Proof of irrationality of square root 2 and square root 3

Question 1MediumLevel 17

What is the main reason for assuming (\sqrt{3}=\frac{p}{q}) in lowest form?

ATo prove (p=q)BSo that (p) and (q) are coprimeCSo that (q=0) can be writtenDSo that decimal can be found

Question 2MediumLevel 17

In contradiction method, which initial assumption is taken for (\sqrt{2})?

A(\sqrt{2}) is irrationalB(\sqrt{2}) is an integerC(\sqrt{2}) is rationalD(\sqrt{2}) is zero

Question 3MediumLevel 17

In the proof of (\sqrt{3}), why can we write (p=3k) from (p^2=3q^2)?

ABecause (p=q)BBecause (q=0)CBecause (p) is evenDBecause (p^2) is divisible by (3), so (p) is divisible by (3)

Question 4MediumLevel 17

While proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?

ABoth \(p\) and \(q\) are divisible by 3.BOnly \(p\) is even.C\(q\) is greater than \(p\).DBoth \(p\) and \(q\) are prime.

Question 5MediumLevel 17

In the proof of (\sqrt{3}), after getting (q^2=3k^2), what is the next conclusion toward final contradiction?

A(q) is evenB(q) is divisible by (3)C(q=0)D(p=q)

Question 6MediumLevel 17

A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?

A\(p\) is divisible by 3, but no conclusion can be drawn about \(q\).B\(q\) is divisible by 3, but no conclusion can be drawn about \(p\).C\(p\) is divisible by 3; on putting \(p=3k\), \(q\) is also proved divisible by 3, contradicting coprimality.DBoth \(p\) and \(q\) must be odd integers.

Question 7MediumLevel 17

Which statement is a wrong conclusion in the proof of \(\sqrt{3}\)?

AFrom \(p^2=3q^2\), \(p^2\) is divisible by \(3\)BIf \(p^2\) is divisible by \(3\), then \(p\) is divisible by \(3\)CEven if both \(p\) and \(q\) are divisible by \(3\), they are coprimeDGetting common factor \(3\) in both is a contradiction

Question 8MediumLevel 17

Which of the following numbers has an irrational positive square root?

A16B36C50D64

Question 9MediumLevel 17

In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q^2=3k^2). What will be the final conclusion?

A(\sqrt{3}) is rationalB(\sqrt{3}) is zeroC(\sqrt{3}) is irrationalD(\sqrt{3}) is an integer

Question 10MediumLevel 17

Which option shows the correct logical chain in the proof of (√2)?

A(m²=2n²), (m) even, (m=2r), (n) evenB(m²=3n²), (m=3r), (n) divisible by (3)C(m=n), (n=0), contradictionD(m) odd, (n) odd, conclusion

Question 11MediumLevel 17

Which option shows the correct logical chain in the proof of (√3)?

A(p²=2q²), (p) even, (q) evenB(p=q), (q=0), conclusionC(p²=3q²), (p) divisible by (3), (p=3k), (q) divisible by (3)D(p) negative, (q) negative, contradiction

Question 12MediumLevel 17

In a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?

Aq cannot be a multiple of 3 because q² is a squareBk and q must be coprimeCp² must be equal to q²Dq is divisible by 3; therefore p and q have a common factor 3

Question 13MediumLevel 17

A student says, “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” Which step is needed to complete this argument?

AShowing that \(\sqrt{3}\) is positiveBShowing that 3 lies between 1 and 4CWriting 3 in decimal formDAssuming \(\sqrt{3}=\frac{m}{n}\) in lowest terms and deriving a contradiction

Question 14MediumLevel 17

In the proof of (\sqrt{2}), which condition is necessary along with assuming (\sqrt{2}=\frac{m}{n})?

A(m,n) coprime and (n\neq0)BBoth (m,n) evenC(n=0)D(m=n)

Question 15MediumLevel 17

In a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?

AHe must also prove that \(n\) is even, contradicting that \(m\) and \(n\) are coprime.BHe must prove that \(m\) is odd.CHe must prove that \(n\) is odd.DHe must write the decimal expansion of \(\sqrt{2}\) up to at least 20 places.

