Important Formulas for CBSE Class 12 Maths Board Exam 2026

CBSE Class 12 Maths Board Exam 2026 is scheduled to be conducted on 9 March 2026, making it important for students to begin focused revision of all key formulas. As the exam date approaches, a strong command over important formulas from chapters like relations and functions, matrices, determinants, continuity and differentiability, integration, vectors, 3D geometry, and probability becomes essential for scoring well. This formula collection is designed to help students quickly revise and confidently prepare for the board exam.

Class 12 Maths Board Exam 2026 Formulas

Preparing for the CBSE Class 12 Maths Board Exam requires a strong grasp of essential formulas.  It covers foundational topics like relations and functions, inverse trigonometric functions, matrices, and determinants, extending to calculus with differentiation and integration. Vector algebra, three-dimensional geometry, and probability are also included to ensure thorough revision for all key areas.

Inverse Trigonometric Functions

If a negative sign is present inside cos⁻¹x, sec⁻¹x, or cot⁻¹x, it results in π - cos⁻¹x, π - sec⁻¹x, and π - cot⁻¹x, respectively. For the other three inverse trigonometric functions (sin⁻¹x, tan⁻¹x, cosec⁻¹x), the negative sign comes out directly (e.g., -sin⁻¹x). (Memory Tip: For 'cos⁻¹x, sec⁻¹x, cot⁻¹x', a negative argument leads to 'π minus' the function. For 'sin⁻¹x, tan⁻¹x, cosec⁻¹x', the negative argument leads to a 'minus sign outside').

If an inverse trigonometric function is applied to its corresponding trigonometric function (e.g., sin(sin⁻¹x)), the argument x comes out, provided x is within the function's domain.

Domains and Principal Value Ranges

• sin⁻¹x: Domain [-1, 1], Range [-π/2, π/2]

• cos⁻¹x: Domain [-1, 1], Range [0, π]

• tan⁻¹x: Domain (-∞, ∞), Range (-π/2, π/2)

• cot⁻¹x: Domain (-∞, ∞), Range (0, π)

• cosec⁻¹x: Domain (-∞, -1] ∪ [1, ∞), Range [-π/2, π/2] (except 0)

• sec⁻¹x: Domain (-∞, -1] ∪ [1, ∞), Range [0, π] (except π/2)

Properties of Inverse Trigonometric Functions

• Negative Arguments:

• sin⁻¹(-x) = -sin⁻¹x

• cosec⁻¹(-x) = -cosec⁻¹x

• tan⁻¹(-x) = -tan⁻¹x

• cos⁻¹(-x) = π - cos⁻¹x

• sec⁻¹(-x) = π - sec⁻¹x

• cot⁻¹(-x) = π - cot⁻¹x

• Reciprocal Relations:

• sin⁻¹(1/x) = cosec⁻¹x

• cos⁻¹(1/x) = sec⁻¹x

• tan⁻¹(1/x) = cot⁻¹x

• Complementary Relations:

• sin⁻¹x + cos⁻¹x = π/2 (for x ∈ [-1, 1])

• tan⁻¹x + cot⁻¹x = π/2 (for x ∈ (-∞, ∞))

• cosec⁻¹x + sec⁻¹x = π/2 (for x ∈ (-∞, -1] ∪ [1, ∞))

• Identity with Same Trig Function (Principal Intervals):

• sin⁻¹(sinx) = x (if x ∈ [-π/2, π/2])

• cos⁻¹(cosx) = x (if x ∈ [0, π])

• tan⁻¹(tanx) = x (if x ∈ (-π/2, π/2])

• Addition/Subtraction Formulas:

• tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)) if xy < 1

• tan⁻¹x - tan⁻¹y = tan⁻¹((x-y)/(1+xy))

• Double Angle Formulas (2tan⁻¹x):

• 2tan⁻¹x = sin⁻¹(2x/(1+x²))

• 2tan⁻¹x = cos⁻¹((1-x²)/(1+x²))

• 2tan⁻¹x = tan⁻¹(2x/(1-x²))

Relation and Function

Types of Relations

• Reflexive Relation: For every element a in the set, (a, a) must belong to the relation.

• Symmetric Relation: If (a, b) belongs to the relation, then (b, a) must also belong.

