Summary Mathematics formula sheet for cbse 12 student

CBSE CLASS 12 MATHEMATICS
Unit-wise Notes (Code 041) — Session 2025–26
Theory: 80 marks | Internal Assessment: 20 marks | 6 Units, 13 Chapters


UNIT I — RELATIONS AND FUNCTIONS (8 Marks)
1. Relations and Functions
Types of Relations
• Empty relation: R = ∅ (no element related to any element)
• Universal relation: R = A × A (every element related to every element)
• Reflexive: (a, a) ∈ R for every a ∈ A
• Symmetric: (a, b) ∈ R ⟹ (b, a) ∈ R
• Transitive: (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R
• Equivalence relation: relation that is reflexive, symmetric AND transitive
• Equivalence classes partition the set A into disjoint subsets

Types of Functions
• One-one (injective): f(x₁) = f(x₂) ⟹ x₁ = x₂
• Onto (surjective): every element of codomain has a pre-image
• Bijective: both one-one and onto (invertible)
• A function is invertible iff it is bijective

Key Tips
• Use horizontal-line test conceptually: injective if no horizontal line cuts graph more than once
• For onto, check if range = codomain
• Composition of functions: (gof)(x) = g(f(x)); check domain compatibility

2. Inverse Trigonometric Functions
Principal Value Branches
• sin⁻¹x : domain [-1,1], range [-π/2, π/2]
• cos⁻¹x : domain [-1,1], range [0, π]
• tan⁻¹x : domain R, range (-π/2, π/2)
• cot⁻¹x : domain R, range (0, π)
• sec⁻¹x : domain R−(-1,1), range [0, π] − {π/2}
• cosec⁻¹x : domain R−(-1,1), range [-π/2, π/2] − {0}

Important Identities
• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• sec⁻¹x + cosec⁻¹x = π/2
• tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)], when xy < 1
• tan⁻¹x − tan⁻¹y = tan⁻¹[(x−y)/(1+xy)]
• 2 tan⁻¹x = sin⁻¹[2x/(1+x²)] = tan⁻¹[2x/(1−x²)] (|x| < 1)
• sin⁻¹(−x) = −sin⁻¹x ; cos⁻¹(−x) = π − cos⁻¹x (odd/even behaviour differs)

, UNIT II — ALGEBRA (10 Marks)
3. Matrices
Basics
• Order m × n means m rows, n columns
• Types: row, column, square, diagonal, scalar, identity, zero matrix
• Symmetric: A' = A ; Skew-symmetric: A' = −A (diagonal elements = 0)
• Every square matrix = symmetric part + skew-symmetric part: A = ½(A+A') + ½(A−A')

Operations
• Addition/subtraction defined only for same order matrices
• Matrix multiplication: possible only if columns of A = rows of B; AB ≠ BA generally
• (AB)' = B'A' ; (A')' = A ; (kA)' = kA'

Invertibility
• A square matrix A is invertible iff |A| ≠ 0
• A⁻¹ = (1/|A|) · adj(A)
• (AB)⁻¹ = B⁻¹A⁻¹ ; (A⁻¹)⁻¹ = A

4. Determinants
Key Formulas
• For 2×2: |A| = a₁₁a₂₂ − a₁₂a₂₁
• Expansion of 3×3 determinant along any row/column
• Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃): use ½|determinant| form
• Collinearity of 3 points: area = 0 ⟺ determinant = 0

Adjoint & Inverse
• adj(A) = transpose of cofactor matrix
• A · adj(A) = adj(A) · A = |A| · I
• |adj A| = |A|ⁿ⁻¹ for n×n matrix

Solving Linear Equations (Cramer's Rule / Matrix Method)
• System AX = B ⟹ X = A⁻¹B, valid only when |A| ≠ 0 (unique solution)
• If |A| = 0 and (adj A)B = 0 → infinitely many solutions; if (adj A)B ≠ 0 → no solution


UNIT III — CALCULUS (35 Marks — Highest Weightage)
5. Continuity and Differentiability
Continuity
• f is continuous at x = a if lim(x→a) f(x) = f(a)
• Check LHL = RHL = f(a) for piecewise functions

Differentiability
• Differentiable ⟹ continuous (converse not always true, e.g. |x| at 0)
• Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
• Logarithmic differentiation used for xˣ-type or product/quotient of many functions
• Implicit differentiation: differentiate both sides w.r.t. x, treat y as function of x