ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Sets, Functions and Groups (Set 2)

Take the ECAT Mathematics — Sets, Functions and Groups (Set 2): 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. The converse of the conditional p → q is:

   • A. q → p ✓
   • B. ~q → ~p
   • C. p → ~q
   • D. ~p → ~q

   Answer: A. q → p

   Explanation: The converse swaps hypothesis and conclusion: q → p. (~q → ~p is the contrapositive and ~p → ~q the inverse.)

2. Q2. In a group G, if ab = ac for elements a, b, c ∈ G, then:

   • A. b = c ✓
   • B. b = a⁻¹c only when G is abelian
   • C. b and c need not be equal
   • D. a = e

   Answer: A. b = c

   Explanation: Multiply on the left by a⁻¹: a⁻¹(ab) = a⁻¹(ac) ⇒ b = c (left cancellation law, valid in every group).

3. Q3. The biconditional p ↔ q is true exactly when:

   • A. at least one of p, q is true
   • B. p and q are both false only
   • C. p is true and q is false
   • D. p and q have the same truth value ✓

   Answer: D. p and q have the same truth value

   Explanation: p ↔ q is true when both are true or both are false, i.e. when they have the same truth value.

4. Q4. A truth table for a compound statement involving three simple statements p, q and r has how many rows?

   • A. 3
   • B. 8 ✓
   • C. 6
   • D. 9

   Answer: B. 8

   Explanation: Each statement can be T or F, giving 2³ = 8 combinations.

5. Q5. The domain of the real function f(x) = √(4 − x²) is:

   • A. [−2, 2] ✓
   • B. (−2, 2)
   • C. [0, 2]
   • D. ℝ − {−2, 2}

   Answer: A. [−2, 2]

   Explanation: We need 4 − x² ≥ 0, i.e. x² ≤ 4, so −2 ≤ x ≤ 2.

The other 25 questions

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6. Q6. If A = {prime numbers less than 10} and B = {odd natural numbers less than 10}, then A ∩ B is:

   • A. {1, 3, 5, 7, 9}
   • B. {3, 5, 7, 9}
   • C. {2, 3, 5, 7}
   • D. {3, 5, 7}

7. Q7. Which statement about a group is always true?

   • A. The identity element is unique
   • B. The group is commutative
   • C. Every element is its own inverse
   • D. The group has finitely many elements

8. Q8. The set of all 2 × 2 non-singular real matrices under matrix multiplication is:

   • A. a group only under addition
   • B. a non-abelian group
   • C. an abelian group
   • D. not closed, so not a group

9. Q9. A binary operation * on ℤ is defined by a * b = a + b − 2. The inverse of 5 under * is:

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

10. Q10. Consider the absolute-value mapping g(x) = |x| from the set of real numbers to itself. Which description of g is correct?

    • A. neither one-one nor onto
    • B. a bijection
    • C. onto but not one-one
    • D. one-one but not onto

11. Q11. The compound statement p ∧ ~p is:

    • A. a contradiction (absurdity)
    • B. equivalent to p
    • C. a contingency
    • D. a tautology

12. Q12. If n(A) = 10, n(B) = 12, n(C) = 15, n(A ∩ B) = 4, n(B ∩ C) = 5, n(A ∩ C) = 6 and n(A ∩ B ∩ C) = 2, then n(A ∪ B ∪ C) is:

    • A. 22
    • B. 26
    • C. 37
    • D. 24

13. Q13. If U = {1, 2, 3, …, 10} and A = {2, 4, 6, 8, 10}, then A′ is:

    • A. {1, 3, 5, 7, 9}
    • B. {2, 4, 6, 8, 10}
    • C. {1, 3, 5, 7}
    • D. ∅

14. Q14. The negation of p ∧ q is logically equivalent to:

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

15. Q15. Which of the following is NOT a group under ordinary multiplication?

    • A. ℤ − {0}
    • B. ℚ − {0}
    • C. ℝ − {0}
    • D. {1, −1}

16. Q16. The power set of the set {∅} is:

    • A. {∅}
    • B. {{∅}}
    • C. ∅
    • D. {∅, {∅}}

17. Q17. For any sets A and B, the difference A − B is equal to:

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

18. Q18. If f(x) = 2x + 1 and g(x) = x − 3, then (f∘g)⁻¹(5) equals:

    • A. −2
    • B. 5
    • C. 3
    • D. 11

19. Q19. In a group of 50 people, 30 drink tea, 25 drink coffee and 10 drink both. How many drink exactly one of the two?

    • A. 25
    • B. 40
    • C. 35
    • D. 45

20. Q20. If (x + 1, y − 2) = (3, 1), then x + y equals:

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

21. Q21. How many distinct mappings (functions) exist from a set containing 3 elements into a set containing 2 elements?

    • A. 64
    • B. 8
    • C. 9
    • D. 6

22. Q22. Of 100 students, 60 play cricket, 50 play hockey and 30 play both. How many students play neither game?

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

23. Q23. The number of one-one functions from a set with 3 elements into a set with 4 elements is:

    • A. 12
    • B. 64
    • C. 81
    • D. 24

24. Q24. If f(x) = 2x − 1 and g(x) = x², then (g∘f)(3) is:

    • A. 25
    • B. 17
    • C. 35
    • D. 9

25. Q25. The set identity A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) is known as the:

    • A. associative law
    • B. commutative law
    • C. distributive law
    • D. De Morgan's law

26. Q26. How many different binary operations can be defined on a set with 2 elements?

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

27. Q27. The range of the real function f(x) = 1/(x − 1), x ≠ 1, is:

    • A. ℝ − {0}
    • B. (0, ∞)
    • C. ℝ
    • D. ℝ − {1}

28. Q28. If the power set of A has 64 elements, then the number of elements in A is:

    • A. 6
    • B. 32
    • C. 8
    • D. 7

29. Q29. The function f: ℕ → ℕ defined by f(n) = 2n is:

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

30. Q30. If f(x) = (x − 1)/(x + 2), x ≠ −2, then f⁻¹(x) is:

    • A. (1 − 2x)/(x − 1)
    • B. (2x − 1)/(1 + x)
    • C. (2x + 1)/(1 − x)
    • D. (x + 2)/(x − 1)