Chapter 1: Relations and Functions — Online MCQ Test
MATHS · Grade 12 · CBSE(NCERT)
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This chapter covers: Chapter 1: Relations and Functions CBSC.
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Chapter 1: Relations and Functions — Important Questions & Answers
A relation R from set A to set B is defined as:
- A. A subset of A × B
- B. A subset of A ∪ B
- C. A subset of A ∩ B
- D. Equal to A × B
Answer: A. A subset of A × B
By definition in NCERT, a relation from set A to set B is any subset of the Cartesian product A × B.
By definition in NCERT, a relation from set A to set B is any subset of the Cartesian product A × B.
If A = {1, 2, 3} and B = {4, 5}, then the number of elements in A × B is:
- A. 5
- B. 6
- C. 8
- D. 10
Answer: B. 6
The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.
The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.
A function f: A → B is defined as a relation where:
- A. Every element of A has at least one image in B
- B. Every element of A has exactly one image in B
- C. Every element of B has a pre-image in A
- D. Some elements of A may not have images in B
Answer: B. Every element of A has exactly one image in B
A function requires that each element in the domain has exactly one corresponding element in the codomain.
A function requires that each element in the domain has exactly one corresponding element in the codomain.
Which of the following is NOT a characteristic of a function?
- A. Every element in domain has exactly one image
- B. Domain must be non-empty
- C. An element in the codomain can have multiple pre-images
- D. An element in the domain can have multiple images
Answer: D. An element in the domain can have multiple images
For a relation to be a function, each domain element must have exactly one image; multiple images violate the definition.
For a relation to be a function, each domain element must have exactly one image; multiple images violate the definition.
The range of a function is:
- A. The set of all possible inputs
- B. The set of all actual outputs
- C. The set of all possible outputs
- D. Always equal to the codomain
Answer: B. The set of all actual outputs
Range is the set of all actual output values (f(A)), which is a subset of the codomain.
Range is the set of all actual output values (f(A)), which is a subset of the codomain.
A relation R on set A is reflexive if:
- A. (a, b) ∈ R for all a, b ∈ A
- B. (a, a) ∈ R for all a ∈ A
- C. If (a, b) ∈ R then (b, a) ∈ R
- D. If (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R
Answer: B. (a, a) ∈ R for all a ∈ A
A relation is reflexive when every element is related to itself, i.e., (a, a) ∈ R for all a ∈ A.
A relation is reflexive when every element is related to itself, i.e., (a, a) ∈ R for all a ∈ A.
For the relation R = {(1, 2), (2, 3), (1, 3)} on A = {1, 2, 3}, which property does R satisfy?
- A. Reflexive
- B. Symmetric
- C. Transitive
- D. Equivalence
Answer: C. Transitive
R is transitive because (1, 2) ∈ R and (2, 3) ∈ R implies (1, 3) ∈ R. It's not reflexive or symmetric.
R is transitive because (1, 2) ∈ R and (2, 3) ∈ R implies (1, 3) ∈ R. It's not reflexive or symmetric.
If f: A → B is a one-one (injective) function, then:
- A. f(a₁) = f(a₂) implies a₁ = a₂
- B. Every element in B has at least one pre-image in A
- C. f is necessarily onto (surjective)
- D. Range of f equals codomain B
Answer: A. f(a₁) = f(a₂) implies a₁ = a₂
A one-one function means different inputs map to different outputs: if f(a₁) = f(a₂) then a₁ = a₂.
A one-one function means different inputs map to different outputs: if f(a₁) = f(a₂) then a₁ = a₂.
If f: A → B and g: B → C are functions, for (g ∘ f)⁻¹ to exist, which condition must hold?
- A. Both f and g must be injective
- B. Both f and g must be surjective
- C. Both f and g must be bijective
- D. f must be injective and g must be surjective
Answer: C. Both f and g must be bijective
For the composition to have an inverse, both functions must be bijective so that (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹ exists.
For the composition to have an inverse, both functions must be bijective so that (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹ exists.
Consider the relation R = {(a, b) | a and b are natural numbers and a divides b}. Which property does R NOT have?
- A. Reflexive
- B. Symmetric
- C. Transitive
- D. All of the above
Answer: B. Symmetric
R is reflexive (a|a) and transitive (if a|b and b|c then a|c), but not symmetric (2|4 but 4∤2).
R is reflexive (a|a) and transitive (if a|b and b|c then a|c), but not symmetric (2|4 but 4∤2).