Choose the correct order in the proof of √3.
Answer and explanation
Correct answer: Rational assumption → p² = 3q² → p divisible by 3 → q divisible by 3 → contradiction
The correct answer is A. Begin by assuming that √3 is rational and express it as p/q in lowest terms, with p and q coprime and q nonzero. Squaring produces p² = 3q², so p² is divisible by 3. Since 3 is prime, divisibility of p² by 3 implies that p itself is divisible by 3; put p = 3k. Substituting and cancelling gives q² = 3k², so q is also divisible by 3. This contradicts the assertion that p and q have no common factor. The proof therefore establishes irrationality. The other options do not derive the required divisibility chain and cannot produce a valid contradiction.
Frequently asked questions
What is the correct answer to this question?
Rational assumption → p² = 3q² → p divisible by 3 → q divisible by 3 → contradiction
Why is this the correct answer?
The correct answer is A. Begin by assuming that √3 is rational and express it as p/q in lowest terms, with p and q coprime and q nonzero. Squaring produces p² = 3q², so p² is divisible by 3. Since 3 is prime, divisibility of p² by 3 implies that p itself is divisible by 3; put p = 3k. Substituting and cancelling gives q² = 3k², so q is also divisible by 3. This contradicts the assertion that p and q have no common factor. The proof therefore establishes irrationality. The other options do not derive the required divisibility chain and cannot produce a valid contradiction.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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