CA Foundation P3 · Chapter 7

Sets, Relations and Functions, Basics of Limits and Continuity of Functions MCQs with Answers

9 multiple-choice questions on Sets, Relations and Functions, Basics of Limits and Continuity of Functions for CA Foundation P3 Quantitative Aptitude. Try each one before revealing the answer and explanation.

1. Question 1

   The number of proper subsets of the set A = {1, 2, 3, 4} is:

   • A) 16
   • B) 15
   • C) 14
   • D) 8
   Show answer & explanation

   Answer: B) 15

   A set with n elements has 2ⁿ subsets. Here 2⁴ = 16. Excluding the set itself, the number of proper subsets is 16 − 1 = 15.

2. Question 2

   In a group of 200 people, 120 drink tea, 90 drink coffee and 40 drink both. The number of people who drink neither tea nor coffee is:

   • A) 70
   • B) 40
   • C) 30
   • D) 10
   Show answer & explanation

   Answer: C) 30

   n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 120 + 90 − 40 = 170. Neither = 200 − 170 = 30. Forgetting to subtract the overlap gives 200 − 210, which is impossible, and is a sign the intersection was missed.

3. Question 3

   The relation R = {(1, 1), (2, 2), (3, 3), (1, 2)} on the set {1, 2, 3} is:

   • A) An equivalence relation
   • B) Symmetric only
   • C) Reflexive and transitive but not symmetric
   • D) Reflexive only
   Show answer & explanation

   Answer: C) Reflexive and transitive but not symmetric

   R contains (1,1), (2,2), (3,3), so it is reflexive. It is not symmetric because (1, 2) ∈ R but (2, 1) ∉ R. For transitivity, the only chains are (1,1),(1,2) → (1,2) and (1,2),(2,2) → (1,2), both in R, so it is transitive. Hence it is not an equivalence relation.

4. Question 4

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

   • A) 49
   • B) 7
   • C) 16
   • D) 11
   Show answer & explanation

   Answer: D) 11

   (f∘g)(2) = f(g(2)) = f(4) = 2(4) + 3 = 11. The value 49 is (g∘f)(2) = g(7) = 49, which reverses the order of composition.

5. Question 5

   The inverse of the function f(x) = (3x − 5)/2 is:

   • A) f⁻¹(x) = (2x + 5)/3
   • B) f⁻¹(x) = (2x − 5)/3
   • C) f⁻¹(x) = 2/(3x − 5)
   • D) f⁻¹(x) = (3x + 5)/2
   Show answer & explanation

   Answer: A) f⁻¹(x) = (2x + 5)/3

   Let y = (3x − 5)/2. Then 2y = 3x − 5, so x = (2y + 5)/3. Interchanging, f⁻¹(x) = (2x + 5)/3. Check: f(f⁻¹(x)) = (2x + 5 − 5)/2 = x. Note that 1/f(x) is not the inverse function.

6. Question 6

   The value of lim (x→3) (x² − 9)/(x − 3) is:

   • A) 0
   • B) 6
   • C) 3
   • D) 9
   Show answer & explanation

   Answer: B) 6

   Direct substitution gives 0/0. Factorising, (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3. Hence the limit is 3 + 3 = 6.

7. Question 7

   The value of lim (x→∞) (4x² + 3x)/(2x² − 7) is:

   • A) 0
   • B) 2
   • C) ∞
   • D) 1/2
   Show answer & explanation

   Answer: B) 2

   Divide numerator and denominator by x²: (4 + 3/x)/(2 − 7/x²). As x → ∞, 3/x → 0 and 7/x² → 0, so the limit is 4/2 = 2. Inverting the ratio of leading coefficients gives 1/2, and concluding 0 or ∞ ignores that both degrees are equal.

8. Question 8

   The function f(x) = kx + 1 for x ≤ 2 and f(x) = 3x − 1 for x > 2 is continuous at x = 2. The value of k is:

   • A) 2
   • B) 3
   • C) 5/2
   • D) 1
   Show answer & explanation

   Answer: A) 2

   For continuity, the left limit, right limit and f(2) must be equal. Left value: 2k + 1. Right limit: 3(2) − 1 = 5. So 2k + 1 = 5 and k = 2.

9. Question 9

   If A has 3 elements and B has 5 elements, the number of one-one functions from A to B is:

   • A) 125
   • B) 243
   • C) 60
   • D) 10
   Show answer & explanation

   Answer: C) 60

   A one-one function assigns distinct images to the 3 elements of A: 5 choices for the first, 4 for the second and 3 for the third, i.e. ⁵P₃ = 60. The total number of functions is 5³ = 125, 3⁵ = 243 reverses domain and co-domain, and ⁵C₃ = 10 ignores order.