ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 6)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 6): 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) = 28, n(B) = 25, n(C) = 20, n(A ∩ B) = 10, n(B ∩ C) = 8, n(A ∩ C) = 7 and n(A ∩ B ∩ C) = 4. Then n(A ∪ B ∪ C) is:

   • A. 56
   • B. 60
   • C. 52 ✓
   • D. 48

   Answer: C. 52

   Explanation: n(A ∪ B ∪ C) = 28 + 25 + 20 − 10 − 8 − 7 + 4 = 52.

2. Q2. How many subsets of a 6-element set contain two particular given elements?

   • A. 16 ✓
   • B. 15
   • C. 64
   • D. 32

   Answer: A. 16

   Explanation: Both given elements are forced into the subset, so only the other 4 elements are free: 2⁴ = 16 subsets.

3. Q3. How many symmetric relations can be defined on a set with 4 elements?

   • A. 512
   • B. 65536
   • C. 4096
   • D. 1024 ✓

   Answer: D. 1024

   Explanation: A symmetric relation is fixed by choosing, independently, each of the 4 diagonal pairs and each of the 6 unordered pairs {a, b} with a ≠ b: 4 + 6 = 10 free choices, so 2¹⁰ = 1024.

4. Q4. How many functions f from {1, 2, 3} to {1, 2, 3, 4, 5} satisfy f(1) < f(2) < f(3)?

   • A. 125
   • B. 20
   • C. 60
   • D. 10 ✓

   Answer: D. 10

   Explanation: Choosing 3 distinct values from 5 and assigning them in increasing order gives exactly one valid function per choice: C(5, 3) = 10.

5. Q5. The number of onto functions from a 4-element set onto a 2-element set is:

   • A. 8
   • B. 16
   • C. 12
   • D. 14 ✓

   Answer: D. 14

   Explanation: All 2⁴ = 16 functions minus the 2 constant (non-onto) ones gives 14.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

6. Q6. The range of the function f(x) = x² − 4x + 5 for real x is:

   • A. [0, ∞)
   • B. (−∞, 1]
   • C. [5, ∞)
   • D. [1, ∞)

7. Q7. The domain of the real function f(x) = √[(x − 1)/(x − 4)] is:

   • A. (−∞, 1) ∪ [4, ∞)
   • B. [1, 4)
   • C. [1, 4]
   • D. (−∞, 1] ∪ (4, ∞)

8. Q8. If f(x) = x² + 1 and g(x) = √(x − 1), then (g ∘ f)(−3) equals:

   • A. −3
   • B. 3
   • C. √9 − 1
   • D. √10

9. Q9. The inverse of f(x) = 1 + √(x − 2), defined for x ≥ 2, is:

   • A. f⁻¹(x) = (x − 1)² + 2 for x ≥ 1
   • B. f⁻¹(x) = (x − 2)² + 1 for x ≥ 2
   • C. f⁻¹(x) = (x + 1)² + 2 for x ≥ −1
   • D. f⁻¹(x) = (x − 1)² − 2 for x ≥ 1

10. Q10. In the group Z₆ = {0, 1, 2, 3, 4, 5} under addition modulo 6, the order of the element 4 is:

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

11. Q11. In the group of non-zero residues modulo 7 under multiplication, the order of the element 3 is:

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

12. Q12. How many elements of the additive group Z₁₂ generate the whole group?

    • A. 2
    • B. 4
    • C. 12
    • D. 6

13. Q13. On the set of real numbers define a ∗ b = a + b − ab. Then the identity element and the inverse of 3 are respectively:

    • A. 0 and 3/2
    • B. 0 and −3
    • C. 1 and 1/3
    • D. 1 and 3/2

14. Q14. On the set of real numbers define a ∗ b = a + b + 2. The inverse of 5 under ∗ is:

    • A. −2
    • B. −5
    • C. −9
    • D. −7

15. Q15. Which statement is logically equivalent to the implication p → q?

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

16. Q16. In a universal set of 50 elements, n(A) = 22, n(B) = 18 and n(A ∩ B) = 6. Then n(A′ ∩ B′) is:

    • A. 22
    • B. 16
    • C. 34
    • D. 10

17. Q17. If n(A) = 12, n(B) = 9 and n(A ∩ B) = 5, the number of elements in the symmetric difference A Δ B = (A − B) ∪ (B − A) is:

    • A. 6
    • B. 16
    • C. 11
    • D. 21

18. Q18. On the set of integers, define a R b to mean ab > 0. The relation R is:

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

19. Q19. The number of different equivalence relations that can be defined on a set of 3 elements is:

    • A. 5
    • B. 8
    • C. 3
    • D. 6

20. Q20. If A has 3 elements and B has 4 elements, how many subsets of A × B contain exactly 2 ordered pairs?

    • A. 24
    • B. 12
    • C. 132
    • D. 66

21. Q21. The number of subsets of the set {x : x is an integer and x² < 10} is:

    • A. 64
    • B. 512
    • C. 256
    • D. 128

22. Q22. The function f: ℤ → ℤ defined by f(n) = n + (−1)ⁿ satisfies:

    • A. f is one-one and onto, and f ∘ f is the identity
    • B. f is one-one but not onto
    • C. f is onto but not one-one
    • D. f is one-one and onto, but f ∘ f is not the identity

23. Q23. The range of the real function f(x) = (x² + 1)/(x² + 2) is:

    • A. [0, 1)
    • B. [1/2, 1)
    • C. [1/2, 1]
    • D. (1/2, 1)

24. Q24. If f(x) = 3x + 2 and (f ∘ g)(x) = 12x − 7, then g(2) equals:

    • A. 4
    • B. 6
    • C. 5
    • D. 3

25. Q25. The function f(x) = x³ sin x is:

    • A. neither even nor odd
    • B. odd
    • C. even
    • D. both even and odd

26. Q26. Which of these sets is a group under multiplication modulo 8?

    • A. {0, 1, 3, 5}
    • B. {2, 4, 6}
    • C. {1, 3, 5, 7}
    • D. {1, 2, 3, 4}

27. Q27. Which of the following is a group under multiplication?

    • A. the set of integers
    • B. {0, 1}
    • C. {1, −1, i, −i}
    • D. the set of non-zero integers

28. Q28. Which of these cannot be the order of a subgroup of a group of order 18?

    • A. 6
    • B. 9
    • C. 3
    • D. 4

29. Q29. In the truth table of (p ∨ q) → r, in how many of the 8 rows is the statement true?

    • A. 6
    • B. 4
    • C. 3
    • D. 5

30. Q30. If f(x) = x + 2 and g(x) = 3x, then (g ∘ f)⁻¹(9) equals:

    • A. 5
    • B. 7/3
    • C. 3
    • D. 1