Standard +0.5 This is a standard proof by contradiction following the classic template for proving √n is irrational (assume √7 = p/q in lowest terms, square both sides, show p² divisible by 7 implies p divisible by 7, substitute to show q² also divisible by 7, contradiction). While it requires understanding proof structure and divisibility arguments, it's a well-rehearsed technique that students practice extensively, making it slightly above average difficulty but not particularly challenging.
Assume that \(\sqrt{7}\) is rational ie \(\sqrt{7} = \frac{a}{b}\), where \(a\) and \(b\) have no common factors; \(b\sqrt{7} = a\), so \(7b^2 = a^2\); So \(a^2\) must be a multiple of 7, which means that \(a\) is a multiple of 7 as well, so \(a = 7k\); \(7b^2 = (7k)^2 \Rightarrow 7b^2 = 49k^2 \Rightarrow b^2 = 7k^2\); This implies that \(b\) is a multiple of 7, but it was assumed at start that \(a\) and \(b\) had no common factors, so this contradicts initial assumption.; Hence \(\sqrt{7}\) cannot be written as \(\frac{a}{b}\) so it is irrational.
E1, M1, E1, M1, E1
Proof must start with an assumption for contradiction; Rearrange and square both sides; Identify that \(a = 7k\); Substitute \(a = 7k\) and simplify; Conclude appropriately; Condone not stating that \(a\) and \(b\) have no common factors; Must have stated at start that \(a\) and \(b\) have no common factors
| Assume that $\sqrt{7}$ is rational ie $\sqrt{7} = \frac{a}{b}$, where $a$ and $b$ have no common factors; $b\sqrt{7} = a$, so $7b^2 = a^2$; So $a^2$ must be a multiple of 7, which means that $a$ is a multiple of 7 as well, so $a = 7k$; $7b^2 = (7k)^2 \Rightarrow 7b^2 = 49k^2 \Rightarrow b^2 = 7k^2$; This implies that $b$ is a multiple of 7, but it was assumed at start that $a$ and $b$ had no common factors, so this contradicts initial assumption.; Hence $\sqrt{7}$ cannot be written as $\frac{a}{b}$ so it is irrational. | E1, M1, E1, M1, E1 | Proof must start with an assumption for contradiction; Rearrange and square both sides; Identify that $a = 7k$; Substitute $a = 7k$ and simplify; Conclude appropriately; Condone not stating that $a$ and $b$ have no common factors; Must have stated at start that $a$ and $b$ have no common factors |