CBSE Class 12 Mathematics Formula Sheet (2025-2026): Everything You Need in One Page

Mastering CBSE Class 12 Mathematics requires quick recall of fundamental formulas, theorems, and identities for the 2025-2026 board exam. Whether you are solving complex calculus problems, navigating three-dimensional geometry, or calculating conditional probabilities, having a consolidated, high-yield reference saves revision time and eliminates calculation errors. This comprehensive CBSE Class 12 Maths Formula Sheet: Everything You Need in One Page covers the entire NCERT syllabus, organized unit-by-unit for effective board exam preparation.

Unit 1: Relations, Functions & Inverse Trigonometric Functions

Understanding domains, principal value branches, and properties of binary operations and composite functions forms the bedrock of CBSE Mathematics.

1. Relations & Functions

• Reflexive Relation: (a, a) ∈ R for all a ∈ A.
• Symmetric Relation: (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A.
• Transitive Relation: (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A.
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive simultaneously.
• Injective (One-to-One): f(x₁) = f(x₂) ⇒ x₁ = x₂.
• Surjective (Onto): Range of f = Codomain of f. Every element y in codomain has a pre-image x such that f(x) = y.
• Bijective Function: A function that is both injective and surjective.

2. Inverse Trigonometric Functions (Principal Value Branches)

• sin^-1(x): Domain = [-1, 1], Principal Value Range = [-π/2, π/2]
• cos^-1(x): Domain = [-1, 1], Principal Value Range = [0, π]
• tan^-1(x): Domain = R, Principal Value Range = (-π/2, π/2)
• cot^-1(x): Domain = R, Principal Value Range = (0, π)
• sec^-1(x): Domain = R - (-1, 1), Principal Value Range = [0, π] - {π/2}
• cosec^-1(x): Domain = R - (-1, 1), Principal Value Range = [-π/2, π/2] - {0}
• Key Identity: sin^-1(x) + cos^-1(x) = π/2, tan^-1(x) + cot^-1(x) = π/2 (for x in respective domains).

Unit 2: Algebra (Matrices & Determinants)

Matrix algebra and determinant evaluations are consistent scoring sections in CBSE Class 12 board examinations.

1. Matrices

• Transpose Properties: (A^T)^T = A, (A + B)^T = A^T + B^T, (AB)^T = B^TA^T, (kA)^T = kA^T.
• Symmetric Matrix: A^T = A; Skew-Symmetric Matrix: A^T = -A (Diagonal elements are always zero).
• Any Square Matrix Decomposition: A = (1/2)(A + A^T) + (1/2)(A - A^T), where the first part is symmetric and the second is skew-symmetric.

2. Determinants & Inverses

• Determinant Properties: |AB| = |A||B|, |kA| = kⁿ|A| (where n is the order of square matrix A), |A^T| = |A|.
• Adjoint Property: A · (adj A) = (adj A) · A = |A| I_n.
• Determinant of Adjoint: |adj A| = |A|^n-1, and |adj(adj A)| = |A|^(n-1)2.
• Matrix Inverse: A^-1 = (1 / |A|) · adj(A), valid if and only if |A| ≠ 0 (non-singular matrix).
• Solving Systems of Linear Equations (Matrix Method): AX = B ⇒ X = A^-1B (for unique solution when |A| ≠ 0).

Unit 3: Calculus

Calculus carries the highest weightage in CBSE Class 12 Mathematics, spanning differentiation, definite integrals, and differential equations.

1. Continuity and Differentiability

• Continuity Condition: Function f(x) is continuous at x = c if lim_x→c^- f(x) = lim_x→c⁺ f(x) = f(c).
• Product Rule: d/dx [u · v] = u · (dv/dx) + v · (du/dx).
• Quotient Rule: d/dx [u / v] = [v · (du/dx) - u · (dv/dx)] / v².
• Chain Rule: dy/dx = (dy/du) · (du/dx).
• Logarithmic Differentiation: Applied for functions of type y = [u(x)]^v(x) by taking ln on both sides.
• Parametric Differentiation: dy/dx = (dy/dt) / (dx/dt), where dx/dt ≠ 0.

2. Applications of Derivatives (AOD)

• Increasing/Decreasing Functions: Strictly increasing on (a, b) if f'(x) > 0; strictly decreasing if f'(x) < 0 for all x ∈ (a, b).
• Local Maxima and Minima (Second Derivative Test):
  1. Find critical points where f'(x) = 0. Let critical point be x = c.
  2. If f''(c) < 0, then x = c is a point of local maximum.
  3. If f''(c) > 0, then x = c is a point of local minimum.
  4. If f''(c) = 0, the test is inconclusive; apply the First Derivative Test.

