📕 Ch.1: Real Numbers (वास्तविक संख्याएँ)

Euclid's Division Lemma: a = bq + r (0 ≤ r < b)

HCF × LCM = Product of two numbers

Fundamental Theorem: Every composite number = product of primes (uniquely)

📕 Ch.2: Polynomials (बहुपद)

Quadratic: ax² + bx + c, Zeroes α, β

Sum of zeroes: α + β = -b/a

Product of zeroes: αβ = c/a

📕 Ch.3: Linear Equations (रैखिक समीकरण युग्म)

a₁/a₂ ≠ b₁/b₂ → Unique solution (intersecting lines)

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution (parallel lines)

a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinite solutions (coincident lines)

📕 Ch.4: Quadratic Equations (द्विघात समीकरण) — ⭐ MOST IMPORTANT

Standard Form: ax² + bx + c = 0

Quadratic Formula: x = (-b ± √(b²-4ac)) / 2a

Discriminant: D = b² - 4ac

D > 0: Two distinct real roots | D = 0: Two equal roots | D < 0: No real roots

📕 Ch.5: Arithmetic Progressions (समांतर श्रेणी / AP) — ⭐ MOST IMPORTANT

nth term: aₙ = a + (n-1)d

Sum of n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 [a + l]

Common difference: d = a₂ - a₁ = a₃ - a₂

a = first term, d = common difference, l = last term, n = number of terms

📐 Ch.6: Triangles (त्रिभुज)

BPT (Basic Proportionality Theorem): DE ∥ BC → AD/DB = AE/EC

Pythagoras: H² = P² + B² (In right triangle)

Similar Triangles: Ratio of areas = (Ratio of sides)²

AAA, SAS, SSS — Similarity criteria

📐 Ch.7: Coordinate Geometry (निर्देशांक ज्यामिति)

Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]

Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)

Section Formula: P = ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂))

Area of Triangle: = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

📷 Distance Formula & Section Formula Diagram

📏 Ch.8-9: Trigonometry (त्रिकोणमिति)

sin θ = P/H, cos θ = B/H, tan θ = P/B (P = लम्ब, B = आधार, H = कर्ण)

Identity 1: sin²θ + cos²θ = 1

Identity 2: 1 + tan²θ = sec²θ

Identity 3: 1 + cot²θ = cosec²θ

Height & Distance: tan θ = ऊँचाई/दूरी (Angle of elevation/depression)

Complementary: sin(90°-θ) = cosθ, cos(90°-θ) = sinθ, tan(90°-θ) = cotθ

📐 Ch.10: Circles (वृत्त)

Tangent ⊥ Radius at point of contact

Two tangents from external point: Equal length (PA = PB)

Angle: ∠APB + ∠AOB = 180° (A, B = tangent points, O = centre)

📦 Ch.11: Areas Related to Circles (वृत्तों से संबंधित क्षेत्रफल)

Area of Circle: πr²

Circumference: 2πr

Area of Sector: (θ/360°) × πr²

Length of Arc: (θ/360°) × 2πr

Area of Segment: Area of Sector - Area of Triangle

📦 Ch.12: Surface Areas & Volumes (पृष्ठीय क्षेत्रफल)

Cylinder: CSA = 2πrh | TSA = 2πr(r+h) | V = πr²h

Cone: CSA = πrl | TSA = πr(r+l) | V = ⅓πr²h | l = √(r²+h²)

Sphere: SA = 4πr² | V = ⁴⁄₃πr³

Hemisphere: CSA = 2πr² | TSA = 3πr² | V = ²⁄₃πr³

Frustum: CSA = π(r₁+r₂)l | V = ⅓πh(r₁²+r₂²+r₁r₂)

📊 Ch.13-14: Statistics & Probability

Mean (Direct): x̄ = Σfᵢxᵢ / Σfᵢ

Mean (Assumed): x̄ = a + Σfᵢdᵢ / Σfᵢ

Median: l + [(n/2 - cf)/f] × h

Mode: l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h

Probability: P(E) = Favourable outcomes / Total outcomes

Range: 0 ≤ P(E) ≤ 1 | P(E) + P(not E) = 1

Playing Cards: 52 cards = 4 suits × 13 | Face cards = 12 | Aces = 4