NCERT Solutions Class 12 Maths Chapter 1 helps you understand Relations and Functions through step-by-step solutions to all 35 questions across 3 exercises. You’ll learn how to determine if relations are reflexive, symmetric, or transitive, prove functions are one-one or onto, find inverse functions, and work with binary operations. These concepts are crucial for calculus and appear frequently in CBSE board exams and competitive tests like JEE.
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Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 1 Relations and Functions – Complete Guide
NCERT Class 12 Chapter 1 Relations and Functions introduces you to one of the most fundamental concepts in mathematics that bridges algebra and calculus. You’ll explore how relations connect elements between sets and understand when a relation qualifies as a function. This chapter builds upon your Class 11 knowledge and takes it to an advanced level, covering equivalence relations, types of functions, and their properties.
📊 CBSE Class 12 Maths Chapter 1 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 4 Marks |
| Unit Name | Relations and Functions |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will learn to identify and prove whether relations are reflexive, symmetric, transitive, or equivalence relations through systematic approaches. The chapter extensively covers different types of functions—one-one (injective), onto (surjective), and bijective functions—which are crucial for understanding inverse functions. You’ll also master the composition of functions and learn to find inverse functions, skills that are essential for solving complex calculus problems.
For CBSE board exams, this chapter typically carries 4 marks and includes a mix of 2-mark and 4-mark questions. You can expect problems asking you to prove whether given relations are equivalence relations, determine the type of function, find the composition of functions, or verify if a function is invertible. These questions test both your theoretical understanding and problem-solving abilities.
Quick Facts – Class 12 Chapter 1
| 📖 Chapter Number | Chapter 1 |
| 📚 Chapter Name | Relations and Functions |
| ✏️ Total Exercises | 3 Exercises |
| ❓ Total Questions | 35 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Relations and Functions is critical because these concepts appear throughout your Class 12 mathematics journey, especially in calculus, probability, and linear programming. The logical thinking and proof techniques you develop here will strengthen your mathematical reasoning for competitive exams like JEE and other entrance tests. With consistent practice of NCERT solutions and understanding the underlying concepts, you’ll find this chapter both manageable and rewarding.
NCERT Solutions Class 12 Maths Chapter 1 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 1.1 | Complete step-by-step solutions for 16 questions | 📥 Download PDF |
| EXERCISE 1.2 | Complete step-by-step solutions for 12 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 1 | Complete step-by-step solutions for 7 questions | 📥 Download PDF |
Relations and Functions – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Reflexive Relation \((a, a) \in R\) for all \(a \in A\) | A relation R on a set A is reflexive if every element of A is related to itself. Note: All elements of A MUST be related to themselves. If even one element is not related to itself, the relation is NOT reflexive. | Checking if a given relation is reflexive. |
| Symmetric Relation If \((a, b) \in R\), then \((b, a) \in R\) | A relation R on a set A is symmetric if whenever (a, b) is in R, then (b, a) is also in R. Note: If (a, b) is NOT in R, then the condition for symmetry is automatically satisfied for that pair. Only if (a, b) IS in R, then (b, a) MUST be in R. | Checking if a given relation is symmetric. |
| Transitive Relation If \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\) | A relation R on a set A is transitive if whenever (a, b) and (b, c) are in R, then (a, c) is also in R. Note: If (a, b) and (b, c) are in R, but (a, c) is NOT in R, then the relation is NOT transitive. If either (a, b) or (b, c) is not in R, then the condition is satisfied for those pairs. | Checking if a given relation is transitive. |
| Equivalence Relation R is Reflexive AND Symmetric AND Transitive | A relation R is an equivalence relation if it is reflexive, symmetric, and transitive. Note: You MUST prove all three properties (reflexivity, symmetry, and transitivity) to show that a relation is an equivalence relation. | Identifying equivalence relations and proving that a given relation is an equivalence relation. |
| Number of Equivalence Relations Bell Numbers (Difficult to calculate directly for large sets) | Finding the total number of equivalence relations on a set. This is related to the Bell numbers. Note: Bell numbers represent the number of ways to partition a set. The first few Bell numbers are 1, 1, 2, 5, 15… | For small sets, list all possibilities. For larger sets, this is a more advanced topic. |
| One-to-One Function (Injective) If \(f(x_1) = f(x_2)\), then \(x_1 = x_2\) | A function f: A -> B is one-to-one if different elements in A have different images in B. Alternatively, horizontal line test. Note: Contrapositive: If \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\). Sometimes easier to prove the contrapositive. | Proving that a function is one-to-one. If \(x_1\) and \(x_2\) are different, their function values must be different. |
| Onto Function (Surjective) For every \(b \in B\), there exists \(a \in A\) such that \(f(a) = b\) | A function f: A -> B is onto if every element in B has a pre-image in A. Range = Codomain. Note: Range of f = Codomain of f. Check if the range of the function is equal to the entire codomain. | Proving that a function is onto. Show that for any element in the codomain, you can find an element in the domain that maps to it. |
| Bijective Function f is One-to-One AND Onto | A function f: A -> B is bijective if it is both one-to-one and onto. Note: A bijective function has an inverse function. | Proving that a function is a bijection (one-to-one correspondence). |
| Composition of Functions \((f \circ g)(x) = f(g(x))\) | The composition of function f with g. Apply g first, then apply f to the result. Note: The domain of \(f \circ g\) is the set of all x in the domain of g such that g(x) is in the domain of f. Order matters! | Evaluating composite functions and finding the formula for a composite function. |
| Inverse of a Function If \(f(x) = y\), then \(f^{-1}(y) = x\) | The inverse of a function f, denoted by \(f^{-1}\). Note: A function has an inverse if and only if it is bijective. To find the inverse, swap x and y in the equation and solve for y. | Finding the inverse of a function and proving that a function is invertible. |
| Inverse Composition \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\) | Composition of a function with its inverse results in the identity function. Note: If \((f \circ g)(x) = x\) AND \((g \circ f)(x) = x\), then g(x) = \(f^{-1}(x)\). | Verifying that two functions are inverses of each other. |
Frequently Asked Questions – NCERT Class 12 Maths Chapter 1
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