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Detailed Solution
Basic Arithmetic Formulas
Fundamental Operations
| Operation | Formula | Example |
| Addition | a + b | 5 + 3 = 8 |
| Subtraction | a - b | 8 - 3 = 5 |
| Multiplication | a × b or a · b | 4 × 3 = 12 |
| Division | a ÷ b or a/b | 12 ÷ 3 = 4 |
Percentage Formulas
| Type | Formula | Usage |
| Percentage | (Part/Whole) × 100 | Finding percentage of a number |
| Percentage Increase | ((New Value - Original)/Original) × 100 | Calculating growth rate |
| Percentage Decrease | ((Original - New Value)/Original) × 100 | Calculating reduction rate |
Average and Ratio
| Concept | Formula | Description |
| Average (Mean) | (Sum of all values)/(Number of values) | Central tendency measure |
| Ratio | a:b = a/b | Comparison between quantities |
| Proportion | a/b = c/d → ad = bc | Equal ratios |
Algebraic Formulas
Basic Algebraic Identities
| Identity | Formula | Expansion |
| Square of Sum | (a + b)² | a² + 2ab + b² |
| Square of Difference | (a - b)² | a² - 2ab + b² |
| Difference of Squares | a² - b² | (a + b)(a - b) |
| Cube of Sum | (a + b)³ | a³ + 3a²b + 3ab² + b³ |
| Cube of Difference | (a - b)³ | a³ - 3a²b + 3ab² - b³ |
| Sum of Cubes | a³ + b³ | (a + b)(a² - ab + b²) |
| Difference of Cubes | a³ - b³ | (a - b)(a² + ab + b²) |
Quadratic Formulas
| Type | Formula | Usage |
| Standard Form | ax² + bx + c = 0 | General quadratic equation |
| Quadratic Formula | x = (-b ± √(b² - 4ac))/(2a) | Finding roots |
| Discriminant | Δ = b² - 4ac | Nature of roots |
| Sum of Roots | α + β = -b/a | Relationship between roots |
| Product of Roots | αβ = c/a | Relationship between roots |
Logarithmic Formulas
| Property | Formula | Description |
| Basic Definition | log_a(x) = y ⟺ a^y = x | Logarithm definition |
| Product Rule | log_a(xy) = log_a(x) + log_a(y) | Logarithm of product |
| Quotient Rule | log_a(x/y) = log_a(x) - log_a(y) | Logarithm of quotient |
| Power Rule | log_a(x^n) = n·log_a(x) | Logarithm of power |
| Change of Base | log_a(x) = log_b(x)/log_b(a) | Converting bases |
Geometry Formulas
2D Shapes - Area and Perimeter
| Shape | Area Formula | Perimeter Formula |
| Rectangle | A = length × width | P = 2(l + w) |
| Square | A = side² | P = 4 × side |
| Triangle | A = ½ × base × height | P = a + b + c |
| Circle | A = πr² | C = 2πr or πd |
| Parallelogram | A = base × height | P = 2(a + b) |
| Trapezoid | A = ½(b₁ + b₂) × h | P = a + b₁ + c + b₂ |
| Rhombus | A = ½ × d₁ × d₂ | P = 4a |
3D Shapes - Surface Area and Volume
| Shape | Surface Area | Volume |
| Cube | SA = 6a² | V = a³ |
| Rectangular Prism | SA = 2(lw + lh + wh) | V = l × w × h |
| Cylinder | SA = 2πr(r + h) | V = πr²h |
| Cone | SA = πr(r + l) | V = ⅓πr²h |
| Sphere | SA = 4πr² | V = ⅔πr³ |
| Pyramid | SA = base area + lateral area | V = ⅓ × base area × height |
Special Triangle Formulas
| Type | Formula | Application |
| Pythagorean Theorem | a² + b² = c² | Right triangles |
| Heron's Formula | A = √(s(s-a)(s-b)(s-c)) | Area using three sides |
| Semi-perimeter | s = (a + b + c)/2 | Used in Heron's formula |
Coordinate Geometry
Distance and Slope Formulas
| Concept | Formula | Description |
| Distance Formula | d = √[(x₂-x₁)² + (y₂-y₁)²] | Distance between two points |
| Midpoint Formula | M = ((x₁+x₂)/2, (y₁+y₂)/2) | Midpoint of line segment |
| Slope Formula | m = (y₂-y₁)/(x₂-x₁) | Slope of line |
| Point-Slope Form | y - y₁ = m(x - x₁) | Equation of line |
| Slope-Intercept Form | y = mx + b | Linear equation |
| Standard Form | Ax + By = C | General linear equation |
Circle Equations
| Type | Formula | Description |
| Standard Form | (x-h)² + (y-k)² = r² | Circle with center (h,k) |
| General Form | x² + y² + Dx + Ey + F = 0 | Expanded circle equation |
Trigonometry Formulas
Basic Trigonometric Ratios
