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

NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.1 help students understand the basic concepts of Relations and Functions in a clear and exam-oriented manner. This exercise focuses on the definition, properties, and types of relations and functions, providing a solid foundation for more advanced topics in mathematics.

Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 2.1 covers basic concepts like ordered pairs, Cartesian product, and the definition of a relation between sets. Students will also learn how to represent relations using set notation and how to check whether a relation is reflexive, symmetric, or transitive.

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

Relations and Functions

Easy

Q.

$\text{If\hspace{0.33em}}\left(\frac{\text{x}}{\text{3}}\text{+1, y-}\frac{\text{2}}{\text{3}}\right)\text{ = }\left(\frac{\text{5}}{\text{3}}\text{, }\frac{\text{1}}{\text{3}}\right)\text{, find the values of x and y.}$

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Relations and Functions

Medium

Q.

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.

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Relations and Functions

Medium

Q.

Let A = {1, 2, 3, 4, 6}. Let R be the relation on
A defined by {(a, b): a, b ∈ A, b is exactly divisible by a}.

(i) Write R in roster form
(ii) Find the domain of R
(iii) Find the range of R.

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Relations and Functions

Easy

Q.

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0),
(ii) f (7),
(iii) f (–3).

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Relations and Functions

Easy

Q.

$\begin{array}{l}\text{The function 't' which maps temperature in degree Celsius\hspace{0.33em}}\mathrm{into}\\ \text{temperature in degree Fahrenheit is defined by t}\left(\text{C}\right)=\frac{\text{9C}}{\text{5}}\text{+32.}\\ \text{Find}\\ \text{(i) \hspace{0.33em}t(0)}\\ \text{(ii)\hspace{0.33em}t(28)}\\ \text{(iii)\hspace{0.33em}t(–10)}\\ \text{(iv)\hspace{0.33em}The value of C, when t(C) = 212.}\end{array}$

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Relations and Functions

Difficult

Q.

$\begin{array}{l}\text{Let f = }\left\{\left(\text{x,\hspace{0.17em} }\frac{{\text{x}}^{\text{2}}}{{\text{1+x}}^{\text{2}}}\right)\text{ : \hspace{0.17em}x}\in \text{R}\right\}\text{ be a function from R to R.}\\ \text{Determine the range of f.}\end{array}$

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Relations and Functions

Medium

Q.

$\begin{array}{l}\text{Let f, g : R}\mathrm{\to }\text{R be defined, respectively by f(x) = x + 1,}\\ \text{g(x) = 2x - 3. Find f + g, f - g and }\frac{\text{f}}{\text{g}}\text{.}\end{array}$

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Relations and Functions

Medium

Q.

 $\begin{array}{l}{Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from }\\ {Z to Z defined by f(x) = ax + b, for some integers a, b.}\\ {Determine a, b.}\end{array}$ 

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Relations and Functions

Easy

Q.

State whether each of the following statements are 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 ∩ ∅) = ∅ .

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Relations and Functions

Difficult

Q.

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.

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Relations and Functions

Difficult

Q.

Find the range of each of the following functions.

(i) f (x) = 2 – 3x, x ∈ R, x > 0.
(ii) f (x) = x² + 2, x is a real number.
(iii) f (x) = x, x is a real number.

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Relations and Functions

Difficult

Q.

Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

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The solutions are explained in a step-by-step manner to help students easily grasp the concepts and solve problems confidently.

Question 1: If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.

• Answer: x = 2 and y = 1.
• Steps: By equating corresponding elements: x/3 + 1 = 5/3 leads to x = 2. Similarly, y - 2/3 = 1/3 leads to y = 1.

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

• Answer: 9.
• Steps: Since set A has 3 elements and set B also has 3 elements , the total elements in the product is 3 * 3 = 9.

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

• G x H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}.
• H x G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}.

Question 4: State whether True or False.

1. (i) If P = {m, n} and Q = {n, m}, then P x Q = {(m, n), (n, m)}.
   • Answer: False.
   • Correct Statement: P x Q = {(m, m), (m, n), (n, m), (n, n)}.
2. (ii) If A and B are non-empty sets, then A x B is a non-empty set of ordered pairs (x, y) such that x is in A and y is in B.
   • Answer: True.
3. (iii) If A = {1, 2}, B = {3, 4}, then A x (B intersection empty set) = empty set.
   • Answer: True.

Question 5: If A = {-1, 1}, find A x A x A.

• Answer: {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)}.

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

• Answer: A = {a, b} and B = {x, y}.
• Note: A is the set of all first elements and B is the set of all second elements.

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

• (i) A x (B intersection C) = (A x B) intersection (A x C)
  • Verification: Since B intersection C is an empty set, both the Left Hand Side and Right Hand Side equal the empty set.
• (ii) A x C is a subset of B x D
  • Verification: A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}. Since every element of A x C is also in B x D, it is a subset.

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

• A x B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
• Number of subsets: 16 (Calculated as 2 to the power of 4).
• Subsets: empty set, {(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)}.

Question 9: Let n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, find A and B.

• Answer: A = {x, y, z} and B = {1, 2}.

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

• Answer: A = {-1, 0, 1}.
• Remaining elements: (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1).

FAQs – Class 11 Maths Chapter 2 Exercise 2.1 Relations and Functions

Q1. What is the focus of Exercise 2.1?
Exercise 2.1 focuses on the basic concepts of relations and functions, including Cartesian products and the properties of relations (reflexive, symmetric, transitive).

Q2. What are the key concepts covered in this exercise?

• Ordered pairs

• Cartesian product of sets

• Definition of a relation

• Properties of relations (reflexive, symmetric, transitive)

Q3. How do we represent relations between sets?
Relations between sets are represented using set notation and by listing ordered pairs. For example, a relation from set A to set B can be represented as a set of ordered pairs like {(a, b), (c, d)}.

Q4. Why is Exercise 2.1 important for board exams?
This exercise lays the foundation for understanding relations and functions, which are crucial concepts in advanced topics such as compositions of functions and inverse functions.

Q5. How can students prepare effectively for Exercise 2.1?
Students should practice the definitions and properties of relations, solve problems involving Cartesian products, and work on understanding how relations are formed between sets.