ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 3)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 3): 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 class, 30 study Physics, 25 Chemistry and 20 Biology. 10 study both Physics and Chemistry, 8 both Chemistry and Biology, 7 both Physics and Biology, and 4 study all three. Every student studies at least one subject. The class strength is:

   • A. 75
   • B. 58
   • C. 54 ✓
   • D. 49

   Answer: C. 54

   Explanation: n(P∪C∪B) = 30 + 25 + 20 − 10 − 8 − 7 + 4 = 54.

2. Q2. Of 100 people, 60 drink tea, 50 drink coffee and 20 drink neither. The number who drink both is:

   • A. 10
   • B. 30 ✓
   • C. 20
   • D. 40

   Answer: B. 30

   Explanation: 80 drink at least one, so 80 = 60 + 50 − n(both), giving n(both) = 30.

3. Q3. The number of reflexive relations on the set {1, 2} is:

   • A. 8
   • B. 2
   • C. 4 ✓
   • D. 16

   Answer: C. 4

   Explanation: A reflexive relation must contain (1,1) and (2,2); each of the other two ordered pairs (1,2), (2,1) may be included or not: 2² = 4.

4. Q4. The number of onto functions from a set with 3 elements to a set with 2 elements is:

   • A. 2
   • B. 4
   • C. 6 ✓
   • D. 8

   Answer: C. 6

   Explanation: Total functions = 2³ = 8; subtract the 2 constant functions (not onto): 8 − 2 = 6.

5. Q5. The number of one-one functions from {a, b, c} to a set with 5 elements is:

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

   Answer: D. 60

   Explanation: Images must be distinct: 5 × 4 × 3 = 60.

The other 25 questions

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6. Q6. If f(x) = 2x + 3 and g(x) = x², then (g∘f)(2) is:

   • A. 25
   • B. 11
   • C. 13
   • D. 49

7. Q7. If f(x) = (2x + 1)/(x − 3), x ≠ 3, then f⁻¹(5) equals:

   • A. 16/3
   • B. 5/2
   • C. −14/3
   • D. 11/7

8. Q8. The domain of f(x) = √(x² − 5x + 6) over ℝ is:

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

9. Q9. The range of f(x) = x² + 4x + 7, x ∈ ℝ, is:

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

10. Q10. On A = {1, 2, 3}, the relation R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)} is:

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

11. Q11. The statement ~p → ~q is called the _____ of the statement p → q.

    • A. inverse
    • B. negation
    • C. contrapositive
    • D. converse

12. Q12. The compound statement (p ∧ q) → p is:

    • A. a contradiction
    • B. a tautology
    • C. true only when q is true
    • D. true only when p is false

13. Q13. The statement 'If a number is divisible by 6, then it is divisible by 3' has the contrapositive:

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

14. Q14. If U = {1, 2, …, 10}, A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, the number of elements in the symmetric difference (A − B) ∪ (B − A) is:

    • A. 2
    • B. 7
    • C. 3
    • D. 5

15. Q15. The number of elements in P(P(P(∅))), where P denotes the power set, is:

    • A. 1
    • B. 2
    • C. 8
    • D. 4

16. Q16. On ℝ, the operation a * b = a + b − 3 has an identity element, which is:

    • A. −3
    • B. 1
    • C. 3
    • D. 0

17. Q17. For the operation a * b = a + b − 3 on ℝ (identity 3), the inverse of 5 is:

    • A. −5
    • B. −2
    • C. 3/5
    • D. 1

18. Q18. The number of generators of the cyclic group (ℤ₆, +₆) is:

    • A. 2
    • B. 3
    • C. 6
    • D. 1

19. Q19. In the group (ℤ₁₂, +₁₂), the order of the element 8 is:

    • A. 4
    • B. 3
    • C. 6
    • D. 12

20. Q20. Which of the following is NOT a group?

    • A. (ℚ − {0}, ×), the non-zero rationals under multiplication
    • B. (ℝ, +), the reals under addition
    • C. (ℤ, ×), the integers under multiplication
    • D. (ℤ, +), the integers under addition

21. Q21. In the group of non-zero residues modulo 7 under multiplication, the inverse of 3 is:

    • A. 4
    • B. 2
    • C. 6
    • D. 5

22. Q22. In a group (G, *) with identity e, if an element a satisfies a * a = a, then:

    • A. a is a generator of G
    • B. a has order 2
    • C. a does not exist
    • D. a = e

23. Q23. In a group, the inverse of the product ab is:

    • A. a⁻¹b⁻¹
    • B. ab
    • C. b⁻¹a⁻¹
    • D. ba

24. Q24. The function f: ℝ → ℝ defined by f(x) = eˣ is:

    • A. both one-one and onto
    • B. one-one but not onto
    • C. neither one-one nor onto
    • D. onto but not one-one

25. Q25. If f and g are invertible functions, (g∘f)⁻¹ equals:

    • A. f⁻¹∘g⁻¹
    • B. f∘g
    • C. g∘f
    • D. g⁻¹∘f⁻¹

26. Q26. If A ⊂ B, then A − B equals:

    • A. B
    • B. ∅
    • C. B − A
    • D. A

27. Q27. If f(x) = 3 − 2x, then f⁻¹(−5) equals:

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

28. Q28. In the group G = {1, −1, i, −i} under multiplication, the number of elements of order 2 is:

    • A. 2
    • B. 3
    • C. 1
    • D. 0

29. Q29. If n(A) = 2 and n(B) = 2, the number of subsets of A × B is:

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

30. Q30. Let A = {x : x² − 3x + 2 = 0} and B = {x ∈ ℕ : x < 3}. Then:

    • A. B ⊂ A but A ≠ B
    • B. A ⊂ B but A ≠ B
    • C. A = B
    • D. A ∩ B = ∅