Essential formulas for CBSE Class 10 Maths covering Real Numbers, Polynomials, Linear/Quadratic Equations, AP, Triangles, Coordinate Geometry, Trigonometry, Circles, and Statistics. Master these formulas for board exams and competitive entrance tests.
a = bq + r, where 0 ≤ r < bFor finding HCF of two numbersEvery positive integer > 1 is a product of primes, unique up to orderUsed for HCF and LCMx = (-b ± √(b² - 4ac)) / 2aFor ax² + bx + c = 0, discriminant D = b² - 4acα + β = -b/aFor quadratic ax² + bx + c = 0αβ = c/aFor quadratic ax² + bx + c = 0p(x) = g(x) × q(x) + r(x)where deg(r) < deg(g)ax + by + c = 0General form; a ≠ 0 or b ≠ 0y = mx + cm = slope, c = y-interceptFor a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ (unique solution)If ratios equal, lines are parallel or identicalx = D₁/D, y = D₂/D where D = a₁b₂ - a₂b₁For system of two linear equationsaₙ = a + (n - 1)da = first term, d = common differenceSₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l)l = last term = a + (n - 1)dd = aₙ₊₁ - aₙDifference between consecutive termsA = (a + b)/2Mean of two numbers a and bsin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacentcot θ = 1/tan θ, sec θ = 1/cos θ, cosec θ = 1/sin θsin² θ + cos² θ = 1Fundamental trigonometric identitytan θ = sin θ/cos θ, cot θ = cos θ/sin θRelations between trigonometric ratios1 + tan² θ = sec² θ, 1 + cot² θ = cosec² θDerived from sin² θ + cos² θ = 1sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1Also sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, tan(90° - θ) = cot θFor complementary anglestan θ = height/distanceUsed in angle of elevation and depression problemsd = √[(x₂ - x₁)² + (y₂ - y₁)²]Distance between points (x₁, y₁) and (x₂, y₂)P = ((m × x₂ + n × x₁)/(m + n), (m × y₂ + n × y₁)/(m + n))Point P divides line segment in ratio m:n internallyM = ((x₁ + x₂)/2, (y₁ + y₂)/2)Midpoint of line segment joining (x₁, y₁) and (x₂, y₂)Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|Coordinates of vertices: (x₁, y₁), (x₂, y₂), (x₃, y₃)m = (y₂ - y₁)/(x₂ - x₁)Slope between points (x₁, y₁) and (x₂, y₂)C = 2πrr = radiusA = πr²r = radiusl = (θ/360°) × 2πr = (θ/180°) × πr (in radians: l = rθ)θ in degrees; r = radiusA = (θ/360°) × πr²θ in degreesA = r²/2 (θ - sin θ) where θ is in radiansArea between chord and arcIf △ABC ~ △DEF, then a/d = b/e = c/f = k (linear ratio), and Area(△ABC)/Area(△DEF) = k²k = scale factor; areas in square of linear ratioSurface Area = 6a², Volume = a³a = side lengthSurface Area = 2(lb + bh + hl), Volume = l × b × hl = length, b = breadth, h = heightCurved Surface Area = 2πrh, Total Surface Area = 2πr(r + h), Volume = πr²hr = radius, h = heightCurved Surface Area = πrl, Total Surface Area = πr(r + l), Volume = 1/3 πr²hr = radius, h = height, l = slant height = √(r² + h²)Surface Area = 4πr², Volume = 4/3 πr³r = radiusCurved Surface Area = 2πr², Total Surface Area = 3πr², Volume = 2/3 πr³r = radiusMean = (Σx)/n or Mean = (ΣfᵢXᵢ)/(Σfᵢ)Σx = sum of all observations, n = number of observations, fᵢ = frequencyMode = most frequently occurring valueFor grouped data: Mode = l + (f₁ - f₀)/((2f₁ - f₀ - f₂)) × hFor ungrouped: arrange in order, median is middle value. For grouped: Median = l + ((n/2 - cf)/f) × hl = lower boundary of median class, cf = cumulative frequency, f = frequency of median class, h = class widthσ² = (Σ(xᵢ - mean)²)/nMeasure of dispersionσ = √[(Σ(xᵢ - mean)²)/n]Square root of varianceP(E) = (Number of favourable outcomes)/(Total number of outcomes)0 ≤ P(E) ≤ 1; P(E) + P(not E) = 1