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Class 10MathematicsReal NumbersProof of irrationality of √2, √3, √5HardLevel 16
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In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q=3r). Why does this make the original assumption false?

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Answer and explanation

Correct answer: Because (\frac{p}{q}) was assumed in lowest form, but common factor (3) was found

Step 1: In the rational assumption, (\frac{p}{q}) was taken in lowest form. Step 2: (p=3k) and (q=3r) show common factor (3) in both. Step 3: This contradicts lowest form, so the rational assumption is false.

Related tags

Sqrt3 ProofRational AssumptionContradictionHard

Frequently asked questions

What is the correct answer to this question?

Because (\frac{p}{q}) was assumed in lowest form, but common factor (3) was found

Why is this the correct answer?

Step 1: In the rational assumption, (\frac{p}{q}) was taken in lowest form. Step 2: (p=3k) and (q=3r) show common factor (3) in both. Step 3: This contradicts lowest form, so the rational assumption is false.

Which subject and chapter does this question cover?

This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Proof of irrationality of √2, √3, √5.

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10 more MCQs
  1. 1Which assumption is taken first to prove that (\sqrt{2}) is irrational?
  2. 2If (\sqrt{2}=\frac{p}{q}), what is obtained after squaring both sides?
  3. 3From the equation (p^2=2q^2), what do we learn about (p^2)?
  4. 4If (p^2) is even, which conclusion about (p) is correct?
  5. 5If (p) is even, in which form can (p) be written?
  6. 6What conclusion is obtained by putting (p=2k) in (p^2=2q^2)?
  7. 7What is the final contradiction in the proof of (\sqrt{2})?
  8. 8What is the correct final conclusion of the proof of (\sqrt{2})?
  9. 9If (\sqrt{3}) is assumed rational, in which form is it written?
  10. 10If (\sqrt{3}=\frac{p}{q}), which equation is obtained after squaring?
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