Question 1The logic of arguments
TMUA Logic of Arguments — Practice Questions by Topic
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- ArgThe logic of arguments
- 113 questions113 on Paper 2 only
- 9 free solutionsThe rest show the correct letter only
Covers: necessary and sufficient conditions, counterexamples, quantifiers, which of several statements must be true.
What this topic tests
The logic of arguments covers necessary and sufficient conditions, counterexamples, quantifiers, and which of several statements must be true. It is Section 2, with proof and with identifying errors.
How it is assessed
This topic is examined on Paper 2 only. Each question has five options. Paper 2 is 20 questions in 75 minutes, with no calculator.
This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.
Key methods
Necessary and sufficient
p is sufficient for q when p forces q. p is necessary for q when q cannot hold without p. “If p then q” says that p is sufficient for q, and that q is necessary for p.
Quantifiers
The negation of “for every x, P(x)” is “there exists an x such that not P(x)”. It is not “for every x, not P(x)”.
Common mistakes
Negating a quantifier by keeping “every”
“Not every cat fears water” means some cat does not. It does not mean no cat does.
A counterexample that misses the hypothesis
The hypothesis has to hold. A pair that already fails the “if” part does not refute the implication.
Worked example
This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.
Consider the statement: if a whole number is a multiple of 6, then it is even. Which option is the contrapositive of this statement?
Answer: B. If a whole number is odd, then it is not a multiple of 6.
Worked solution. The statement has the form “if P, then Q”, with P meaning “multiple of 6” and Q meaning “even”. The contrapositive is “if not Q, then not P”. Not even means odd, and not a multiple of 6 stays as it is. So the contrapositive is: if a whole number is odd, then it is not a multiple of 6.
Why the other options look right. A is the converse, which swaps the hypothesis and the conclusion. C is the inverse, which negates both parts and keeps their order. D keeps the hypothesis and negates only the conclusion. E swaps the two parts but negates only “even”, forgetting to negate “a multiple of 6”.
Paper 2 questions
Paper 2 is Mathematical Reasoning. This topic is examined on Paper 2 only.
Q1. Consider the statement: if a whole number is a multiple of 6, then it is even. Which option is the contrapositive of this statement?
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Answer and worked solution
Answer: B. If a whole number is odd, then it is not a multiple of 6.
Worked solution. The statement has the form “if P, then Q”, with P meaning “multiple of 6” and Q meaning “even”. The contrapositive is “if not Q, then not P”. Not even means odd, and not a multiple of 6 stays as it is. So the contrapositive is: if a whole number is odd, then it is not a multiple of 6.
Why the other options look right. A is the converse, which swaps the hypothesis and the conclusion. C is the inverse, which negates both parts and keeps their order. D keeps the hypothesis and negates only the conclusion. E swaps the two parts but negates only “even”, forgetting to negate “a multiple of 6”.
Q2. Consider the claim: if a positive integer is divisible by 9, then it is divisible by 6. Which of the following is a counterexample? I: 9. II: 18. III: 12.
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Answer and worked solution
Answer: B. I only
Worked solution. A counterexample must be divisible by 9 and not divisible by 6. The number 9 is divisible by 9 but not by 2, hence not by 6, so I is a counterexample. The number 18 is divisible by both 9 and 6, so the implication holds for it. The number 12 is not divisible by 9, so the hypothesis is false and it is not a counterexample. Only I works.
Why the other options look right. A overlooks that 9 is not divisible by 2, assumes it is divisible by 6, and so finds no counterexample. C looks for a number that satisfies both parts of the claim, confusing a counterexample with an example, and picks 18. D counts every listed number divisible by 9 as a counterexample without checking whether it fails divisibility by 6, so it accepts 18 as well as 9. E counts any number for which exactly one of the two divisibility properties holds, so it accepts 9 and also 12, which is divisible by 6 but not by 9.
Q3. P is the statement x > 0. Q is the statement x > 4. Which of the following is true?
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Answer and worked solution
Answer: B. P is necessary but not sufficient for Q.
Worked solution. If Q is true, then x > 4, so x > 0 and P is true. Thus P is necessary for Q. P is not sufficient for Q, because x = 1 makes P true and Q false. So P is necessary but not sufficient for Q.
Why the other options look right. A would require P and Q to be equivalent, but x = 1 separates them. C reverses the direction: x > 0 does not force x > 4. D misses that x > 4 does force x > 0. E swaps the roles and the direction: Q is sufficient for P, and it is not necessary for P.
