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Class 12 MathematicsChapter Notes
2 chapters · Definitions, key points, formulas & exam tips
Ch 1
Relations and Functions
Key Definitions
Relation: A subset of the Cartesian product A × B. Represents a connection between elements of two sets.
Function: A relation where every element of domain has exactly one image in codomain.
Bijective Function: A function that is both injective (one-one) and surjective (onto).
Key Points to Remember
- →Reflexive: (a,a) ∈ R for all a ∈ A.
- →Symmetric: (a,b) ∈ R ⟹ (b,a) ∈ R.
- →Transitive: (a,b) ∈ R and (b,c) ∈ R ⟹ (a,c) ∈ R.
- →Equivalence relation: reflexive + symmetric + transitive.
- →One-one (injective): f(a) = f(b) ⟹ a = b.
- →Onto (surjective): every element of codomain has at least one pre-image.
- →Invertible function must be bijective.
Exam Tips
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To prove a function is one-one: assume f(x₁) = f(x₂) and show x₁ = x₂.
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To prove onto: let y be in codomain, find x in domain such that f(x) = y.
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Composition of functions: (f∘g)(x) = f(g(x)).
Ch 2
Inverse Trigonometric Functions
Key Definitions
Principal Value: The value of inverse trig function that lies in the defined principal range.
Key Points to Remember
- →Range of sin⁻¹: [−π/2, π/2]. Range of cos⁻¹: [0, π]. Range of tan⁻¹: (−π/2, π/2).
- →sin⁻¹(sin x) = x only if x ∈ [−π/2, π/2].
- →sin⁻¹(−x) = −sin⁻¹(x). cos⁻¹(−x) = π − cos⁻¹(x).
- →sin⁻¹x + cos⁻¹x = π/2. tan⁻¹x + cot⁻¹x = π/2.
- →2tan⁻¹x = sin⁻¹(2x/1+x²) = cos⁻¹(1−x²/1+x²) = tan⁻¹(2x/1−x²).
Exam Tips
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Always check domain before writing inverse trig value.
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Identity problems: use compound angle formulas for tan⁻¹.
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Simplification: convert to single inverse trig function.