0🧮 1. Algebra
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(a + b)² = a² + 2ab + b²
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(a − b)² = a² − 2ab + b²
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a² − b² = (a + b)(a − b)
📐 2. Trigonometry
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sin²θ + cos²θ = 1
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1 + tan²θ = sec²θ
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1 + cot²θ = cosec²θ
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sin(A ± B) = sinA cosB ± cosA sinB
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cos(A ± B) = cosA cosB ∓ sinA sinB
📈 3. Differentiation
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d/dx (xⁿ) = n·xⁿ⁻¹
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d/dx (sin x) = cos x
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d/dx (cos x) = −sin x
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d/dx (eˣ) = eˣ
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d/dx (ln x) = 1/x
∫ 4. Integration
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∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1)
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∫1/x dx = ln|x| + C
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∫eˣ dx = eˣ + C
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∫sin x dx = −cos x + C
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∫cos x dx = sin x + C
📊 5. Matrices
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(A·B)ᵗ = Bᵗ·Aᵗ
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Determinant of 2×2 matrix: |A| = ad − bc
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Inverse of matrix A: A⁻¹ = adj(A)/|A|
🧭 6. Geometry / Coordinate Geometry
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Distance between (x₁, y₁) and (x₂, y₂):
√[(x₂ − x₁)² + (y₂ − y₁)²] -
Midpoint:
((x₁ + x₂)/2 , (y₁ + y₂)/2) -
Slope of a line (m):
m = (y₂ − y₁)/(x₂ − x₁)
📐 7. Mensuration (2D & 3D)
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Area of Circle = πr²
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Circumference = 2πr
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Surface Area of Sphere = 4πr²
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Volume of Sphere = (4/3)πr³
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Volume of Cylinder = πr²h
📏 8. Vector Algebra
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|a| = √(a₁² + a₂² + a₃²)
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Dot product: a·b = |a||b|cosθ
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Cross product: |a × b| = |a||b|sinθ