• Transitive Relation: If (a, b) and (b, c) belong to the relation, then (a, c) must also belong.

• Equivalence Relation: A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive.

Types of Functions

• One-to-one (Injective) Function: If f(x₁) = f(x₂) implies x₁ = x₂.

• Onto (Surjective) Function: If the range of the function is equal to its codomain.

Matrices

Basic Concepts

• Order of a Matrix: m x n (m rows, n columns).

• Number of Elements: m * n.

• Addition/Subtraction: Orders must be the same; corresponding elements are added/subtracted.

• Multiplication (AB): If A is m x n and B is p x q, multiplication is possible only if n = p. Resultant matrix is m x q.

Matrix Types and Properties

• Transpose of a Matrix (Aᵀ): Rows and columns are interchanged.

• Symmetric Matrix: A matrix A is symmetric if Aᵀ = A (aᵢⱼ = aⱼᵢ).

• Skew-Symmetric Matrix: A matrix A is skew-symmetric if Aᵀ = -A (aᵢⱼ = -aⱼᵢ).

• All diagonal elements are always zero.

• The determinant of an odd order skew-symmetric matrix is zero.

Determinants

Area of Triangle

For vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the area of the triangle is Area = (1/2) |det(X)| where X is the matrix [[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]]. (Memory Tip: Area is always positive, so use modulus).

• Collinearity Condition: If three points are collinear, the area of the triangle formed by them is zero.

Adjoint and Inverse of a Matrix

• Adjoint of a Matrix (adj(A)): The transpose of the cofactor matrix. (Cofactor Cᵢⱼ = (-1)ⁱ⁺ʲ * Mᵢⱼ, where Mᵢⱼ is the minor).

• Inverse of a Matrix (A⁻¹): A⁻¹ = (1 / det(A)) * adj(A).

• An inverse exists only if det(A) ≠ 0 (invertible matrix).

• If det(A) = 0, the inverse does not exist.

Continuity and Differentiability

Continuity

A function f(x) is continuous at x = a if lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a) (Left Hand Limit = Right Hand Limit = Value of the function).

Differentiability

A function f(x) is differentiable at x = a if the limit lim (h→0) [f(a+h) - f(a)] / h exists.

Basic Differentiation Formulas

• d/dx (xⁿ) = nxⁿ⁻¹

• d/dx (c) = 0

• d/dx (sinx) = cosx

• d/dx (cosx) = -sinx

• d/dx (tanx) = sec²x

• d/dx (secx) = secx tanx

• d/dx (cotx) = -cosec²x

• d/dx (cosecx) = -cosecx cotx

• d/dx (eˣ) = eˣ

• d/dx (aˣ) = aˣ logₑa

• d/dx (log|x|) = 1/x

• d/dx (sin⁻¹x) = 1/√(1-x²)

• d/dx (cos⁻¹x) = -1/√(1-x²)

• d/dx (tan⁻¹x) = 1/(1+x²)

• d/dx (cot⁻¹x) = -1/(1+x²)

• d/dx (sec⁻¹x) = 1/(|x|√(x²-1))

• d/dx (cosec⁻¹x) = -1/(|x|√(x²-1))

Rules of Differentiation

• Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

• Product Rule: If y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x).

• Quotient Rule: If y = u(x)/v(x), then dy/dx = [u'(x)v(x) - u(x)v'(x)] / [v(x)]².

Application of Derivatives

Rate of Change

dy/dx represents the rate of change of y with respect to x.

Increasing and Decreasing Functions

• If f'(x) > 0, the function is strictly increasing.

• If f'(x) ≥ 0, the function is increasing.

• If f'(x) < 0, the function is strictly decreasing.

• If f'(x) ≤ 0, the function is decreasing.

Maxima and Minima

• First Derivative Test: Critical points α occur where f'(x) = 0 or is undefined.

• f'(x) changes from positive to negative at α means α is a local maxima.

• f'(x) changes from negative to positive at α means α is a local minima.

• Second Derivative Test: For critical point α where f'(α) = 0:

• If f''(α) > 0, α is a local minima.

• If f''(α) < 0, α is a local maxima.

• If f''(α) = 0, the test fails.

