📐 CBSE Class 10 Maths: Master Formula Handbook

🔢 1. Number Systems & Polynomials

A. Real Numbers:

• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed (factorised) as a product of primes, apart from the order in which prime factors occur.
• HCF and LCM Relation: For any two positive integers a and b:
  HCF(a, b) × LCM(a, b) = a × b
• Irrationality: If p is a prime number and p divides a², then p divides a (where a is a positive integer). Used to prove √2, √3, and √5 are irrational.

B. Polynomials:

• For quadratic polynomial P(x) = ax² + bx + c with zeros α and β:
  • Sum of zeros: α + β = -b / a
  • Product of zeros: α · β = c / a
• Forming quadratic polynomial: k · [x² - (Sum of zeros)x + (Product of zeros)] = k · [x² - (α + β)x + αβ].

⚖️ 2. Algebra: Linear & Quadratic Equations

A. Pair of Linear Equations in Two Variables:

For system a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

• Unique Solution (Consistent & Intersecting Lines): a₁/a₂ ≠ b₁/b₂.
• Infinitely Many Solutions (Consistent & Coincident Lines): a₁/a₂ = b₁/b₂ = c₁/c₂.
• No Solution (Inconsistent & Parallel Lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

B. Quadratic Equations:

Standard form: ax² + bx + c = 0 (where a ≠ 0).

• Quadratic Formula (Sridharacharya's Rule): x = (-b ± √(b² - 4ac)) / (2a)
• Discriminant: D = b² - 4ac
  • If D > 0: Two distinct real roots.
  • If D = 0: Two equal real roots (x = -b / 2a).
  • If D < 0: No real roots (roots are imaginary).

📈 3. Arithmetic Progressions (AP)

An Arithmetic Progression is a sequence where each term is obtained by adding a fixed common difference d to the preceding term.

• Common Difference: d = a₂ - a₁ = aₙ - aₙ₋₁ (Can be positive, zero, or negative).
• nth Term of an AP: aₙ = a + (n - 1)d (where a is the first term, n is number of terms).
• nth Term from End: l - (n - 1)d (where l is the last term).
• Sum of First n Terms:
  Sₙ = (n / 2) · [2a + (n - 1)d] OR Sₙ = (n / 2) · [a + l]
• Relation between Sₙ and aₙ: aₙ = Sₙ - Sₙ₋₁.

📍 4. Coordinate Geometry

Let point A(x₁, y₁) and B(x₂, y₂) lie in a Cartesian plane:

• Distance Formula: AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Distance from Origin (0, 0): OP = √(x² + y²)
• Section Formula (Internal Division in ratio m₁ : m₂):
  P(x, y) = [ (m₁x₂ + m₂x₁) / (m₁ + m₂), (m₁y₂ + m₂y₁) / (m₁ + m₂) ]
• Midpoint Formula: M(x, y) = [ (x₁ + x₂) / 2, (y₁ + y₂) / 2 ]
• Centroid of Triangle ABC: G(x, y) = [ (x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3 ]

📐 5. Complete Trigonometry Handbook

In right-angled triangle ABC (right-angled at B):

• Basic Ratios: sin θ = Perpendicular / Hypotenuse, cos θ = Base / Hypotenuse, tan θ = Perpendicular / Base, cosec θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ.
• Fundamental Trigonometric Identities:
  1. sin²θ + cos²θ = 1 → sin²θ = 1 - cos²θ, cos²θ = 1 - sin²θ
  2. 1 + tan²θ = sec²θ → sec²θ - tan²θ = 1
  3. 1 + cot²θ = cosec²θ → cosec²θ - cot²θ = 1
• Standard Angle Values Table:
  • sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
  • cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
  • tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = Not Defined

📦 6. Surface Areas, Volumes & Statistics

A. 3D Geometry Formula Sheet:

• Right Circular Cylinder: CSA = 2πrh | TSA = 2πr(r + h) | Volume = πr²h
• Right Circular Cone: Slant height l = √(r² + h²) | CSA = πrl | TSA = πr(l + r) | Volume = (1/3)πr²h
• Sphere: Surface Area = 4πr² | Volume = (4/3)πr³
• Hemisphere: CSA = 2πr² | TSA = 3πr² | Volume = (2/3)πr³

B. Statistics & Probability:

• Direct Mean: x̄ = Σ(fᵢxᵢ) / Σfᵢ
• Assumed Mean Method: x̄ = a + [ Σ(fᵢdᵢ) / Σfᵢ ] (where dᵢ = xᵢ - a)
• Mode of Grouped Data: Mode = l + [ (f₁ - f₀) / (2f₁ - f₀ - f₂) ] × h
• Median of Grouped Data: Median = l + [ ((n/2) - cf) / f ] × h
• Empirical Relationship: 3 × Median = Mode + 2 × Mean
• Probability: P(E) = Number of outcomes favourable to E / Total number of possible outcomes. (0 ≤ P(E) ≤ 1 and P(E) + P(not E) = 1).

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