3. Indefinite & Definite Integrals

• Standard Integrals:
  • ∫ 1/(x² + a²) dx = (1/a) tan^-1(x/a) + C
  • ∫ 1/(x² - a²) dx = (1/2a) ln|(x - a)/(x + a)| + C
  • ∫ 1/√(a² - x²) dx = sin^-1(x/a) + C
  • ∫ 1/√(x² ± a²) dx = ln|x + √(x² ± a²)| + C
  • ∫ √(a² - x²) dx = (x/2)√(a² - x²) + (a²/2) sin^-1(x/a) + C
• Integration by Parts (ILATE rule): ∫ u · v dx = u ∫ v dx - ∫ [ (du/dx) ∫ v dx ] dx
• Special Exponential Form: ∫ e^x [ f(x) + f'(x) ] dx = e^x f(x) + C
• King's Property of Definite Integrals: ∫_a^b f(x) dx = ∫_a^b f(a + b - x) dx
• Symmetry Properties:
  • ∫_-a^a f(x) dx = 2 ∫₀^a f(x) dx if f(-x) = f(x) (Even function).
  • ∫_-a^a f(x) dx = 0 if f(-x) = -f(x) (Odd function).

4. Differential Equations

• Order & Degree: Order is the highest derivative present. Degree is the highest power of the highest order derivative after making the equation polynomial in derivatives.
• Linear Differential Equation (First Order): dy/dx + P(x)y = Q(x)
  1. Integrating Factor, I.F. = e^{∫ P(x) dx}
  2. General Solution: y · (I.F.) = ∫ [ Q(x) · (I.F.) ] dx + C
• Homogeneous Differential Equation: dy/dx = F(y/x) ⇒ Substitute y = vx, dy/dx = v + x(dv/dx).

Unit 4: Vectors & Three-Dimensional Geometry

Vector algebra and 3D geometry involve direct formula substitutions in board exam 12 questions.

1. Vectors

• Dot Product (Scalar Product): a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
• Cross Product (Vector Product): a × b = |a||b| sin θ n̂.
• Projection of Vector a on Vector b: Projection = (a · b) / |b|.
• Area of Triangle: (1/2) |a × b| where a and b are adjacent sides.
• Area of Parallelogram: |a × b| (adjacent sides) or (1/2) |d₁ × d₂| (diagonals).

2. Three-Dimensional Geometry

• Direction Cosines & Ratios: If direction cosines are l, m, n, then l² + m² + n² = 1. If direction ratios are a, b, c, then l = ± a / √(a² + b² + c²).
• Vector Equation of Line passing through a point and parallel to b: r = a + λb.
• Cartesian Equation of Line: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c.
• Shortest Distance Between Skew Lines:

  d = | ((a₂ - a₁) · (b₁ × b₂)) / |b₁ × b₂| |

• Distance Between Parallel Lines (r = a₁ + λb and r = a₂ + μb):

  d = | (b × (a₂ - a₁)) / |b| |

Unit 5: Linear Programming & Probability

1. Linear Programming Problem (LPP)

• Feasible Region: The common region determined by all constraints including non-negativity restrictions (x ≥ 0, y ≥ 0).
• Corner Point Theorem: An optimal value (maximum or minimum) of the objective function Z = ax + by occurs at one of the corner points (vertices) of the feasible region.
• Bounded vs. Unbounded Region: If the feasible region is unbounded, an optimal value M is confirmed as maximum only if the open half-plane ax + by > M has no points in common with the feasible region.

2. Probability

• Conditional Probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
• Multiplication Theorem: P(A ∩ B) = P(B) · P(A|B) = P(A) · P(B|A).
• Independent Events: Events A and B are independent if and only if P(A ∩ B) = P(A) · P(B).
• Theorem of Total Probability: P(A) = ∑_i=1ⁿ P(E_i) · P(A|E_i), where {E₁, E₂, ..., E_n} is a partition of sample space S.
• Bayes' Theorem:

  P(E_k|A) = [ P(E_k) · P(A|E_k) ] / [ ∑_i=1ⁿ P(E_i) · P(A|E_i) ]

Important CBSE Questions with Answers

Below are frequently asked NCERT exemplar and CBSE board questions demonstrating the direct application of standard formulas.

Question 1 (Calculus - Definite Integrals)

Question: Evaluate the definite integral ∫₀^π/2 (sin⁴x / (sin⁴x + cos⁴x)) dx.