• Absolute Maxima and Minima in [a, b]:

1. Find critical points in (a, b).

2. Evaluate f(x) at critical points and endpoints a, b.

3. The largest value is absolute maximum, smallest is absolute minimum.

Integrals

Basic Integration Formulas

• ∫xⁿ dx = xⁿ⁺¹ / (n+1) + C (for n ≠ -1)

• ∫1 dx = x + C

• ∫cosx dx = sinx + C

• ∫sinx dx = -cosx + C

• ∫sec²x dx = tanx + C

• ∫cosec²x dx = -cotx + C

• ∫secx tanx dx = secx + C

• ∫cosecx cotx dx = -cosecx + C

• ∫eˣ dx = eˣ + C

• ∫(1/x) dx = log|x| + C

• ∫aˣ dx = aˣ / logₑa + C

• ∫(1/√(1-x²)) dx = sin⁻¹x + C

• ∫(1/(1+x²)) dx = tan⁻¹x + C

• ∫(1/(x√(x²-1))) dx = sec⁻¹x + C

Special Integrals

• ∫tanx dx = log|secx| + C or -log|cosx| + C

• ∫cotx dx = log|sinx| + C or -log|cosecx| + C

• ∫secx dx = log|secx + tanx| + C

• ∫cosecx dx = log|cosecx - cotx| + C

Integrals of Rational and Irrational Functions (Special Forms)

• ∫(1/(x² - a²)) dx = (1/(2a)) log|(x-a)/(x+a)| + C

• ∫(1/(a² - x²)) dx = (1/(2a)) log|(a+x)/(a-x)| + C

• ∫(1/(x² + a²)) dx = (1/a) tan⁻¹(x/a) + C

• ∫(1/√(a² - x²)) dx = sin⁻¹(x/a) + C

• ∫(1/√(x² + a²)) dx = log|x + √(x² + a²)| + C

• ∫(1/√(x² - a²)) dx = log|x + √(x² - a²)| + C

Integration by Parts

∫u v dx = u ∫v dx - ∫[ (du/dx) ∫v dx ] dx. The choice of u and v follows the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).

Partial Fractions

Applicable when the degree of the numerator is less than the denominator.

• Non-repeated Linear Factors: (x-a)(x-b) => A/(x-a) + B/(x-b)

• Repeated Linear Factors: (x-a)²(x-b) => A/(x-a) + B/(x-a)² + C/(x-b)

• Irreducible Quadratic Factor: (x²+a)(x-b) => (Ax+B)/(x²+a) + C/(x-b)

Integrals of Forms √(a² ± x²), √(x² ± a²)

• ∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) log|x + √(x² + a²)| + C

• ∫√(x² - a²) dx = (x/2)√(x² - a²) - (a²/2) log|x + √(x² - a²)| + C

• ∫√(a² - x²) dx = (x/2)√(a² - x²) + (a²/2) sin⁻¹(x/a) + C

Definite Integrals

Properties of Definite Integrals

• P0 (Change of Variable): ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt

• P1 (Interchange of Limits): ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx

• P2 (Breaking the Interval): ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜₐ f(x) dx (where a < c < b).

• P3 (f(a+b-x) Property): ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx

• Special Case (P4: f(a-x) Property): ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx

• P5 (0 to 2a Property): ∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a-x) dx

• If f(2a-x) = f(x), then ∫₀²ᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx

• If f(2a-x) = -f(x), then ∫₀²ᵃ f(x) dx = 0

• P6 (-a to a Property: Even/Odd Functions): ∫₋ᵃᵃ f(x) dx

• If f(x) is even (f(-x) = f(x)), then ∫₋ᵃᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx.

• If f(x) is odd (f(-x) = -f(x)), then ∫₋ᵃᵃ f(x) dx = 0.

Application of Integrals

Area Bounded by a Curve and Axes

• With respect to x-axis: Area = ∫ₐᵇ |f(x)| dx.

• (Memory Tip: Area is always positive, so use modulus).

• With respect to y-axis: Area = ∫ₐᵇ |g(y)| dy.

Area Between Two Curves

• With respect to x-axis: Area = ∫ₐᵇ |f_upper(x) - f_lower(x)| dx.

• With respect to y-axis: Area = ∫ₐᵇ |g_right(y) - g_left(y)| dy.

