Essential Maths Formulas for Class 10 Students optimized SEO

1. Algebra

• Polynomials

  • If α, β are roots of ax²+bx+c=0 →
    α + β = -b/a , αβ = c/a

  • Relation between coefficients and zeros:
    x² - (sum of zeros)x + (product of zeros)

• Identity Formulas

  • (a+b)² = a² + 2ab + b²

  • (a-b)² = a² - 2ab + b²

  • (x+a)(x+b) = x² + (a+b)x + ab

  • a³ + b³ = (a+b)(a² - ab + b²)

  • a³ - b³ = (a-b)(a² + ab + b²)

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2. Linear & Quadratic Equations

• Pair of Linear Equations (ax+by+c=0)

  • Consistent condition: a1/a2 ≠ b1/b2

  • Infinite solutions: a1/a2 = b1/b2 = c1/c2

  • No solution: a1/a2 = b1/b2 ≠ c1/c2

• Quadratic Equations

  • General: ax² + bx + c = 0

  • Roots: x = (-b ± √(b²-4ac)) / 2a

  • Discriminant: D = b² - 4ac

    • D>0 → 2 real, distinct roots

    • D=0 → real, equal roots

    • D<0 → no real roots

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3. Arithmetic Progression (AP)

• nth term: an = a + (n-1)d

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

• Alternative: Sn = n/2 (a + l)

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4. Geometry

• Triangle Similarity Theorem:
  Δ1/Δ2 = (side1/side2)²

• Pythagoras Theorem:
  Hypotenuse² = Base² + Perpendicular²

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5. Trigonometry

• Ratios
  sinθ = Opp/Hyp, cosθ = Adj/Hyp, tanθ = Opp/Adj
  cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ

• Identities

  • sin²θ + cos²θ = 1

  • 1 + tan²θ = sec²θ

  • 1 + cot²θ = cosec²θ

• Values (θ=0°,30°,45°,60°,90°)
  Memorize sin, cos, tan, cot, sec, cosec table.

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6. Mensuration (Surface Area & Volume)

• Sphere:
  SA = 4πr², V = (4/3)πr³

• Hemisphere:
  CSA = 2πr², TSA = 3πr², V = (2/3)πr³

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

• Cone:
  CSA = πrl, TSA = πr(l+r), V = (1/3)πr²h

• Cuboid:
  SA = 2(lb+bh+hl), V = l×b×h

• Cube:
  SA = 6a², V = a³

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7. Coordinate Geometry

• Distance formula: √((x₂-x₁)² + (y₂-y₁)²)

• Section formula: ( (mx₂+nx₁)/(m+n) , (my₂+ny₁)/(m+n) )

• Midpoint formula: ( (x₁+x₂)/2 , (y₁+y₂)/2 )

• Slope (extra, from RD Sharma): m = (y₂-y₁)/(x₂-x₁)

• Equation of line: y - y₁ = m(x - x₁)

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8. Probability

• P(E) = Favourable outcomes / Total outcomes

• 0 ≤ P(E) ≤ 1

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9. Statistics

• Mean (Direct method): Σfixi / Σfi

• Median: L + [(N/2 - CF)/f] × h

• Mode: L + [(f₁-f₀)/(2f₁-f₀-f₂)] × h

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10. Extra Important Results

• Sum of first n natural numbers: n(n+1)/2

• Sum of squares: n(n+1)(2n+1)/6

• Sum of cubes: (n(n+1)/2)²

• Heron’s formula (triangle area): √[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2

• Distance between parallel lines ax+by+c₁=0 & ax+by+c₂=0 → |c₁-c₂| / √(a²+b²)

• Distance from point (x₁,y₁) to line ax+by+c=0 → |ax₁+by₁+c| / √(a²+b²)