| Function | Ratio | Abbreviation |
| Sine | opposite/hypotenuse | sin θ |
| Cosine | adjacent/hypotenuse | cos θ |
| Tangent | opposite/adjacent | tan θ |
| Cosecant | hypotenuse/opposite | csc θ = 1/sin θ |
| Secant | hypotenuse/adjacent | sec θ = 1/cos θ |
| Cotangent | adjacent/opposite | cot θ = 1/tan θ |
Trigonometric Identities
| Type | Identity |
| Pythagorean | sin²θ + cos²θ = 1 |
| Pythagorean | 1 + tan²θ = sec²θ |
| Pythagorean | 1 + cot²θ = csc²θ |
| Double Angle | sin(2θ) = 2sin θ cos θ |
| Double Angle | cos(2θ) = cos²θ - sin²θ |
| Sum Formula | sin(A + B) = sin A cos B + cos A sin B |
| Sum Formula | cos(A + B) = cos A cos B - sin A sin B |
Standard Angle Values
| Angle | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
Statistics and Probability
Central Tendency
| Measure | Formula | Description |
| Mean | x̄ = (Σx)/n | Average value |
| Median | Middle value when ordered | Central position |
| Mode | Most frequently occurring value | Most common value |
Dispersion Measures
| Measure | Formula | Description |
| Range | R = Highest - Lowest | Spread of data |
| Variance | σ² = Σ(x - x̄)²/n | Average squared deviation |
| Standard Deviation | σ = √(variance) | Square root of variance |
Probability Formulas
| Type | Formula | Description |
| Basic Probability | P(E) = (Number of favorable outcomes)/(Total outcomes) | Fundamental probability |
| Addition Rule | P(A ∪ B) = P(A) + P(B) - P(A ∩ B) | Union of events |
| Multiplication Rule | P(A ∩ B) = P(A) × P(B | A) |
| Conditional Probability | P(A | B) = P(A ∩ B)/P(B) |
Calculus Formulas
Derivatives
| Function | Derivative |
| Constant | d/dx(c) = 0 |
| Power Rule | d/dx(xⁿ) = nx^(n-1) |
| Sum Rule | d/dx(f + g) = f' + g' |
| Product Rule | d/dx(fg) = f'g + fg' |
| Quotient Rule | d/dx(f/g) = (f'g - fg')/g² |
| Chain Rule | d/dx(f(g(x))) = f'(g(x)) × g'(x) |
Common Derivatives
| Function | Derivative |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
Integration
| Function | Integral |
| Constant | ∫c dx = cx + C |
| Power Rule | ∫xⁿ dx = x^(n+1)/(n+1) + C |
| Exponential | ∫eˣ dx = eˣ + C |
| Trigonometric | ∫sin x dx = -cos x + C |
| Trigonometric | ∫cos x dx = sin x + C |
Physics-Related Math Formulas
Motion Formulas
| Type | Formula | Variables |
| Velocity | v = d/t | v = velocity, d = distance, t = time |
| Acceleration | a = (v₂ - v₁)/t | a = acceleration, v = velocity |
| Distance | s = ut + ½at² | s = distance, u = initial velocity |
| Velocity² | v² = u² + 2as | Final velocity equation |
Energy and Work
| Type | Formula | Description |
| Kinetic Energy | KE = ½mv² | Energy of motion |
| Potential Energy | PE = mgh | Energy of position |
| Work | W = F × d × cos θ | Work done by force |
| Power | P = W/t | Rate of doing work |
Grade-Specific Formula Charts
Class 8 Essential Formulas
Algebra:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- a² - b² = (a + b)(a - b)
Geometry:
- Area of rectangle = l × w
- Area of square = a²
- Area of triangle = ½ × base × height
- Volume of cube = a³
- Volume of cuboid = l × w × h
Class 9 Key Formulas
Algebra:
- All Class 8 identities plus:
- (x + a)(x + b) = x² + (a + b)x + ab
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Geometry:
- Heron's formula: A = √(s(s-a)(s-b)(s-c))
- Surface area of sphere = 4πr²
- Volume of sphere = (4/3)πr³
Class 10 Important Formulas
Quadratic Equations:
- ax² + bx + c = 0
- x = (-b ± √(b² - 4ac))/(2a)
Coordinate Geometry:
- Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
- Section formula: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
Trigonometry:
- sin²θ + cos²θ = 1
- All standard angle values
Quick Reference Tips
Memory Aids
- SOHCAHTOA for trigonometric ratios
- PEMDAS for order of operations
- FOIL for binomial multiplication
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