NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.1 Question Answers

NCERT Solutions for Class 11 Maths Chapter 2 Relations And Functions Exercise 2.1

Below is the NCERT Solutions for Class 11 Maths Chapter 2 Relations And Functions Exercise 2.1:

1. If , find the values of x and y .

Solution:

Given, As the ordered pairs are equal, the corresponding elements should also be equal. Thus, x/3 + 1 = 5/3 and y – 2/3 = 1/3 Solving, we get x + 3 = 5 and 3y – 2 = 1 [Taking L.C.M and adding] x = 2 and 3y = 3 Therefore, x = 2 and y = 1

2. If set A has 3 elements and set B = {3, 4, 5}, then find the number of elements in (A × B).

Solution:

Given, set A has 3 elements, and the elements of set B are {3, 4, and 5}. So, the number of elements in set B = 3 Then, the number of elements in (A × B) = (Number of elements in A) × (Number of elements in B) = 3 × 3 = 9 Therefore, the number of elements in (A × B) will be 9.

3. If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.

Solution:

Given, G = {7, 8} and H = {5, 4, 2} We know that The Cartesian product of two non-empty sets, P and Q, is given as P × Q = {( p , q ): p ∈ P, q ∈ Q} So, G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)} H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

4. State whether each of the following statements is true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = { m , n } and Q = { n , m }, then P × Q = {( m , n ), ( n , m )}.

(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs ( x , y ) such that x ∈ A and y ∈ B.

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.

Solution:

(i) The statement is False. The correct statement is If P = { m , n } and Q = { n , m }, then P × Q = {( m , m ), ( m , n ), ( n, m ), ( n , n )} (ii) True (iii) True

5. If A = {–1, 1}, find A × A × A.

Solution:

The A × A × A for a non-empty set A is given by A × A × A = {( a , b , c ): a , b , c ∈ A} Here, it is given A = {–1, 1} So, A × A × A = {(–1, –1, –1), (–1, –1, 1), (–1, 1, –1), (–1, 1, 1), (1, –1, –1), (1, –1, 1), (1, 1, –1), (1, 1, 1)}

6. If A × B = {( a , x ), ( a , y ), ( b , x ), ( b , y )}. Find A and B.

Solution:

Given, A × B = {( a , x ), ( a, y ), ( b , x ), ( b , y )} We know that the Cartesian product of two non-empty sets, P and Q, is given by P × Q = {( p , q ): p ∈ P, q ∈ Q} Hence, A is the set of all first elements, and B is the set of all second elements. Therefore, A = { a , b } and B = { x , y }

7. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × C is a subset of B × D

Solution:

Given, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} (i) To verify: A × (B ∩ C) = (A × B) ∩ (A × C) Now, B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = Φ Thus, L.H.S. = A × (B ∩ C) = A × Φ = Φ Next, A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)} A × C = {(1, 5), (1, 6), (2, 5), (2, 6)} Thus, R.H.S. = (A × B) ∩ (A × C) = Φ Therefore, L.H.S. = R.H.S. Hence, verified (ii) To verify: A × C is a subset of B × D First, A × C = {(1, 5), (1, 6), (2, 5), (2, 6)} And, B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)} Now, it’s clearly seen that all the elements of set A × C are the elements of set B × D. Thus, A × C is a subset of B × D. Hence, verified

8. Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.

Solution:

Given, A = {1, 2} and B = {3, 4} So, A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} Number of elements in A × B is n (A × B) = 4 We know that If C is a set with n (C) = m , then n [P(C)] = 2 ^m . Thus, the set A × B has 2 ⁴ = 16 subsets And these subsets are as below: Φ, {(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}, {(1, 4), (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 3), (2, 3), (2, 4)}, {(1, 4), (2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3), (2, 4)}

9. Let A and B be two sets such that n (A) = 3 and n (B) = 2. If ( x , 1), ( y , 2), ( z , 1) are in A × B, find A and B, where x , y and z are distinct elements.

Solution:

Given, n (A) = 3 and n (B) = 2; and ( x , 1), ( y , 2), ( z , 1) are in A × B. We know that A = Set of first elements of the ordered pair elements of A × B B = Set of second elements of the ordered pair elements of A × B. So, clearly, x , y , and z are the elements of A; and 1 and 2 are the elements of B. As n (A) = 3 and n (B) = 2, it is clear that set A = { x , y , z } and set B = {1, 2}.

10. The Cartesian product A × A has 9 elements, among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.

Solution:

We know that If n (A) = p and n (B) = q, then n (A × B) = pq . Also, n (A × A) = n (A) × n (A) Given, n (A × A) = 9 So, n (A) × n (A) = 9 Thus, n (A) = 3 Also, given that the ordered pairs (–1, 0) and (0, 1) are two of the nine elements of A × A. And, we know in A × A = {( a, a ): a ∈ A}. Thus, –1, 0, and 1 have to be the elements of A. As n (A) = 3, clearly A = {–1, 0, 1} Hence, the remaining elements of set A × A are as follows: (–1, –1), (–1, 1), (0, –1), (0, 0), (1, –1), (1, 0), and (1, 1)

Benefits of Solving NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.1

• Clear Understanding of Concepts: The exercise focuses on Relations and Functions, and solving these problems helps students develop a deep understanding of important mathematical concepts like domain, codomain, and range.
• Improved Problem-Solving Skills : By practicing the problems, students learn how to apply theoretical knowledge to practical problems, enhancing their problem-solving abilities.
• Stronger Foundation : This exercise builds a solid foundation for advanced topics in mathematics, especially those related to functions, which are crucial for further studies in calculus, algebra, and other branches of mathematics.
• Boosts Exam Preparation : Regularly solving exercises from NCERT helps students familiarize themselves with the types of questions commonly asked in exams, improving accuracy and speed during the actual exam.
• Step-by-Step Solutions : The NCERT solutions provide detailed, step-by-step explanations, making it easier for students to understand how to approach each problem and ensuring they don’t miss any important steps.
• Enhances Confidence : As students solve more problems correctly, their confidence in tackling related topics grows, which positively impacts their overall learning and performance in mathematics.