Class 12 MathematicsChapter Notes

Key Definitions

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

Key Definitions

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