Answer & Step-by-Step Solution:

1. Let I = ∫₀^π/2 (sin⁴x / (sin⁴x + cos⁴x)) dx   --- (Equation 1)
2. Apply King's Property ∫₀^a f(x) dx = ∫₀^a f(a - x) dx:
   I = ∫₀^π/2 [sin⁴(π/2 - x) / (sin⁴(π/2 - x) + cos⁴(π/2 - x))] dx
   Since sin(π/2 - x) = cos x and cos(π/2 - x) = sin x, we get:
   I = ∫₀^π/2 (cos⁴x / (cos⁴x + sin⁴x)) dx   --- (Equation 2)
3. Adding Equations (1) and (2):
   2I = ∫₀^π/2 [(sin⁴x + cos⁴x) / (sin⁴x + cos⁴x)] dx = ∫₀^π/2 1 dx = [x]₀^π/2 = π/2.
4. Therefore, I = π/4.

Question 2 (Vectors & 3D Geometry)

Question: Find the shortest distance between the lines r = (i + 2j + k) + λ(i - j + k) and r = (2i - j - k) + μ(2i + j + 2k).

Answer & Step-by-Step Solution:

1. Identify position vectors and direction vectors:
   • a₁ = i + 2j + k, b₁ = i - j + k
   • a₂ = 2i - j - k, b₂ = 2i + j + 2k
2. Calculate (a₂ - a₁) = (2 - 1)i + (-1 - 2)j + (-1 - 1)k = i - 3j - 2k.
3. Compute cross product b₁ × b₂:
   | i   j   k |
   | 1 -1   1 | = i(-2 - 1) - j(2 - 2) + k(1 - (-2)) = -3i + 0j + 3k = -3i + 3k.
   Magnitude |b₁ × b₂| = √((-3)² + 0² + 3²) = √(9 + 9) = 3√2.
4. Compute (a₂ - a₁) · (b₁ × b₂) = (1)(-3) + (-3)(0) + (-2)(3) = -3 - 6 = -9.
5. Shortest distance d = | -9 / (3√2) | = 3/√2 = (3√2)/2 units.

Question 3 (Probability - Bayes' Theorem)

Question: An insurance company insured 2000 scooter drivers, 4000 car drivers, and 6000 truck drivers. The probability of an accident is 0.01, 0.03, and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Answer & Step-by-Step Solution:

1. Total drivers = 2000 + 4000 + 6000 = 12000.
   • P(E₁) [Scooter] = 2000/12000 = 1/6
   • P(E₂) [Car] = 4000/12000 = 1/3
   • P(E₃) [Truck] = 6000/12000 = 1/2
2. Probabilities of accident given driver category:
   • P(A|E₁) = 0.01 = 1/100
   • P(A|E₂) = 0.03 = 3/100
   • P(A|E₃) = 0.15 = 15/100
3. Apply Bayes' Theorem for P(E₁|A):
   P(E₁|A) = [ P(E₁) · P(A|E₁) ] / [ P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + P(E₃)P(A|E₃) ]
   Numerator = (1/6)(1/100) = 1/600
   Denominator = (1/6)(1/100) + (1/3)(3/100) + (1/2)(15/100) = 1/600 + 6/600 + 45/600 = 52/600
   P(E₁|A) = (1/600) / (52/600) = 1/52.

How to Prepare and Memorize Formulas for Board Exam 12

Achieving a 100/100 score in CBSE Mathematics depends on precise recall and active formula application rather than passive reading. Follow these proven teacher-tested techniques:

1. Derive Key Formulas Once: Memorizing becomes intuitive when you understand how standard integrals (e.g., integration by parts, partial fractions) and 3D shortest distance formulas originate geometrically.
2. Active Recall Through Self-Quizzing: Dedicate 15 minutes each morning to writing down the core formulas of one unit (e.g., Matrix inverse properties, definite integral properties) without looking at notes.
3. Maintain a Sign & Limit Warning List: Common mark deductions in board exam 12 occur from missing "+ C" in indefinite integrals, swapping direction ratios, or ignoring principal value branches in Inverse Trigonometric Functions.
4. Solve 3 Step-by-Step Questions per Formula: Immediately after revising a formula block (such as Bayes' theorem or shortest distance between skew lines), solve 3 past-year board questions to reinforce the substitution patterns.

Where to Practice More

To master the application of this complete formula sheet under timed examination conditions, practice with official syllabus-aligned question banks, chapter-wise sample papers, and verified step-by-step marking schemes. Visit qptool.theorify.in to generate customized CBSE Class 12 Mathematics mock tests, practice papers, and previous year board questions with detailed answer keys.