Question 16MediumLevel 17

If \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?

A\(p\) is divisible by 3, but no conclusion can be drawn about \(q\).BBoth \(p\) and \(q\) are divisible by 3.C\(q\) is divisible by 3, but \(p\) is not divisible by 3.DNeither \(p\) nor \(q\) is divisible by 3.

Question 17MediumLevel 17

Which option gives the correct reasoning from (p²) to (p) in the proof of (√3)?

AIf (p²) is divisible by (3), then (p) is divisible by (2)BIf (p²) is divisible by (3), then (p) is divisible by (3)CIf (p²) is divisible by (3), then (p=0)DIf (p²) is divisible by (3), then (p=q)

Question 18MediumLevel 17

On what basis does the final conclusion in irrationality of (√2) come?

ARational assumption makes both (m) and (n) evenBDecimal terminatesC(m=n) is provedD(n=0) is proved

Question 19MediumLevel 17

On what basis does the final conclusion in irrationality of (\sqrt{3}) come?

ADecimal terminatesBRational assumption makes both (p) and (q) divisible by (3)C(p=q) is provedD(q=0) is proved

Question 20MediumLevel 17

In the proof of (\sqrt{2}), if a student writes directly from (m^2=2n^2) that (n) is even, what is the correct comment?

AIt is correct because (n) is even firstBIt is incomplete because (m) must be proved even firstCIt is correct because (n=0)DIt is correct because (m=n)

Proof of irrationality of square root 2 and square root 3

Quiz this set

What is the main reason for assuming (\sqrt{3}=\frac{p}{q}) in lowest form?

In contradiction method, which initial assumption is taken for (\sqrt{2})?

In the proof of (\sqrt{3}), why can we write (p=3k) from (p^2=3q^2)?

While proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?

In the proof of (\sqrt{3}), after getting (q^2=3k^2), what is the next conclusion toward final contradiction?

A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?

Which statement is a wrong conclusion in the proof of \(\sqrt{3}\)?

Which of the following numbers has an irrational positive square root?

In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q^2=3k^2). What will be the final conclusion?

Which option shows the correct logical chain in the proof of (√2)?

Which option shows the correct logical chain in the proof of (√3)?

In a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?

A student says, “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” Which step is needed to complete this argument?

In the proof of (\sqrt{2}), which condition is necessary along with assuming (\sqrt{2}=\frac{m}{n})?

In a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?

If \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?

Which option gives the correct reasoning from (p²) to (p) in the proof of (√3)?

On what basis does the final conclusion in irrationality of (√2) come?

On what basis does the final conclusion in irrationality of (\sqrt{3}) come?

In the proof of (\sqrt{2}), if a student writes directly from (m^2=2n^2) that (n) is even, what is the correct comment?

Quiz result

Proof of irrationality of square root 2 and square root 3

Quiz this set

What is the main reason for assuming (\sqrt{3}=\frac{p}{q}) in lowest form?

In contradiction method, which initial assumption is taken for (\sqrt{2})?

In the proof of (\sqrt{3}), why can we write (p=3k) from (p^2=3q^2)?

While proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?

In the proof of (\sqrt{3}), after getting (q^2=3k^2), what is the next conclusion toward final contradiction?

A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?

Which statement is a wrong conclusion in the proof of \(\sqrt{3}\)?

Which of the following numbers has an irrational positive square root?

In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q^2=3k^2). What will be the final conclusion?

Which option shows the correct logical chain in the proof of (√2)?

Which option shows the correct logical chain in the proof of (√3)?

In a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?

A student says, “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” Which step is needed to complete this argument?

In the proof of (\sqrt{2}), which condition is necessary along with assuming (\sqrt{2}=\frac{m}{n})?

In a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?

If \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?

Which option gives the correct reasoning from (p²) to (p) in the proof of (√3)?

On what basis does the final conclusion in irrationality of (√2) come?

On what basis does the final conclusion in irrationality of (\sqrt{3}) come?

In the proof of (\sqrt{2}), if a student writes directly from (m^2=2n^2) that (n) is even, what is the correct comment?

Quiz result

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