Class 10 Maths Complete Formula Sheet: Chapters 1-14 Revision

Real Numbers Formulas

Fundamental Theorem of Arithmetic states every composite number can be uniquely expressed as prime factors. Remember: HCF × LCM = Product of Two Numbers. For rational numbers p/q, decimal expansion terminates only if q's prime factors are 2ⁿ or 5ᵐ. Non-terminating recurring decimals indicate rational numbers, while non-recurring signify irrationals like √2. Key relationship: If prime p divides a², then p divides a.

Polynomials Essentials

Linear polynomials (degree 1) have one zero, quadratics (degree 2) have up to two zeros, and cubics (degree 3) have up to three. For quadratic ax² + bx + c = 0:

Sum of zeros = -b/a
Product of zeros = c/a
For cubic ax³ + bx² + cx + d:
Sum of zeros = -b/a
Sum of product pairs = c/a
Product of zeros = -d/a
Essential identities:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)

Linear Equations Mastery

Pair of equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Consistency analysis:

Intersecting lines (unique solution): a₁/a₂ ≠ b₁/b₂
Coincident lines (infinite solutions): a₁/a₂ = b₁/b₂ = c₁/c₂
Parallel lines (no solution): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Solve via substitution or elimination. For reducible equations, use substitution like 1/x = p and 1/y = q.

Quadratic Equations Breakdown

Standard form: ax² + bx + c = 0. Solutions:

Factorization method
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Discriminant D = b² - 4ac determines nature:
D > 0: Two distinct real roots
D = 0: Equal real roots
D < 0: No real roots
Root relationships: Sum = -b/a, Product = c/a.

Arithmetic Progressions Key Formulas

General form: a, a+d, a+2d,...

nth term: aₙ = a + (n - 1)d
Sum of first n terms: Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l) where l = last term
Common problem types: Finding middle terms in symmetric APs or sums of specific sequences.

Geometry Essentials

Triangles

Similarity criteria: AAA, SSS, SAS
Basic Proportionality Theorem: Line parallel to one side divides others proportionally
Pythagoras Theorem: AC² = AB² + BC² in right ΔABC

Coordinate Geometry

Distance formula: √[(x₂ - x₁)² + (y₂ - y₁)²]
Section formula: (m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)
Midpoint: [(x₁ + x₂)/2, (y₁ + y₂)/2]
Area of triangle: 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Circles

Tangent properties: Perpendicular to radius at point of contact
Lengths of tangents from external point are equal

Trigonometry Fundamentals

Basic ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Reciprocals:
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Key identities:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Table values:
θ | 0° | 30° | 45° | 60° | 90°
sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1
cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0

Applications and Measurements

Areas and Volumes

Circle: Area = πr², Circumference = 2πr
Sector: Length of arc = (θ/360) × 2πr, Area = (θ/360) × πr²
Cuboid: Volume = l × b × h, TSA = 2(lb + bh + hl)
Sphere: Volume = 4/3πr³, TSA = 4πr²

Statistics Formulas

Mean: Direct method, assumed mean, or step deviation
Median for grouped data: l + [(n/2 - cf)/f] × h
Mode: l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h
Empirical relation: Mode ≈ 3 Median - 2 Mean

Probability Basics

Probability = Number of favorable outcomes / Total outcomes
Sample space considerations: Coins, dice, cards scenarios

Revision Checklist and Tips

Daily practice: Solve 5 problems from each chapter weekly
Concept mapping: Create formula trees connecting related chapters
NCERT focus: Prioritize theorem proofs and examples
Error log: Track mistakes in dedicated notebook
Timed tests: Simulate exam pressure monthly

Recommended resources:

RD Sharma Solutions for varied problem sets
Oswaal Question Banks for exam patterns
Khan Academy for visual explanations

Final Preparation Strategy

Mastering these formulas requires understanding their derivations, not just memorization. After analyzing this video, I emphasize practicing applied problems over theoretical recall—board exams increasingly test conceptual application. Which formula do you find most challenging? Share in comments for personalized tips!