ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 4)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 4): 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.

1. Q1. In a survey, n(A) = 30, n(B) = 25, n(C) = 20, n(A ∩ B) = 10, n(B ∩ C) = 8, n(A ∩ C) = 6 and n(A ∩ B ∩ C) = 4. The number of elements belonging to A only is:

   • A. 20
   • B. 14
   • C. 18 ✓
   • D. 24

   Answer: C. 18

   Explanation: A only = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C) = 30 − 10 − 6 + 4 = 18. Forgetting to add back the triple intersection gives 14.

2. Q2. The power set of the power set of {a, b}, that is P(P({a, b})), has how many elements?

   • A. 16 ✓
   • B. 4
   • C. 32
   • D. 8

   Answer: A. 16

   Explanation: P({a, b}) has 2² = 4 elements, so P(P({a, b})) has 2⁴ = 16 elements.

3. Q3. Sets A and B contain 3 and 2 elements respectively. How many distinct subsets of A × B exist (each one being a relation from A to B)?

   • A. 8
   • B. 6
   • C. 32
   • D. 64 ✓

   Answer: D. 64

   Explanation: A × B has 6 elements and every subset of A × B is a relation, giving 2⁶ = 64 relations.

4. Q4. The number of onto functions from {1, 2, 3} to {p, q} is:

   • A. 2
   • B. 3
   • C. 8
   • D. 6 ✓

   Answer: D. 6

   Explanation: There are 2³ = 8 functions in all; the two constant functions are not onto. So 8 − 2 = 6 are onto.

5. Q5. The range of f(x) = x² − 4x + 7, with domain ℝ, is:

   • A. (−∞, 3]
   • B. [2, ∞)
   • C. [7, ∞)
   • D. [3, ∞) ✓

   Answer: D. [3, ∞)

   Explanation: f(x) = (x − 2)² + 3 ≥ 3, with equality at x = 2, and f takes every value above 3. The range is [3, ∞).

The other 25 questions

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6. Q6. Find the largest subset of ℝ on which f(x) = √(x² − 5x + 6) is defined.

   • A. [2, 3]
   • B. (−∞, 2) ∪ (3, ∞)
   • C. (2, 3)
   • D. (−∞, 2] ∪ [3, ∞)

7. Q7. If f(x) = 2x + 3 and g(x) = x², the sum of all real values of x for which (f∘g)(x) = (g∘f)(x) is:

   • A. −3
   • B. 6
   • C. 0
   • D. −6

8. Q8. If f(x) = (2x + 1)/(x − 3), then f⁻¹(5) equals:

   • A. 16
   • B. 16/3
   • C. 11/2
   • D. 2

9. Q9. In the group (ℤ₆, +₆), the order of the element 4 is:

   • A. 4
   • B. 3
   • C. 6
   • D. 2

10. Q10. The number of generators of the cyclic group (ℤ₁₂, +₁₂) is:

    • A. 4
    • B. 2
    • C. 11
    • D. 6

11. Q11. On the set ℝ∖{−1}, the operation * is defined by a * b = a + b + ab. The inverse of 3 under * is:

    • A. −3/4
    • B. −1/3
    • C. −3
    • D. 1/4

12. Q12. Which of the following is a group under the stated operation?

    • A. (ℕ, +)
    • B. (ℚ∖{0}, ×)
    • C. (ℤ, −)
    • D. (ℤ, ×)

13. Q13. The statement (p → q) ∧ ¬q is logically equivalent to:

    • A. p ∧ ¬q
    • B. ¬p ∨ q
    • C. ¬p ∧ ¬q
    • D. p ∧ q

14. Q14. The truth table of a compound statement involving four distinct simple statements has how many rows?

    • A. 12
    • B. 8
    • C. 4
    • D. 16

15. Q15. The contrapositive of “If a number is divisible by 6, then it is divisible by 3” is:

    • A. If a number is not divisible by 6, then it is not divisible by 3
    • B. If a number is divisible by 3, then it is divisible by 6
    • C. If a number is not divisible by 3, then it is not divisible by 6
    • D. A number is divisible by 6 and not divisible by 3

16. Q16. Identify the tautology among the following compound statements.

    • A. p ∧ q
    • B. p → (p ∨ q)
    • C. p ∧ ¬p
    • D. p ∨ q

17. Q17. For sets A and B, (A ∩ B′)′ equals:

    • A. A ∪ B′
    • B. A′ ∪ B
    • C. A′ ∩ B
    • D. A′ ∪ B′

18. Q18. Which of the following identities holds for all sets A, B, C?

    • A. A − (B ∩ C) = (A − B) ∩ (A − C)
    • B. A − (B − C) = (A − B) − C
    • C. A − (B ∪ C) = (A − B) ∪ (A − C)
    • D. A − (B ∪ C) = (A − B) ∩ (A − C)

19. Q19. If n(A) = 12, n(B) = 9 and n(A ∩ B) = 4, the number of elements in the symmetric difference AΔB (elements in exactly one of the sets) is:

    • A. 13
    • B. 21
    • C. 17
    • D. 9

20. Q20. How many elements does the set {x : x ∈ ℤ, x² < 30} contain?

    • A. 5
    • B. 10
    • C. 11
    • D. 6

21. Q21. If A × B has 12 elements and n(A) = 3, the number of different relations from B to A is:

    • A. 144
    • B. 24
    • C. 12
    • D. 4096

22. Q22. Which of the following functions from ℝ to ℝ is one-one?

    • A. f(x) = x³ + 5
    • B. f(x) = |x|
    • C. f(x) = x/(1 + x²)
    • D. f(x) = x²

23. Q23. If f(x) = 1/(1 − x) for x ≠ 1, then f(f(f(x))) equals:

    • A. x
    • B. 1/x
    • C. 1 − x
    • D. −x

24. Q24. In the Cayley table of a finite group, each element of the group appears:

    • A. as many times as the order of the element
    • B. exactly once in every row and every column
    • C. exactly once on the main diagonal only
    • D. only in the first row and first column

25. Q25. Which of the following is a monoid but NOT a group?

    • A. (ℤ, +)
    • B. (W, +), where W = {0, 1, 2, …}
    • C. ({1, −1}, ×)
    • D. (ℚ∖{0}, ×)

26. Q26. A group of order 12 cannot have a subgroup of order:

    • A. 4
    • B. 3
    • C. 8
    • D. 6

27. Q27. The number of one-one functions from {1, 2, 3} to {a, b, c, d, e} is:

    • A. 125
    • B. 60
    • C. 243
    • D. 15

28. Q28. On ℤ, the relation x R y defined by |x − y| ≤ 1 is:

    • A. reflexive and transitive but not symmetric
    • B. reflexive and symmetric but not transitive
    • C. symmetric and transitive but not reflexive
    • D. an equivalence relation

29. Q29. The function f(x) = x² + 2 has domain [0, ∞). The value of f⁻¹(11) is:

    • A. 9
    • B. 3
    • C. √13
    • D. ±3

30. Q30. In a universal set of 50 elements, n(A) = 22, n(B) = 18 and n(A ∩ B) = 7. The number of elements in (A ∪ B)′ is:

    • A. 27
    • B. 10
    • C. 24
    • D. 17