1.01b Logical connectives: congruence, if-then, if and only if

29 questions

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OCR MEI C1 Q4
5 marks Moderate -0.3
  1. Prove that 12 is a factor of \(3n^2 + 6n\) for all even positive integers \(n\). [3]
  2. Determine whether 12 is a factor of \(3n^2 + 6n\) for all positive integers \(n\). [2]
AQA AS Paper 1 Specimen Q2
1 marks Moderate -0.8
Consider the two statements, A and B, below. A: \(x^2 - 6x + 8 > 0\) B: \(x > 4\) Choose the most appropriate option below. Circle your answer. [1 mark] \(A \Rightarrow B\) \(A \Leftarrow B\) \(A \Leftrightarrow B\) There is no connection between A and B
AQA Paper 2 2018 June Q1
1 marks Easy -2.5
Which of these statements is correct? Tick one box. [1 mark] \(x = 2 \Rightarrow x^2 = 4\) \(x^2 = 4 \Rightarrow x = 2\) \(x^2 = 4 \Leftrightarrow x = 2\) \(x^2 = 4 \Rightarrow x = -2\)
AQA Paper 2 Specimen Q7
4 marks Standard +0.3
A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers. He claims that this will always be true. He attempts to prove his claim as follows: Step 1: Check first few cases \(3 + 5 = 8\) and \(8 = 3^2 - 1^2\) \(5 + 7 = 12\) and \(12 = 4^2 - 2^2\) \(7 + 9 = 16\) and \(16 = 5^2 - 3^2\) Step 2: Use pattern to predict and check a large example \(101 + 103 = 204\) subtract 1 and divide by 2 for the first number Add 1 and divide by two for the second number \(52^2 - 50^2 = 204\) it works! Step 3: Conclusion The first few cases work and there is a pattern, which can be used to predict larger numbers. Therefore, it must be true for all consecutive odd numbers.
  1. Explain what is wrong with the student's "proof". [1 mark]
  2. Prove that the student's claim is correct. [3 marks]