Differential Equations

Order and Degree

• Order: The highest order derivative present.

• Degree: The power of the highest order derivative (if in polynomial form).

Methods of Solving Differential Equations

• Variable Separable Method: Separate x terms with dx and y terms with dy, then integrate.

• Homogeneous Differential Equation: Substitute y = vx for dy/dx or x = vy for dx/dy.

• Linear Differential Equation (dy/dx + Py = Q or dx/dy + Px = Q):

• Integrating Factor (IF) = e^(∫P dx) (for dy/dx form) or e^(∫P dy) (for dx/dy form).

• Solution for dy/dx + Py = Q: y * IF = ∫(Q * IF) dx + C.

Vectors

Vector Representation and Magnitude

• Position Vector (P(x, y, z)): r = xî + yĵ + zk̂.

• Magnitude: |r| = √(x² + y² + z²).

• Unit Vector: â = a / |a|.

Direction Cosines and Direction Angles

• α, β, γ are direction angles. cosα, cosβ, cosγ are direction cosines (l, m, n).

• l² + m² + n² = 1.

• For r = xî + yĵ + zk̂, direction cosines are x/|r|, y/|r|, z/|r|.

Scalar (Dot) Product

• a ⋅ b = |a||b|cosθ.

• cosθ = (a ⋅ b) / (|a||b|).

• Perpendicularity Condition: If a ⋅ b = 0.

Vector (Cross) Product

• a × b = |a||b|sinθ n̂.

• |a × b| = |a||b|sinθ.

• Parallelism Condition: If a × b = 0 (implies a₁/b₁ = a₂/b₂ = a₃/b₃).

• Calculated as the determinant of a 3x3 matrix.

Projection

Projection of vector a on vector b: (a ⋅ b̂) = (a ⋅ b) / |b|.

Area Applications

• Area of a Triangle: (1/2) |a × b| (where a, b are adjacent sides).

• Area of a Parallelogram:

• Adjacent sides a, b: |a × b|.

• Diagonals d₁, d₂: (1/2) |d₁ × d₂|.

Three-Dimensional Geometry

Direction Cosines and Ratios

• Direction Cosines (l, m, n): l² + m² + n² = 1.

• Direction Ratios (a, b, c): Numbers proportional to direction cosines.

• l = ±a/√(a²+b²+c²), etc.

Angle Between Two Lines

For direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂):

cosθ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁²+b₁²+c₁²) * √(a₂²+b₂²+c₂²)]

• Perpendicular Lines: If a₁a₂ + b₁b₂ + c₁c₂ = 0.

• Parallel Lines: If a₁/a₂ = b₁/b₂ = c₁/c₂.

Equation of a Line

• Through (x₁, y₁, z₁) parallel to (a, b, c):

• Vector Form: r = a + λq.

• Cartesian Form: (x - x₁) / a = (y - y₁) / b = (z - z₁) / c.

• Through (x₁, y₁, z₁) and (x₂, y₂, z₂):

• Vector Form: r = a + λ(b - a).

• Cartesian Form: (x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁).

Shortest Distance Between Two Lines

• Skew Lines (r = a₁ + λb₁, r = a₂ + μb₂):
  Shortest Distance = |(a₂ - a₁) ⋅ (b₁ × b₂)| / |b₁ × b₂|.

• Parallel Lines (r = a₁ + λb, r = a₂ + μb):
  Shortest Distance = |(a₂ - a₁) × b| / |b|.

Probability

Fundamental Formulas

• Conditional Probability: P(A|B) = P(A ∩ B) / P(B) (if P(B) ≠ 0).

• Independent Events: P(A ∩ B) = P(A) * P(B).

• Multiplication Theorem: P(A ∩ B) = P(A) * P(B|A).

• Addition Theorem (Union): P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Total Probability Theorem

If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events, and A is any event:

P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + … + P(Eₙ)P(A|Eₙ).

(Memory Tip: This finds the total probability of an outcome by considering all possible paths leading to it).

Bayes' Theorem

If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events, and A is any event:

P(Eᵢ|A) = [P(Eᵢ)P(A|Eᵢ)] / P(A), where P(A) is from the Total Probability Theorem.

(Memory Tip: This is used when the outcome is known, to find the probability of a specific cause).