Class 12 Maths Chapter 1: Relations and Functions – Concepts, NCERT Solutions & PYQs with Answers (2025–26)

  Unit I – Relations & Functions from Class 12 CBSE Maths, including deeper insights, conceptual clarity, exercise solutions, and top-scoring CBSE answers:

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🧠 1. In-Depth Concept Review

📌 Relations

• A relation on a set $A$ is any subset of $A \times A$. Key types include :

  • Empty relation: no pair included.

  • Universal relation: all possible pairs included.

• ✅ Reflexive: ∀ $x$, $(x,x) \in R$.

• ✅ Symmetric: if $(x,y) \in R$, then $(y,x) \in R$.

• ✅ Transitive: if $(x,y), (y,z) \in R$, then $(x,z)\in R$.

• ✅ Equivalence relation: satisfies all three; partitions $A$ into equivalence classes.

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📌 Functions

• A relation is a function if each element in the domain corresponds to exactly one value in the codomain.

• Types of functions:

  • Injective (one‑to‑one): distinct inputs yield distinct outputs.

  • Surjective (onto): every element in the codomain is hit.

  • Bijective: both injective and surjective ⟶ has an inverse.

• Inverse function $f^{-1}$ exists only if $f$ is bijective 

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📌 Inverse Trigonometric Functions

Key principal domains and properties 

Function	Domain	Range
$\sin^{-1} x$	$[-1,1]$	$[-π/2, π/2]$
$\cos^{-1} x$	$[-1,1]$	$[0, π]$
$\tan^{-1} x$	ℝ	$(-π/2, π/2)$
$\cot^{-1} x$	ℝ	$(0, π)$
$\sec^{-1} x$	(	x
$\cosec^{-1} x$	(	x

• 🧩 Odd/even properties:

  • $\sin^{-1}(-x) = -\sin^{-1} x$,

  • $\cos^{-1}(-x) = π - \cos^{-1} x$, etc.

• 📌 Identities: $\sin^{-1} x + \cos^{-1} x = π/2$, etc. 

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📘 2. NCERT Exercise Overview

• Ex 1.1 – Relations: Check reflexivity, symmetry, transitivity; find equivalence relations and classes.

• Ex 1.2 – Functions: Define functions, analyze domain/range, determine injective, surjective, bijective.

• Ex 1.3 – Inverse & Composite: Find $f^{-1}$; verify $f(g(x))$, $g(f(x))$.

• Ex 1.4 – Binary Operations: Define operations, test associativity, commutativity, identity elements.

📚 Detailed NCERT solutions available on many platforms (BYJU'S, Cuemath).

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🏆 3. CBSE PYQs with Best-Scoring Answers

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🔹 2023: Minimal Equivalence Relation

• Q. Smallest equivalence relation on $A=\{1,2,3\}$ containing $(1,2)$.

• Answer: Add reflexivity and symmetry:

  $$\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$$

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🔹 2022: Bijective Function & Inverse

• Q. Show $f(x)=x-4$ is bijective; find $f^{-1}(x)$.

• Answer: Proved injective and surjective; inverse is $f^{-1}(x)=x+4$.

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🔹 2021: Nature of $f(x)=x^2$

• Answer: Injective on $\mathbb{N}$, not surjective → so it is only injective.

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🔹 2020: Range of a Quadratic

• Q. Find range of $x^2 -4x +5$.

• Answer: Completing square gives $(x-2)^2 +1 \ge 1$ ⇒ [1, ∞)).

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🔑 4. Exam-Acing Tips

• Notation: Always define domain and codomain clearly.

• Proof strategy: Use “Suppose..., then...” for injectivity/surjectivity.

• Inverse functions: Show $f(g(x)) = x$ and $g(f(x)) = x$.

• For Inverse Trigo: Always mention domain, range, and key identities.

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📌 #Tags:

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