ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 7)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review.…

What this test covers

30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

Q1. In a survey, n(A) = 30, n(B) = 28, n(C) = 22, n(A ∩ B) = 12, n(B ∩ C) = 9, n(A ∩ C) = 10 and n(A ∩ B ∩ C) = 5. How many people belong to exactly two of the three sets?
- A. 16 ✓
- B. 11
- C. 26
- D. 21
Answer: A. 16
Explanation: Exactly two = (12 − 5) + (9 − 5) + (10 − 5) = 7 + 4 + 5 = 16. (Subtracting the triple overlap from each pair removes the people counted in all three.)
Q2. On A = {1, 2, 3}, the relation R = {(1, 2), (2, 3)} is not transitive. What is the minimum number of ordered pairs that must be added to R to make it transitive?
- A. 2
- B. 3
- C. 1 ✓
- D. 0
Answer: C. 1
Explanation: Transitivity requires (1, 3) because (1, 2) and (2, 3) are in R. Adding (1, 3) creates no other forced pair, so one pair suffices.
Q3. The domain of the real function f(x) = 1/√(x² − x − 6) is
- A. (−2, 3)
- B. (−∞, −2] ∪ [3, ∞)
- C. (−∞, −3) ∪ (2, ∞)
- D. (−∞, −2) ∪ (3, ∞) ✓
Answer: D. (−∞, −2) ∪ (3, ∞)
Explanation: We need x² − x − 6 = (x − 3)(x + 2) > 0 (strictly, since it is in a denominator under a root). This holds for x < −2 or x > 3.
Q4. The range of the function f(x) = |x − 2| + |x + 1| is
- A. [0, ∞)
- B. [1, ∞)
- C. (−∞, 3]
- D. [3, ∞) ✓
Answer: D. [3, ∞)
Explanation: For −1 ≤ x ≤ 2 the sum is constant, (2 − x) + (x + 1) = 3, and outside that interval it is larger than 3 and grows without bound. So the range is [3, ∞).
Q5. Which of the following functions is an odd function of x (that is, f(−x) = −f(x))?
- A. f(x) = x² + |x|
- B. f(x) = x + 1
- C. f(x) = |x| − x²
- D. f(x) = x|x| ✓
Answer: D. f(x) = x|x|
Explanation: For f(x) = x|x|: f(−x) = (−x)|−x| = −x|x| = −f(x), so it is odd. The other three are even or neither (x + 1 has f(−x) = 1 − x, which is neither f(x) nor −f(x)).

The other 25 questions

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Q6. Let f(x) = 2x for x < 0 and f(x) = x + 1 for x ≥ 0. Then f(f(−1)) + f(f(1)) equals
- A. −1
- B. −4
- C. −2
- D. 1
Q7. For f(x) = x² − 2x with domain x ≥ 1, the inverse function is
- A. f⁻¹(x) = 1 − √(x + 1)
- B. f⁻¹(x) = 1 + √(x + 1)
- C. f⁻¹(x) = 1 + √(x − 1)
- D. f⁻¹(x) = √(x + 1) − 1
Q8. The function f(x) = (ax + 3)/(x − 2), x ≠ 2, is its own inverse (f = f⁻¹) when a equals
- A. 2
- B. 3
- C. −2
- D. −3
Q9. The negation of the statement "For every real number x, x² > x" is
- A. There exists a real number x such that x² ≤ x
- B. There exists a real number x such that x² < x
- C. For every real number x, x² ≤ x
- D. For every real number x, x² < x
Q10. Which of the following compound statements is logically equivalent to p ↔ q?
- A. (p ∨ q) ∧ ~(p ∧ q)
- B. (p ∧ q) ∨ (~p ∧ ~q)
- C. (p ∧ ~q) ∨ (~p ∧ q)
- D. (p → q) ∨ (q → p)
Q11. In the truth table of (p → q) ∧ (q → r), how many of the 8 rows give the value true?
- A. 5
- B. 3
- C. 6
- D. 4
Q12. In the group (ℤ₁₅, +₁₅), the order of the element 9 is
- A. 15
- B. 3
- C. 5
- D. 9
Q13. How many subgroups does the cyclic group (ℤ₁₂, +₁₂) have (including the trivial subgroup and the whole group)?
- A. 5
- B. 4
- C. 12
- D. 6
Q14. In the group of units modulo 9 under multiplication, {1, 2, 4, 5, 7, 8}, the inverse of 4 is
- A. 2
- B. 7
- C. 8
- D. 5
Q15. On the set of non-zero real numbers define a ∗ b = ab/2. The inverse of 4 under ∗ is
- A. 2
- B. ½
- C. ¼
- D. 1
Q16. On ℝ define a ∗ b = a + 2b. Which of the following statements about ∗ is correct?
- A. ∗ has a two-sided identity element
- B. ∗ is commutative
- C. ∗ is not associative
- D. ∗ is associative
Q17. A × B has exactly 6 elements and contains (1, 2), (2, 3) and (1, 3). Then n(A) + n(B) equals
- A. 5
- B. 6
- C. 7
- D. 8
Q18. How many subsets of {1, 2, 3, 4, 5, 6} contain at least one odd number?
- A. 63
- B. 48
- C. 56
- D. 60
Q19. In a universal set U, n(A − B) = 12, n(B − A) = 9, n(A ∩ B) = 5 and n(A′ ∩ B′) = 4. Then n(U) equals
- A. 30
- B. 26
- C. 34
- D. 21
Q20. On {1, 2, …, 10} let x R y mean that x − y is divisible by 3. The number of elements in the largest equivalence class is
- A. 4
- B. 5
- C. 10
- D. 3
Q21. How many functions f from {1, 2, 3} to {0, 1} satisfy f(1) + f(2) + f(3) = 2?
- A. 6
- B. 2
- C. 3
- D. 4
Q22. For any sets A and B, the set (A ∪ B) ∩ (A ∪ B′) ∩ (A′ ∪ B) equals
- A. B − A
- B. A ∪ B
- C. A
- D. A ∩ B
Q23. If the conditional p → q is false, which of the following statements must be true?
- A. p ↔ q
- B. p ∧ ~q
- C. ~q → ~p
- D. ~p ∨ q
Q24. How many elements of order 4 are there in the group (ℤ₁₂, +₁₂)?
- A. 4
- B. 3
- C. 2
- D. 1
Q25. On the set {a, b}, how many binary operations ∗ can be defined that have a as an identity element?
- A. 4
- B. 2
- C. 3
- D. 1
Q26. If f(x) = (x − 1)/(x + 1), then f(f(f(f(x)))) equals
- A. x
- B. −x
- C. −1/x
- D. (x − 1)/(x + 1)
Q27. The power set of set A has 112 more elements than the power set of set B. If n(A) = m and n(B) = n, then m + n equals
- A. 13
- B. 11
- C. 9
- D. 10
Q28. Which of the following set identities is NOT true for all sets A, B and C?
- A. (A − B) − C = A − (B ∪ C)
- B. A ∪ (B − C) = (A ∪ B) − (A ∪ C)
- C. A − (B ∪ C) = (A − B) ∩ (A − C)
- D. A − (B ∩ C) = (A − B) ∪ (A − C)
Q29. The function f(x) = x² − 4x is defined on [2, ∞). Which codomain makes f an onto (surjective) function?
- A. ℝ
- B. [0, ∞)
- C. [−4, ∞)
- D. (−4, ∞)
Q30. How many functions f: {1, 2, 3} → {1, 2, 3, 4, 5} satisfy f(1) < f(2) < f(3)?
- A. 15
- B. 20
- C. 10
- D. 60