Lesson A1

A1 Mathematical proof, disproof and contradiction Quiz: AQA Maths, Unit 1

20 questions

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Lesson A1, Mathematical proof, disproof and contradiction: 20 multiple choice questions for the AQA Maths (7357), Unit 1: Proof, written with Revision Ninja.

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The 20 questions

  1. Which method of proof checks every case in a finite list of possibilities?

    • Proof by exhaustion
    • Proof by deduction from axioms
    • Disproof by counter-example
    • Proof by contradiction

  2. What is the purpose of a counter-example?

    • To show that a statement holds only for even numbers
    • To show that a general statement is false
    • To show that the converse of a statement is always true
    • To show that a general statement is true for every case

  3. In a proof by contradiction, what is assumed at the start?

    • The statement to be proved is false
    • The statement to be proved is true for the first case only
    • The statement holds for all integers greater than 1
    • The converse of the statement is also true

  4. Which number has a classic proof of irrationality by contradiction?

    • 22/7
    • 0.75
    • sqrt(9)
    • sqrt(2)

  5. In the proof that sqrt(2) is irrational, what contradiction is reached?

    • p and q are both odd, so their sum is even
    • p is larger than q, so sqrt(2) is greater than 2
    • p and q are both even, but p/q was in lowest terms
    • q equals zero, so the fraction is undefined

  6. What is the standard contradiction method for proving there are infinitely many primes?

    • Show that the primes form an arithmetic sequence
    • Assume every prime is odd and count the odd numbers
    • Assume finitely many primes and build a number not divisible by any of them
    • Check all primes up to one million by exhaustion

  7. What is the smallest positive integer n that disproves 'n^2 + n + 1 is prime for every positive integer n'?

    • 3
    • 4
    • 5
    • 2

  8. What is the smallest non-negative integer n for which n^2 + n + 41 is not prime?

    • 40
    • 39
    • 10
    • 20

  9. Which statement disproves 'every prime number is odd'?

    • 3 is prime and odd, so the claim holds
    • All primes greater than 5 are odd
    • 2 is prime and it is even
    • 7 is prime and odd, so the claim holds

  10. Which algebraic set-up correctly proves that the sum of an odd and an even integer is odd?

    • Let the odd integer be m and the even integer be m+1, so the sum is 2m+1
    • Let the integers be 2m and 2n, so the sum 2(m+n) is odd
    • Let the odd integer be 2m+1 and the even integer be 2n, so the sum is 2(m+n)+1
    • Let the integers be m and n, so the sum m+n is always odd

  11. To prove that n^2 is at most 4n for each integer n from 1 to 4 by exhaustion, what is done?

    • The statement is assumed false and a contradiction found
    • Each of n = 1, 2, 3 and 4 is checked individually
    • Only n = 4 is checked, since it is the largest
    • A general algebraic argument is given for all n

  12. Which is a valid disproof of 'the sum of two irrational numbers is always irrational'?

    • pi + 2 is irrational, so the claim holds
    • 1/2 + 1/3 = 5/6, which is irrational
    • sqrt(2) + sqrt(3) is irrational, so the claim holds
    • sqrt(2) + (-sqrt(2)) = 0, which is rational

  13. Which assumption starts a proof by contradiction that sqrt(3) is irrational?

    • sqrt(3) = p/q, with p and q integers having no common factor
    • sqrt(3) is a terminating decimal with finitely many digits
    • sqrt(3) = p/q, with p and q integers sharing a common factor
    • p and q are both even integers

  14. Why is x = -3 a valid counter-example to 'for all real x, sqrt(x^2) = x'?

    • sqrt(9) = 3, but x = -3, so the two sides differ
    • sqrt(0) = 0 and x = 0, so the two sides agree
    • sqrt(4) = 2 and x = 2, so the statement is confirmed
    • sqrt(9) = 3 and x = 3, so the two sides agree

  15. Which quadrilateral is a counter-example to 'every quadrilateral with equal diagonals is a rectangle'?

    • A square
    • An isosceles trapezium that is not a rectangle
    • A kite with unequal diagonals
    • A rhombus that is not a square

  16. To prove that if n^2 is even then n is even, by contradiction, which assumption is made?

    • That n is even but n^2 is odd
    • That n is a prime number
    • That n^2 is odd
    • That n is odd

  17. In proving x + 1/x >= 2 for x > 0 by contradiction, which step yields the contradiction?

    • Multiplying the assumed inequality by x gives (x-1)^2 < 0
    • Substituting x = 1 gives 2 < 2, which is true
    • Multiplying by x gives x^2 + 1 > 2x, which is always true
    • Assuming x = 0 makes 1/x undefined, so x > 0 fails

  18. Which statement about a deductive proof is correct?

    • Each line is a guess that is verified afterwards
    • Each line follows logically from the given assumptions or from earlier established results
    • Each line is checked by testing a numerical example
    • Each line is assumed true until a contradiction appears

  19. Which is the correct conclusion of a valid disproof of a universal statement?

    • The universal statement is true for all integers except one
    • The converse of the universal statement is false
    • The universal statement is false
    • The universal statement is true but still unproven

  20. Which statement about the proof that sqrt(2) is irrational is correct?

    • It is a disproof using a counter-example such as 1.414
    • It is a proof by exhaustion over the rational numbers
    • It is a proof by contradiction that relies on writing the number in lowest terms
    • It is a proof by induction on the decimal digits

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