Q4. Which of the following is necessary but not sufficient for the statement x2 ≥ 9?
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Answer and worked solution
Answer: D. |x| ≥ 1
Worked solution. x2 ≥ 9 is equivalent to |x| ≥ 3. If |x| ≥ 3, then |x| ≥ 1, so |x| ≥ 1 is necessary. It is not sufficient, because x = 2 satisfies |x| ≥ 1 but not x2 ≥ 9. So |x| ≥ 1 is necessary but not sufficient.
Why the other options look right. A is necessary and sufficient, since it is exactly |x| ≥ 3. B is sufficient but not necessary, because x = −4 also satisfies x2 ≥ 9. C is neither: x = −4 fails x ≥ 0, and x = 1 satisfies x ≥ 0 but not x2 ≥ 9. E is sufficient but not necessary, because x = 4 works and does not satisfy x ≤ −3.
Q5. A real number x satisfies exactly two of the following three conditions. (I) x² < 4. (II) x > 1. (III) x³ > 4x. Which of the following describes all the possible values of x?
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Answer and worked solution
Answer: B. −2 < x < 0, or 1 < x < 2, or x > 2
Worked solution. Condition (I) holds for −2 < x < 2. Condition (III) says x(x − 2)(x + 2) > 0, which holds for −2 < x < 0 and for x > 2. Count the true conditions on each interval. For x ≤ −2 none holds. For −2 < x < 0, (I) and (III) hold: two. At x = 0 and for 0 < x ≤ 1 only (I) holds. For 1 < x < 2, (I) and (II) hold: two. At x = 2 only (II) holds, because 2² < 4 and 2³ > 8 are both false. For x > 2, (II) and (III) hold: two. No x satisfies all three. So the possible values are −2 < x < 0, or 1 < x < 2, or x > 2, which is option B.
Why the other options look right. A includes x = 2, which comes from reading (I) as x² ≤ 4; then x = 2 satisfies (I) and (II), but in fact only (II) holds there. C reverses the sign pattern of x(x − 2)(x + 2), taking (III) to hold for x < −2 and 0 < x < 2; then exactly two conditions hold only for 0 < x ≤ 1 (and all three hold for 1 < x < 2). D divides x³ > 4x by x without reversing the inequality for negative x, so it takes (III) to be x² > 4, that is x < −2 or x > 2; then no x between −2 and 0 satisfies two conditions. E divides by x and reverses the inequality for every x, as if x were negative, so it takes (III) to be x² < 4 with x ≠ 0; then (I) and (III) hold together for −2 < x ≤ 1 with x ≠ 0, and all three hold for 1 < x < 2.
Q6. A student claims that whenever f(x)2 is at most 1 for every x from 0 to 2, the integral of f(x)2 from 0 to 2 is at most the integral of f(x) from 0 to 2. Which of these functions is a counterexample?
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Answer and worked solution
Answer: A. x − 1
Worked solution. The function f(x) = x − 1 runs from −1 to 1 on [0, 2], so its square is at most 1. Its integral is [x2/2 − x] from 0 to 2, which is 0. The integral of (x − 1)2 is [x3/3 − x2 + x] from 0 to 2, which is 2/3. Since 2/3 > 0, the claimed comparison fails, and x − 1 is a counterexample.
Why the other options look right. B is 1/2. Its square integrates to 1/2, which is less than its integral 1, so the claim holds for it. C is x, and x2 reaches 4 at x = 2, so the hypothesis does not apply. D is x/2. It stays inside the bound, and the integral of its square is 2/3, which is less than its integral 1. E is 2x − 1, and (2x − 1)2 reaches 9 at x = 2, so the hypothesis does not apply.
Q7. Consider the three statements: (1) 4p + 1 and 4p − 1 are both prime when p is an odd prime; (2) every prime greater than 3 is one more or one less than a multiple of 6; (3) no multiple of 3 greater than 3 is prime. The result 21 = 3 × 7 can be used to provide a counterexample to which of them?
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Answer and worked solution
Answer: B. 1 only
Worked solution. For the odd prime p = 5, 4p + 1 = 21 = 3 × 7, which is not prime, so statement 1 is false. Statement 2 is about primes: 21 is not prime, and its prime factors do not refute it either (3 is excluded, and 7 = 6 + 1), so this factorisation gives no counterexample to statement 2. Statement 3 says that no multiple of 3 greater than 3 is prime, and 21 is a composite multiple of 3, which agrees with it. Only statement 1 is refuted, which is option B.
Why the other options look right. A misses the failure of statement 1 at p = 5, where 4p + 1 = 21. C takes 21 to refute statement 2 because 21 = 6 × 3 + 3 is not next to a multiple of 6, but 21 is not prime, so it is not a counterexample. D includes statement 2 as well as statement 1, and 21 does not refute statement 2. E includes statements 2 and 3; 21 is a composite multiple of 3, so it agrees with statement 3 rather than refuting it. Only statement 1 is refuted.
Q8. The function f is continuous for all real x, and the integral from 0 to 4 of f(x) dx is positive. Which one of the following must be true?
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Answer and worked solution
Answer: C. f(x) > 0 for some x with 0 ≤ x ≤ 4
Worked solution. Suppose instead that f(x) ≤ 0 for every x with 0 ≤ x ≤ 4. Then the integral from 0 to 4 of f(x) dx would be at most 0, which contradicts the integral being positive. So f(x) > 0 for at least one x in the interval, which is option C. None of the other statements is forced; counterexamples are given below.
Why the other options look right. A assumes that a positive integral forces the middle value f(2) to be positive; f(x) = (x − 2)2 has integral 16/3 from 0 to 4 but f(2) = 0. B takes a positive integral to mean that f is positive throughout; the same f(x) = (x − 2)2 is 0 at x = 2. D confuses the integral of f with the change f(4) − f(0), as if f were the antiderivative; f(x) = 1 has integral 4 but f(4) = f(0). E assumes each half of the interval contributes a positive amount; f(x) = x − 1 has integral 8 − 4 = 4 from 0 to 4 but 2 − 2 = 0 from 0 to 2.
Q9. Consider the claim: if f and g are both strictly increasing functions on the real numbers, then the product f(x)g(x) is strictly increasing on the real numbers. Which one of the following pairs is a counterexample to the claim?
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Answer
Answer: A. f(x) = x and g(x) = ex
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Q10. A set S of whole numbers greater than 1 is called meshed if and only if, for every member a of S, every prime factor of a divides at least one other member of S. Let T be a set of whole numbers greater than 1. Which of the following is true if and only if T is not meshed?
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Answer
Answer: D. There is a member a of T such that some prime factor of a divides no other member of T
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Q11. A password is a rearrangement of the letters a, b, c, d. Each guess must also be a rearrangement of these four letters, and entering a guess reports how many letters are in the correct position. The guesses abcd and badc each score 0. Using the best strategy, how many further attempts are needed to guarantee that the password can be deduced?
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Answer
Answer: C. Two
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Q12. Consider the claim that, for positive integers a and b, if a² divides b³, then a divides b. Which of the following are counterexamples to the claim? I: a = 8, b = 12. II: a = 4, b = 2. III: a = 6, b = 12.
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Answer
Answer: A. I only
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Q13. Which function is a counterexample to the claim that if f'(x) > 0 for every real x, then the equation f(x) = k has a real solution for every real number k?
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Answer
Answer: C. f(x) = 3x
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Q14. A positive integer is called tidy if its digits, read from left to right, never decrease. For example, 7, 1124 and 3399 are tidy, but 132 is not. Which of the following must be true for every tidy number n? I: If n has at least two digits, deleting any one of its digits leaves a tidy number. II: Writing any digit at the right-hand end of n gives a tidy number. III: If no digit of n is 9, then n + 1 is tidy.
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Answer
Answer: A. I and III only
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Q15. Which statement is the negation of this one? There is a positive integer k such that, for every positive integer m, some positive integer n ≤ m makes kn + m a perfect square.
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Answer
Answer: C. For every positive integer k there is a positive integer m such that no positive integer n ≤ m makes kn + m a perfect square.
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Q16. a and b are real numbers. Given that a³ = ab, which of the following must be true? I. b ≥ 0 II. If b < 0, then a = 0. III. If a ≠ 0, then b > 0.
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Answer
Answer: D. II and III only
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Q17. Consider the conjecture: if k is an odd positive integer, then 2k + 1 is prime. Three cases are proposed. I: k = 1, and 21 + 1 = 3, which is prime. II: k = 2, and 22 + 1 = 5, which is prime. III: k = 9, and 29 + 1 = 513 = 3 × 171. Which of these provide a counterexample to the conjecture?
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Answer
Answer: C. III only
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Q18. Consider the following statement about the positive integers a, b and n. (*): n divides a² − b². The condition that n divides a + b is:
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Answer
Answer: E. sufficient but not necessary for (*)
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Q19. A drawer holds 10 red socks, 8 blue socks and 5 green socks. Socks are taken out one at a time without looking. What is the smallest number of socks that must be taken out to be certain of having at least two socks of one colour and at least two socks of a different colour?
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Answer
Answer: A. 13
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Q20. A multiple-choice question about a statement involving a real number x offered these options. A. The statement is true for every x > 2. B. The statement is true only if x > 2. C. The statement is true for some x > 2. D. The statement is false for some x > 2. E. The statement is true if and only if x > 2. Exactly one option was correct. Which one was it?
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Answer
Answer: D. D
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Q21. Consider the statement: every integer N greater than 10 can be written as the sum of two odd composite positive integers. Which of the following values of N are counterexamples? I: N = 9. II: N = 14. III: N = 16.
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Answer
Answer: B. II and III only
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Q22. Six people, numbered 1 to 6, each make one statement. For each k from 1 to 6, person k says: at least k of us six are liars. Each person is either a truth-teller, who always tells the truth, or a liar, who always lies. Which people are the truth-tellers?
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Answer
Answer: E. Persons 1, 2 and 3 only
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Q23. For a function f, the claim 'f(x) > 0 only if x > 1' is read as a statement about every real x. Which one of the following claims is logically equivalent to it?
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Answer: B. f(x) ≤ 0 if x ≤ 1
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Q24. Consider the claim about real numbers x and y: if x + y and xy are both integers, then x and y are both integers. Which of the following are counterexamples to the claim? I: x = √2 and y = −√2. II: x = √2 and y = √2. III: x = 3 − √2 and y = 3 + √2.
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Answer
Answer: E. I and III only
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Q25. Let f(x) be a cubic polynomial. Consider the statements P: f(x) = 0 for exactly one real value of x. Q: f'(x) ≠ 0 for every real x. Which one of the following is correct?
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Answer
Answer: A. P is necessary but not sufficient for Q.
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Q26. Let p(x) be a polynomial, and let a < b. Consider the statement (*): there exists c with a < c < b such that p'(c) = 1. Which one of the following is true of the condition p(b) − p(a) = b − a?
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Answer
Answer: A. It is sufficient but not necessary for (*).
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Q27. Consider the statement about positive integers n: if the sum of the digits of n is divisible by 4, then n is divisible by 4. What is the smallest n that is a counterexample to the statement?
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Answer
Answer: C. 13
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Q28. Consider the statement: if n is prime, then n2 + 4 is not prime. Which of the following is a counterexample? I: n = 2. II: n = 3. III: n = 5.
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Answer
Answer: D. II and III only
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Q29. A line passes through (2, −3) and is neither horizontal nor vertical. Let P be the statement: if its gradient is positive, then its x-intercept is positive. Which of these must be true? I: P. II: the converse of P. III: the inverse of P, namely: if its gradient is not positive, then its x-intercept is not positive.
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Answer
Answer: A. I only
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Q30. For which real k is the following true: for every real x, if x > k then x2 > k?
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Answer: D. k ≤ 0 or k ≥ 1
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Q31. Which of these statements are true? I: For every real x there is a real y with y2 = x. II: For every real y there is a real x with y2 = x. III: There is a real x such that x ≤ y2 for every real y.
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Answer: B. II and III only
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Q32. Eight real numbers a1, ..., a8 and eight real numbers b1, ..., b8 satisfy an ≤ bn for each n. Which statement must be true?
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Answer
Answer: A. the median of the a's is at most the median of the b's
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Q33. R says that k is an integer multiple of π. S says that the integral of sin x from 0 to k equals 0. Which statement is true?
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Answer: D. R is necessary but not sufficient for S.
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Q34. The line ax + by = c is drawn, where a, b and c are nonzero real numbers. Which condition is necessary but not sufficient for the line to meet the circle x² + y² = 1 at two distinct points?
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Answer: A. |c| < |a| + |b|
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Q35. Consider this statement about a quadrilateral Q. (*): If at least one interior angle of Q is 90°, then the four interior angles form an arithmetic progression when arranged in increasing order. Which of (*), its converse and its contrapositive are true?
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Answer: B. none of them
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Q36. Let (*) be the statement “if a quadrilateral is a square, then its diagonals are equal in length”. Which of the following statements are logically equivalent to (*)? I: A quadrilateral is not a square unless its diagonals are equal in length. II: Having diagonals of equal length is a necessary condition for a quadrilateral to be a square. III: Every quadrilateral whose diagonals are equal in length is a square.
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Answer
Answer: C. I and II only
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Q37. Let n be a positive integer. Which one of the following statements is a sufficient condition for exactly three of the other four?
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Answer
Answer: E. n is divisible by 12
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Q38. Five envelopes are numbered 1 to 5. Envelope 1 has a red seal and an 8p stamp. Envelope 2 has a blue seal and a 3p stamp. Envelope 3 has a green seal and a 2p stamp. Envelope 4 has a red seal and a 4p stamp. Envelope 5 has a blue seal and a 10p stamp. Which one of the following statements about these five envelopes is false?
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Answer
Answer: B. Every envelope with a stamp worth less than 5p has a seal that is not red.
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Q39. Consider the statement: every evening this month, Priya will solve at least two logic puzzles or read a chapter of her novel. Which of the following is equivalent to the statement being not true?
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Answer: B. Some evening this month, Priya will solve fewer than two logic puzzles and will not read a chapter of her novel.
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Q40. Let T be a set of positive integers. Call a positive integer n a T-key when some member of T divides every divisor d of n with d > 1. For example, if T consists of 2 and 5, then 8 is a T-key, because 2 divides each of 2, 4 and 8, while 20 is not a T-key, because 2 does not divide 5 and 5 does not divide 2. Thus n is not a T-key if and only if
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Answer
Answer: A. for every member t of T, there is a divisor d of n with d > 1 that t does not divide.
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Q41. Five people each make one statement. Nora and Priya are the two women. Omar, Quinn and Rohan are the three men. Omar: Of these five statements, an odd number are true. Nora: Both statements made by women are true. Quinn: My first name is Quentin, and Omar's statement is false. Priya: No statement made by a man is true. Rohan: Neither statement made by a woman is true. Quinn's first name may or may not be Quentin. How many of the five statements can be simultaneously true?
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Answer
Answer: C. 2 only
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Q42. Statements P and Q have truth values. It is known that ‘P or Q’ is true and ‘P and Q’ is false. Which conclusion must be true?
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Answer and worked solution
Answer: D. Exactly one of P and Q is true.
Worked solution. Because ‘P or Q’ is true, P and Q cannot both be false. Because ‘P and Q’ is false, they cannot both be true. The only remaining truth-value pairs are (true, false) and (false, true). Therefore exactly one of P and Q is true.
Why the other options look right. A need not hold because P may be false while Q is true. B need not hold because Q may be false while P is true. C is the opposite of what the two given facts force: the remaining possibilities have different truth values. E makes ‘P or Q’ false, contradicting the first given fact.
Q43. Nadia tells the truth on Monday, Tuesday and Sunday, and lies on every other day. Hugo tells the truth on Wednesday, Thursday, Saturday and Sunday, and lies on every other day. Asked whether they were lying yesterday, both answer yes. What day is it?
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Answer
Answer: C. Wednesday
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Q44. Five envelopes, A, B, C, D and E, contain the amounts 3, 9, 15, 18 and 27 pounds, one amount in each envelope. The envelopes carry these statements. A: Exactly one of these five statements is false, and the envelope with the false statement contains 3 pounds. B: C contains 15 pounds. C: B contains 27 pounds. D: E contains 18 pounds. E: A contains 18 pounds. Given that A's statement is true, which envelope contains 27 pounds?
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Answer
Answer: B. Envelope B
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Q45. Consider this statement about positive integers x, y and z: x = y + z. Call that statement (*). The condition that x > y and x > z is
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Answer
Answer: A. necessary but not sufficient for (*)
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Q46. A region R in the plane is given by y < x + 5 and y < 9 − x. Which of the following statements is true at every point of R? I. x < 0 II. y < 7 III. x² + y² ≤ 2(x + y) IV. y < (x + 2)(5 − x)
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Answer: B. II only
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Q47. The numbers p and q satisfy 1 < p < q. Which statement must be true?
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Answer: A. logp(q) > logq(p)
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Q48. Suppose x²(y − 1) = y − 1. Which of the following must be true? I. x² = 1 II. x = 1 or y = 1 III. x² = 1 or y = 1 IV. x = y = 1
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Answer
Answer: C. III only
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Q49. Asha tells the truth only on Monday and Tuesday. Ben tells the truth only on Tuesday and Wednesday. On every other day, each of them lies. One day Asha says "I will lie tomorrow" and Ben says "I lied yesterday". What day is it?
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Answer: B. Tuesday
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Q50. For real numbers x and y, consider the equation |x + y| = |x| + |y|. Which of the following statements about conditions for this equation is correct?
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Answer: B. xy > 0 is sufficient but not necessary
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Q51. Ravi tells the truth on Monday, Wednesday and Friday only, and lies on every other day. Sofia tells the truth on Saturday and Sunday only, and lies on every other day. One day, Ravi says "I will lie tomorrow" and Sofia says "I lied yesterday". Which day is it?
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Answer: D. Monday
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Q52. Which condition is sufficient but not necessary for an integer k to be a multiple of 12?
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Answer: C. k is a multiple of 24
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Q53. Which of the following statements is true for every real number x?
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Answer: E. If x > 2, then x² > 4.
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Q54. Fay, Gus and Hal are each either a truth-teller, who always tells the truth, or a liar, who always lies. Fay says: "Gus is a liar." Gus says: "Hal is a liar." Hal says: "Fay and Gus are both liars." Which of them are truth-tellers?
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Answer: B. Gus only
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Q55. Consider the equation |x − 2| − |x − 7| = 5 for real x. Which statement is true?
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Answer: D. It holds if and only if x ≥ 7.
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Q56. Tara tells the truth on Monday, Wednesday and Friday, and lies on every other day. On one day she says "I told the truth yesterday". On the next day she says "I lied yesterday". On which day did she make the first statement?
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Answer: B. Sunday
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Q57. Let n be an integer. For the statement "16 divides n² − 1", the condition "n is odd" is
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Answer: A. necessary but not sufficient
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Q58. Four football teams, Rovers, Town, United and City, each play every other team exactly once. A win earns 3 points, a draw earns 1 point for each team, and a loss earns 0 points. At the end, Rovers have 6 points, Town have 5 points and United have 2 points. How many points do City have?
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Answer: E. 2
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Q59. For integers m and n, consider the statement "mn is odd". The condition "m + n is even" is
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Answer: A. necessary but not sufficient
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Q60. Positive integers a, b and c satisfy a ≤ b ≤ c and abc = 48. Which of the following must be true? I. a ≤ 3 II. a + b + c ≥ 12 III. b ≤ 6
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Answer: D. I and III only
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Q61. Which one of the following is not true for every real number x?
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Answer: D. x² − 3x + 2 ≥ 0
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Q62. Rita tells the truth on Monday, Thursday and Friday only, and lies on every other day. On one day she makes two separate statements: "I lied yesterday" and "I will lie tomorrow". Which day is it?
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Answer: B. Monday
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Q63. Take the statement x² < 2x + 3, for a real number x. Relative to that statement, the condition |x| < 3 is
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Answer: A. necessary but not sufficient
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Q64. Positive integers a, b and c satisfy a ≤ b ≤ c and abc = 30. Which of these must be true? I: a + b ≤ c. II: c is even. III: b ≤ 3.
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Answer: A. I only
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Q65. Four teams, P, Q, R and S, each play every other team exactly once. A win earns 3 points, a draw earns 1 point for each team, and a loss earns 0 points. At the end, P has 7 points, Q has 5, R has 3 and S has 1. Which of the following must be true?
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Answer: D. Q drew with S
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Q66. For positive integers n, consider the claim that n² is a multiple of 8. The condition that n is a multiple of 2 is which of the following?
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Answer: D. necessary but not sufficient
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Q67. The positive integers a, b and c satisfy a < b < c and a + b + c = 15. Which of the following must be true?
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Answer: A. a ≤ 4
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Q68. Which of the following statements is not true for all real numbers x and y?
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Answer: B. |x + y| ≥ |x| + |y|
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Q69. Consider the statement: every rectangle whose side lengths are whole numbers of centimetres, and whose perimeter is 24 cm, has area at least 20 cm2. Which of the following is a counterexample to this statement? I: a 1 cm by 11 cm rectangle. II: a 2 cm by 10 cm rectangle. III: a 6 cm by 6 cm rectangle.
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Answer: B. I only
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Q70. Five boxes each contain the same positive number of counters. Exactly one of the following labels is true. Which label is the true one? The labels are: fewer than 5 counters; 7 or 8 counters; 1 or 8 counters; more than 5 and fewer than 9 counters; 2, 3 or 4 counters.
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Answer: D. more than 5 and fewer than 9 counters
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Q71. Let x be a real number, and consider the inequality |x − 1| ≥ 3. Which of the following conditions on x is necessary but not sufficient for this inequality to hold?
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Answer: C. |x| ≥ 2
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Q72. Consider the following statements about a real number x. P: x3 > x. Q: x > 1. Which of the following is correct?
