Prove that $2 + 3\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.
Step-by-Step Solution
Key Concept: $2 + 3\sqrt{5} = a/b \Rightarrow \sqrt{5} = (a - 2b)/(3b)$.
Let $2 + 3\sqrt{5} = a/b$ (rational). [0.5 Mark]
$3\sqrt{5} = a/b - 2 = (a - 2b)/b \Rightarrow \sqrt{5} = (a - 2b)/(3b)$. [1.5 Marks]
RHS is rational, so $\sqrt{5}$ is rational, contradicting given fact. Hence $2 + 3\sqrt{5}$ is irrational. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Assumption setup: 0.5 Mark
Isolating $\sqrt{5}$: 1.5 Marks
Contradiction statement: 1.0 Mark
Correct Answer: