Counter example to disprove statement

A question is this type if and only if it asks to show a statement is false by finding a specific counter example that contradicts the claim (e.g., 'if p is prime then 2p+1 is prime' or 'n² + 3n + 1 is always prime').

35 questions · Moderate -0.6

1.01c Disproof by counter example
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OCR MEI C3 Q12
2 marks Moderate -0.8
Prove that the following statement is false. For all integers \(n\) greater than or equal to 1, \(n^2 + 3n + 1\) is a prime number. [2]
AQA AS Paper 1 2019 June Q2
1 marks Easy -1.2
Dan believes that for every positive integer \(n\), at least one of \(2^n - 1\) and \(2^n + 1\) is prime. Which value of \(n\) shown below is a counter example to Dan's belief? Circle your answer. [1 mark] \(n = 3\) \(n = 4\) \(n = 5\) \(n = 6\)
AQA AS Paper 2 2024 June Q5
3 marks Moderate -0.8
A student suggests that for any positive integer \(n\) the value of the expression $$4n^2 + 3$$ is always a prime number. Prove that the student's statement is false by finding a counter example. Fully justify your answer. [3 marks]
Edexcel AS Paper 1 Q2
Easy -1.8
Use a counter example to show that the following statement is false. "\(n^2 - n + 5\) is a prime number, for \(2 \leq n \leq 6\)"
OCR MEI Paper 2 2022 June Q5
3 marks Standard +0.3
Tom conjectures that if \(n\) is an odd number greater than 1, then \(2^n - 1\) is prime. Find a counter example to disprove Tom's conjecture. [3]
WJEC Unit 1 2022 June Q6
5 marks Standard +0.3
In each of the two statements below, \(x\) and \(y\) are real numbers. One of the statements is true while the other is false. A: \(x^2 + y^2 \geqslant 2xy\), for all real values of \(x\) and \(y\). B: \(x + y \geqslant 2\sqrt{xy}\), for all real values of \(x\) and \(y\).
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]
WJEC Unit 1 2023 June Q8
3 marks Easy -1.8
Show, by counter example, that the following statement is false. "For all positive integer values of \(n\), \(n^2 + 1\) is a prime number." [3]
WJEC Unit 1 Specimen Q6
5 marks Standard +0.3
In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true while the other is false. A: Given that \((2c + 1)^2 = (2d + 1)^2\), then \(c = d\). B: Given that \((2c + 1)^3 = (2d + 1)^3\), then \(c = d\).
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [5]
SPS SPS SM Pure 2023 September Q13
6 marks Standard +0.8
Prove or disprove each of the following statements:
  1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
  2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
  3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]
OCR AS Pure 2017 Specimen Q6
5 marks Standard +0.3
  1. A student suggests that, for any prime number between 20 and 40, when its digits are squared and then added, the sum is an odd number. For example, 23 has digits 2 and 3 which gives \(2^2 + 3^2 = 13\), which is odd. Show by counter example that this suggestion is false. [2]
  2. Prove that the sum of the squares of any three consecutive positive integers cannot be divided by 3. [3]