ge 6$ . When $v = 3$ , $\frac{9}{3} = 6 \ge 6$ .

Provide a counter-example to show that the engineer's claim is incorrect.

[1]

Given that $x$ and $y$ are positive real numbers such that $\neq 3y$ , use proof by contradiction to prove that:

\frac{x}{y} + \frac{9y}{x} > 6

[4]

1.1.3 Proof by contradiction (A-level only) Questions

Expand and simplify $p + 3q)^2$ .

[1]

An electronics engineer claims that for any non-zero voltage ratio $v$ , the expression $\frac{9}{v}$ is always greater than or equal to 6.

The engineer tests two values: When $v = 9$ , $\frac{9}{9} = 10 \ge 6$ . When $v = 3$ , $\frac{9}{3} = 6 \ge 6$ .

Provide a counter-example to show that the engineer's claim is incorrect.

[1]

Given that $x$ and $y$ are positive real numbers such that $\neq 3y$ , use proof by contradiction to prove that:

\frac{x}{y} + \frac{9y}{x} > 6

[4]