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Answer: C. P is necessary but not sufficient for Q
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Q73. The real numbers a and b are such that exactly one of the following statements is true. Which one is it?
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Answer: A. a ≤ b
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Q74. A statement S depends on a real number x. It is known that S is true whenever x > 2, and that S is false whenever x < 1. Which one of the following must be correct?
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Answer: D. S is true only if x ≥ 1
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Q75. Which of the following statements are true? I. There is a real number y such that y > x2 for every real number x. II. There is a real number x such that x + y2 > 0 for every real number y. III. For every real number x, there is a real number y such that xy = x + y.
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Answer: C. II only
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Q76. A function f takes real values, and a and b are real numbers. Exactly one of the following statements is true. Which one is it?
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Answer: A. (f(a) + f(b))2 ≥ 0
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Q77. Priya tells the truth on Friday, Saturday and Sunday, and lies on every other day. Hugo tells the truth on Monday, Tuesday and Wednesday, and lies on every other day. On one day, each is asked whether they were telling the truth yesterday, and each answers "No." What day is it?
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Answer: A. Monday
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Q78. Five jars, P to T, contain £4, £6, £8, £10 and £12 in some order, and each jar has a label. The labels are: P, "Exactly one label is false, and it is on the jar containing £4." Q, "R contains £8." R, "S contains £12." S, "T contains £4." T, "P contains £6." The label on P is true. Which jar contains £12?
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Answer: D. jar S
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Q79. For real numbers x and y, consider the statement x2 + y2 < 2. The condition that both |x| < 1 and |y| < 1 is:
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Answer: B. sufficient but not necessary
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Q80. Positive integers a, b and c satisfy a ≤ b ≤ c and 1/a + 1/b + 1/c = 1/2. Which of the following must be true? I: a = b = c = 6. II: a ≤ 6. III: b ≥ 4.
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Answer: D. II and III only
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Q81. The real numbers a, b and c satisfy a < b < c. Which of the following must be true? I: a2 < b2 < c2. II: (c − b)(b − a) > 0. III: a + b > c.
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Answer: C. II only
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Q82. Let a and b be real numbers. Which of the following is equivalent to the statement 'a + b > 0 and ab > 0'?
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Answer: B. a > 0 and b > 0
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Q83. A function f satisfies f(x) = f((x − 1)2) for every real number x. Which of the following must be true?
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Answer: B. f(3) − f(−1) = 0
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Q84. Suppose 0 < a < 1 < b. Which of the following must be true?
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Answer: D. b/a > b
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Q85. Let m and n be integers. Exactly three of these four statements are true: mn is even; m + n is even; m2 + n is odd; m + n2 is odd. Which statement is false?
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Answer: B. m + n is even
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Q86. A conjecture says that if every digit of a positive integer is odd, then that integer is not a multiple of 7. Three integers are proposed: I. 35, which equals 5 × 7; II. 1001, which equals 7 × 143; III. 139, which is not a multiple of 7. Which of these are counterexamples to the conjecture?
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Answer: A. I only
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Q87. Let n be a positive integer. Which of the following is necessary and sufficient for 12 + 22 + 32 + ... + n2 to be divisible by n?
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Answer: E. n is odd and not a multiple of 3
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Q88. The arithmetic mean of five consecutive integers is a multiple of 4. Which of the following must be true? I. The largest of the five integers is even. II. The sum of the five integers is a multiple of 4. III. The smallest of the five integers is odd.
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Answer: D. I and II only
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Q89. Consider the claim: if f(x + 2) = f(x) for every real x, then f(x + 1) = f(x) for every real x. Which function is a counterexample?
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Answer: B. f(x) = cos(πx)
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Q90. Let a, b and c be integers with c ≠ 0. Consider the statement that (a2 b)/c is a positive odd integer. Which of the following is necessary but not sufficient for that statement? I. a is odd II. bc > 0 III. c is odd
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Answer: C. II only
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Q91. For all real numbers u and v with u ≤ v, which of these statements must be true? I. u3 ≤ v3 II. u2 ≤ v2 III. 2u + v ≤ u + 2v
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Answer: D. I and III only
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Q92. The numbers x, m and n are positive integers. When x is divided by m, the remainder is n. When x is divided by n, the remainder is m − 3. Which of the following must be true?
