Chapter Exam

Chapter 2 - Relations and Functions — Online MCQ Test

Q: What is a relation in mathematics?

Answer: A. A subset of the Cartesian product of two sets — A relation is formally defined as a subset of the Cartesian product A × B, where A and B are two sets.

Q: If A = {1, 2, 3} and B = {a, b}, how many elements are in A × B?

Answer: C. 6 — The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.

Q: Which of the following is a function?

Answer: B. {(1, 2), (2, 3), (3, 4)} — A function requires each input to map to exactly one output. Only option B satisfies this condition.

Q: If f(x) = x² and g(x) = x + 1, what is (f ∘ g)(2)?

Answer: B. 9 — (f ∘ g)(2) = f(g(2)) = f(3) = 3² = 9. First apply g, then apply f to the result.

Q: What is the range of f(x) = √(x - 2) for x ≥ 2?

Answer: A. [0, ∞) — Since √(x - 2) gives non-negative outputs and can take any non-negative value as x varies from 2 to ∞, the range is [0, ∞).

Q: A relation that is reflexive, antisymmetric, and transitive is called a _____.

Answer: B. Partial order relation — A partial order (or partial ordering) is a relation that is reflexive, antisymmetric, and transitive.

Q: If f: A → B and g: B → C are both one-to-one functions, then (g ∘ f): A → C is _____.

Answer: A. One-to-one — The composition of two one-to-one functions is always one-to-one. If (g ∘ f)(a₁) = (g ∘ f)(a₂), then g(f(a₁)) = g(f(a₂)), which implies f(a₁) = f(a₂) (g is one-to-one), and then a₁ = a₂ (f is one-to-one).

Q: A function f: R → R satisfies f(x + y) = f(x) + f(y) and f(1) = 2. Then f(3.5) is _____.

Answer: A. 7 — From Cauchy's functional equation f(x+y) = f(x) + f(y) with f(1) = 2, we get f(n) = 2n for all integers n. Then f(3.5) = f(7/2) = (7/2)f(1) = (7/2) × 2 = 7 (assuming continuity or linearity on rationals).

MATHS · Grade 11 · CBSE(NCERT)

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Chapter 2 - Relations and Functions — Important Questions & Answers (FAQ)

Frequently asked questions from CBSE(NCERT) Grade 11 MATHS — Chapter 2 - Relations and Functions, with answers and explanations. These are sample questions; the exam has its own separate question set.

What is a relation in mathematics?

A. A subset of the Cartesian product of two sets ✓
B. A function that maps every element to itself
C. A set of ordered pairs with no specific rule
D. An equivalence between two algebraic expressions

Answer: A. A subset of the Cartesian product of two sets
A relation is formally defined as a subset of the Cartesian product A × B, where A and B are two sets.

If A = {1, 2, 3} and B = {a, b}, how many elements are in A × B?

A. 3
B. 5
C. 6 ✓
D. 2

Answer: C. 6
The Cartesian product A × B has |A| × |B| = 3 × 2 = 6 elements.

What is the domain of a function?

A. The set of all output values
B. The set of all input values for which the function is defined ✓
C. The set of all real numbers
D. The set of all integers

Answer: B. The set of all input values for which the function is defined
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Which of the following is a function?

A. {(1, 2), (1, 3), (2, 4)}
B. {(1, 2), (2, 3), (3, 4)} ✓
C. {(1, 1), (1, 2), (2, 2)}
D. {(2, 1), (2, 2), (2, 3)}

Answer: B. {(1, 2), (2, 3), (3, 4)}
A function requires each input to map to exactly one output. Only option B satisfies this condition.

If f(x) = x² and g(x) = x + 1, what is (f ∘ g)(2)?

A. 5
B. 9 ✓
C. 8
D. 7

Answer: B. 9
(f ∘ g)(2) = f(g(2)) = f(3) = 3² = 9. First apply g, then apply f to the result.

What is the range of f(x) = √(x - 2) for x ≥ 2?

A. [0, ∞) ✓
B. [2, ∞)
C. (-∞, ∞)
D. [1, ∞)

Answer: A. [0, ∞)
Since √(x - 2) gives non-negative outputs and can take any non-negative value as x varies from 2 to ∞, the range is [0, ∞).

A relation that is reflexive, antisymmetric, and transitive is called a _____.

A. Equivalence relation
B. Partial order relation ✓
C. Congruence relation
D. Total relation

Answer: B. Partial order relation
A partial order (or partial ordering) is a relation that is reflexive, antisymmetric, and transitive.

If f: A → B and g: B → C are both one-to-one functions, then (g ∘ f): A → C is _____.

A. One-to-one ✓
B. Onto
C. Neither one-to-one nor onto
D. Cannot be determined

Answer: A. One-to-one
The composition of two one-to-one functions is always one-to-one. If (g ∘ f)(a₁) = (g ∘ f)(a₂), then g(f(a₁)) = g(f(a₂)), which implies f(a₁) = f(a₂) (g is one-to-one), and then a₁ = a₂ (f is one-to-one).

Consider the function f: Z → Z defined by f(n) = n/2 if n is even, and f(n) = (n+1)/2 if n is odd. Is f surjective?

A. Yes, f is surjective because every integer has a pre-image ✓
B. No, because f is not defined for all integers
C. Yes, f is surjective because it covers all even integers
D. No, because negative integers cannot be in the range

Answer: A. Yes, f is surjective because every integer has a pre-image
For any integer m in Z, we can find an integer n such that f(n) = m. For example, if m ≥ 0, take n = 2m (even), then f(2m) = m. Thus f is onto (surjective).

A function f: R → R satisfies f(x + y) = f(x) + f(y) and f(1) = 2. Then f(3.5) is _____.

A. 7 ✓
B. 8
C. 6
D. 5

Answer: A. 7
From Cauchy's functional equation f(x+y) = f(x) + f(y) with f(1) = 2, we get f(n) = 2n for all integers n. Then f(3.5) = f(7/2) = (7/2)f(1) = (7/2) × 2 = 7 (assuming continuity or linearity on rationals).

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