Contradiction proof of irrationality

A question is this type if and only if it asks to prove that a specific number (e.g., √7, √3, log₂3) is irrational using proof by contradiction, typically assuming it can be written as p/q.

10 questions · Standard +0.4

1.01d Proof by contradiction

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Edexcel P4 2023 June Q7

4 marks Moderate -0.3

Use proof by contradiction to prove that \(\sqrt { 7 }\) is irrational.
(You may assume that if \(k\) is an integer and \(k ^ { 2 }\) is a multiple of 7 then \(k\) is a multiple of 7 )

OCR C3 Q5

9 marks Standard +0.8

(i) Find, as natural logarithms, the solutions of the equation

$$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$ (ii) Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.

Edexcel Paper 2 Specimen Q14

8 marks Standard +0.3

(i) Kayden claims that

$$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
(ii) Prove that \(\sqrt { 3 }\) is an irrational number.

OCR MEI Further Extra Pure 2024 June Q5

4 marks Standard +0.8

5 In this question you may assume that if \(p\) and \(q\) are distinct prime numbers and \(\mathbf { p } ^ { \alpha } = \mathbf { q } ^ { \beta }\) where \(\alpha , \beta \in \mathbb { Z }\), then \(\alpha = 0\) and \(\beta = 0\).

Prove that it is not possible to find \(a\) and \(b\) for which \(\mathrm { a } , \mathrm { b } \in \mathbb { Z }\) and \(3 = 2 ^ { \frac { \mathrm { a } } { \mathrm { b } } }\).
Deduce that \(\log _ { 2 } 3 \notin \mathbb { Q }\).

OCR H240/01 2018 March Q6

5 marks Standard +0.5

6 Prove by contradiction that \(\sqrt { 7 }\) is irrational.

WJEC Unit 3 2019 June Q15

Use proof by contradiction to show that \(\sqrt { 6 }\) is irrational.

AQA Paper 3 2018 June Q10

10 marks Standard +0.8

Prove by contradiction that \(\sqrt[3]{2}\) is an irrational number. [7 marks]

SPS SPS FM 2020 October Q6

5 marks Moderate -0.3

Prove by contradiction that \(\sqrt{7}\) is irrational. [5]

SPS SPS SM Pure 2022 June Q15

6 marks Standard +0.8

Prove that $$n - 1 \text{ is divisible by } 3 \Rightarrow n^3 - 1 \text{ is divisible by } 9$$ [3 marks]
Show that if \(\log_2 3 = \frac{p}{q}\), then $$2^p = 3^q.$$ Use proof by contradiction to prove that \(\log_2 3\) is irrational. [3 marks]

SPS SPS SM Pure 2023 June Q14

6 marks Standard +0.3

Prove that the sum of the squares of 2 consecutive odd integers is always 2 more than a multiple of 8 [3]
Use proof by contradiction to show that \(\log_2 5\) is irrational. [3]