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Answer: C. n ≥ m − 2
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Q93. Consider this claim about positive integers a, b and c with b > c: if a divides b + c and a divides b − c, then a divides b. Which of the following are counterexamples? I. a = 2, b = 5, c = 3 II. a = 6, b = 9, c = 3 III. a = 5, b = 7, c = 3
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Answer: C. I and II only
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Q94. If |2x − 3| ≤ 5, which of the following must be true? I. x > −1 II. x ≤ 4 III. x ≠ 5
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Answer: A. II and III only
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Q95. It is claimed that every positive integer n for which n + 1 or n + 3 is divisible by 10 must be prime. How many integers n with 0 < n < 40 are counterexamples to this claim?
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Answer: C. 3
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Q96. A cubic is given by f(x) = x3 + ax2 + bx + c, and (x − 1) and (x − 3) are factors. Which of the following must be true? I. a + b + c = −1. II. If c = −6, then f(2) = 0. III. f(x) < 0 for every x < 1.
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Answer: E. I and II only
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Q97. Let n be a positive integer, and let P be the statement “7 divides n² + n + 1”. Which of the following are true? I. “n leaves remainder 2 when divided by 7” is a sufficient condition for P. II. “n leaves remainder 2 when divided by 7” is a necessary condition for P. III. “n is not a multiple of 7” is a necessary condition for P.
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Answer: D. I and III only
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Q98. Which of the following conditions are sufficient but not necessary for |x − 1| < x? I. x > 1 II. x > 0 III. x > 2
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Answer: C. I and III only
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Q99. Consider the statement that there is a real number M such that f(x) ≤ M for every real number x. Which of the following is a negation of this statement?
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Answer: A. For every real number M, there is a real number x such that f(x) > M.
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Q100. Each card has a letter on one side and a number on the other. Five cards show K, G, 6, 4 and 9. Priya says that a card has K on one side if and only if the number on the other side is a multiple of 3. Which cards must be turned over to check her statement?
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Answer
Answer: E. all five cards
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Q101. The numbers x and y are non-zero real numbers. Which of these statements are true? I. If x/y > 2 and y > 0, then x > 2y. II. x/y > 2 if and only if y/x < 1/2. III. If xy > 2 and x > 0, then y > 0.
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Answer: D. I and III only
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Q102. Amir, Bea and Chen each wear one hat. The hats are green, white and yellow, one of each colour. If Bea wears green, then Amir wears white. If Chen does not wear yellow, then Bea wears green. Amir does not wear white. Which colour is Chen wearing?
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Answer: C. yellow
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Q103. Consider the claim: if (f(x))2 = x2 for every real x, then either f(x) = x for every real x or f(x) = −x for every real x. Which function is a counterexample?
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Answer
Answer: A. f(x) = |x|
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Q104. Let f(x) = x/(x + 2) for every integer x ≠ −2. Which of the following must be true? I. f(x + 1) > f(x) whenever both values are defined. II. f(x) > 0 for every integer x ≠ −2. III. f(x) ≠ 1 for every integer x ≠ −2.
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Answer: C. I and III only
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Q105. A function f is called jumpy when, for every real number d > 0, there are real numbers x and y with |x − y| < d and |f(x) − f(y)| ≥ 1. Which statement is true if and only if f is not jumpy?
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Answer
Answer: B. There is a real number d > 0 such that, for all real numbers x and y with |x − y| < d, |f(x) − f(y)| < 1.
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Q106. Which one of the following claims is correct?
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Answer: C. Every prime number greater than 2 is odd.
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Q107. Which statement is true?
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Answer: E. Every prime number greater than 3 leaves remainder 1 or 5 when divided by 6.
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Q108. Which statement must be true?
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Answer: D. If f is continuous, f(1) = 3 and f(4) = −1, then f(x) = 0 has a solution with 1 < x < 4.
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Q109. The function f is differentiable for every real x. Which statement must be true?
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Answer: A. If f'(x) > 0 for every x, then f is strictly increasing on the real line.
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Q110. Consider the claim: if f is continuous on 0 ≤ x ≤ 2 and the integral from 0 to 2 of f(x) dx is 0, then f(1) = 0. Which function is a counterexample to this claim?
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Answer: B. f(x) = cos(πx)
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Q111. The positive integer p is prime, and q is a positive integer. Which statement must be true?
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Answer: D. If the highest common factor of p and q is p, then p divides q.
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Q112. Exactly one of the following statements about the positive integer n is true. Which one is it?
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Answer: E. n is odd.
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Q113. A positive integer is squarefree when it is not divisible by p² for any prime p. Which statement must be true?
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Answer: B. Every positive divisor of a squarefree positive integer is squarefree.
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