ABHYAS · CLASS 12 STUDENTS (CBSE AND STATE BOARDS), JEE AND CUET PREPARATION
Class 12 Maths कक्षा 12 गणित
Unlimited Class 12 board-exam practice: relations, inverse trigonometry, matrices and determinants, continuity and differentiability, applications of derivatives, integrals and areas, differential equations, vectors, 3D geometry, linear programming and probability, with hints and worked steps.
Inverse trigonometric functions
Determinants
- Determinant of a 2 × 2 or 3 × 3 matrix|a b; c d| = ad − bc. For 3 × 3, expand along a row: a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃, where each M is a 2 × 2 minor. For an n × n matrix, |kA| = kⁿ|A|.
- Adjoint and inverse of a 2 × 2 matrixFor A = [a, b; c, d], adj A = [d, −b; −c, a] and A⁻¹ = (1/|A|) adj A, provided |A| = ad − bc ≠ 0.
Continuity and differentiability
- Continuity and differentiabilityf is continuous at c when the left limit, right limit and f(c) are all equal. A sharp corner (like |x| at 0) is continuous but not differentiable.
- Derivative by the chain rule at a pointd/dx [g(x)]ⁿ = n[g(x)]ⁿ⁻¹ × g′(x). Differentiate the outer function, keep the inside, then multiply by the derivative of the inside.