15 extra multiple-choice questions for Relations and Functions (10th Standard Mathematics, Samacheer Kalvi), beyond the ones printed in the textbook — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
If n(A) = 3 and n(B) = 4, how many relations can be defined from set A to set B?
- A. 12
- B. 64
- C. 1024
- D. 4096Correct
Explanation. The total number of relations from a set A of p elements to a set B of q elements is given by \(2^(p x q)\). Here, p = 3 and q = 4, so the number of relations is \(2^12 = 4096\).
Q2
If A x B = {(a, 1), (b, 2), (a, 2), (b, 1)}, then which of the following is set A?
- A. {1, 2}
- B. {a, b}Correct
- C. {a, b, 1, 2}
- D. {a, 1}
Explanation. The Cartesian product A x B consists of ordered pairs where the first coordinate belongs to A. Since the first coordinates in the given pairs are a and b, the set A is {a, b}.
Q3
If n(A) = 4 and n(B) = 3, what is the total number of functions that can be defined from set A to set B?
- A. 12
- B. 64
- C. 81Correct
- D. 128
Explanation. The total number of functions from a set A with p elements to a set B with q elements is given by \(q^p\). Here, p = 4 and q = 3, so the number of functions is \(3^4 = 81\).
Q4
For any two non-empty sets A and B, which of the following statements is always true regarding the cardinality of their Cartesian products?
- A. n(A x B) > n(B x A)
- B. n(A x B) < n(B x A)
- C. n(A x B) = n(B x A)Correct
- D. n(A x B) is not equal to n(B x A)
Explanation. The cardinalities of the Cartesian products A x B and B x A are always equal to the product of the cardinalities of both sets. Thus, n(A x B) is always equal to n(B x A).
Q5
Let A = {1, 2} and B = {3, 4}. If a relation R from A to B contains no elements, what is this relation called?
- A. Identity relation
- B. Null relationCorrect
- C. Universal relation
- D. Reflexive relation
Explanation. A relation from set A to set B is a subset of A x B. If the relation contains no elements, it is denoted by the empty set and is called a null relation.
Q6
A function f from X to Y is defined as f(x) = c for all x in X, where c is a fixed element in Y. What type of function is f?
- A. Identity function
- B. Constant functionCorrect
- C. One-one function
- D. Onto function
Explanation. A function f from X to Y is called a constant function if the range of f contains only a single element, meaning every element of the domain maps to the same constant value.
Q7
What is the domain of the real-valued function f(x) \(= 1/(x^2 - 9)\)?
- A. The set of all real numbers
- B. The set of all real numbers except 3
- C. The set of all real numbers except -3 and 3Correct
- D. The set of all positive real numbers
Explanation. The function f(x) is undefined when the denominator is zero. Since \(x^2 - 9\) is zero at x = 3 and x = -3, these values are excluded from the domain of all real numbers.
Q8
If a function f from A to B is both one-one and onto, then what is it called?
- A. Constant function
- B. Identity function
- C. BijectionCorrect
- D. Into function
Explanation. A function is bijective, or a bijection, if it is both one-one (injective) and onto (surjective). This means every element maps to a unique image and all co-domain elements have pre-images.
Q9
If f(x) = 2x + 1 and g(x) \(= x^2 - 2\), what is the value of the composition (g o f)(1)?
- A. 1
- B. 3
- C. 7Correct
- D. 9
Explanation. First evaluate f(1) = 2(1) + 1 = 3. Then find the composition (g o f)(1) by substituting this result into g(x), giving g(3) \(= 3^2 - 2 = 7\).
Q10
If f(x) = x + 2 and g(x) = 3x - 1, then which of the following represents (f o g)(x)?
- A. 3x + 1Correct
- B. 3x + 5
- C. 3x + 2
- D. 3x - 1
Explanation. To find the composition (f o g)(x), substitute g(x) = 3x - 1 into the function f(x), which gives f(3x - 1) = (3x - 1) + 2 = 3x + 1.
Q11
If f from A to B is a constant function and n(B) = 5, what is the cardinality of the range of f?
- A. 0
- B. 1Correct
- C. 5
- D. Cannot be determined
Explanation. A constant function maps all elements of its domain to exactly one single element in the co-domain. Therefore, the range of any constant function always contains exactly one element.
Q12
If the function f(x) = ax + b represents the identity function on the set of real numbers, then what are the values of a and b?
- A. a = 0, b = 1
- B. a = 1, b = 1
- C. a = 1, b = 0Correct
- D. a = 0, b = 0
Explanation. The identity function on any set maps every element to itself, so f(x) = x. Comparing this with the linear form ax + b gives a = 1 and b = 0.
Q13
If f, g, and h are three real-valued functions, then which of the following properties is always true for their compositions?
- A. Compositions are always commutative
- B. Compositions are always associativeCorrect
- C. Compositions are neither commutative nor associative
- D. Compositions are commutative but not associative
Explanation. Function composition is always associative, meaning (f o g) o h equals f o (g o h) for any three functions. However, composition is generally not commutative.
Q14
If A = {1, 2, 3} and B = {4}, what is the set A x B?
- A. {(1, 4), (2, 4), (3, 4)}Correct
- B. {(4, 1), (4, 2), (4, 3)}
- C. {1, 2, 3, 4}
- D. {(1, 4), (4, 2), (3, 4)}
Explanation. The Cartesian product A x B is the set of all ordered pairs (a, b) where a is in A and b is in B. Here, A = {1, 2, 3} and B = {4}, so we get {(1, 4), (2, 4), (3, 4)}.
Q15
What is the nature of the function f(x) \(= x^3\) defined from the set of real numbers to the set of real numbers?
- A. Linear function
- B. Quadratic function
- C. Cubic functionCorrect
- D. Reciprocal function
Explanation. A function of the form f(x) \(= ax^3 + bx^2 +\) cx + d with a not equal to zero is called a cubic function. Therefore, the function f(x) \(= x^3\) is a cubic function.
