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What are some important formulas I need to know when it comes to math?
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Detailed Solution
Real Numbers
Euclid's Division Algorithm
| Formula | Description |
| a = bq + r | Where 0 ≤ r < b, a is dividend, b is divisor, q is quotient, r is remainder |
Finding HCF and LCM
| Formula | Description |
| HCF × LCM = Product of two numbers | For any two positive integers a and b |
| LCM(a,b) = (a × b) / HCF(a,b) | Relationship between HCF and LCM |
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, and this factorization is unique.
Polynomials
Degree and Types
| Type | Form | Degree |
| Linear | ax + b | 1 |
| Quadratic | ax² + bx + c | 2 |
| Cubic | ax³ + bx² + cx + d | 3 |
Relationship between Zeros and Coefficients
For Quadratic Polynomial ax² + bx + c
| Formula | Description |
| Sum of zeros = -b/a | α + β = -b/a |
| Product of zeros = c/a | αβ = c/a |
For Cubic Polynomial ax³ + bx² + cx + d
| Formula | Description |
| Sum of zeros = -b/a | α + β + γ = -b/a |
| Sum of products taken two at a time = c/a | αβ + βγ + αγ = c/a |
| Product of zeros = -d/a | αβγ = -d/a |
Division Algorithm for Polynomials
p(x) = g(x) × q(x) + r(x)
Where degree of r(x) < degree of g(x) or r(x) = 0
Pair of Linear Equations in Two Variables
Standard Form
| Form | Description |
| a₁x + b₁y + c₁ = 0 | First equation |
| a₂x + b₂y + c₂ = 0 | Second equation |
Conditions for Solutions
| Condition | Type of Solution |
| a₁/a₂ ≠ b₁/b₂ | Unique solution (intersecting lines) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many solutions (coincident lines) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | No solution (parallel lines) |
Methods of Solution
| Method | Formula |
| Cramer's Rule | x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁), y = (a₂c₁ - a₁c₂)/(a₁b₂ - a₂b₁) |
Quadratic Equations
Standard Form
ax² + bx + c = 0 (where a ≠ 0)
Quadratic Formula
| Formula | Description |
| x = [-b ± √(b² - 4ac)] / 2a | Solutions of quadratic equation |
Discriminant
| Discriminant (Δ) | Nature of Roots |
| Δ = b² - 4ac > 0 | Two distinct real roots |
| Δ = b² - 4ac = 0 | Two equal real roots |
| Δ = b² - 4ac < 0 | No real roots |
Sum and Product of Roots
| Formula | Description |
| Sum of roots = -b/a | α + β = -b/a |
| Product of roots = c/a | αβ = c/a |
Arithmetic Progressions
General Form
a, a+d, a+2d, a+3d, ...
Important Formulas
| Formula | Description |
| aₙ = a + (n-1)d | nth term of AP |
| Sₙ = n/2[2a + (n-1)d] | Sum of first n terms |
| Sₙ = n/2[a + l] | Sum of first n terms (using last term) |
| d = (aₙ - a₁)/(n-1) | Common difference |
Sum of Natural Numbers
| Formula | Description |
| 1 + 2 + 3 + ... + n = n(n+1)/2 | Sum of first n natural numbers |
| 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6 | Sum of squares of first n natural numbers |
| 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² | Sum of cubes of first n natural numbers |
Triangles
Similarity Criteria
| Criteria | Description |
| AAA (AA) | All corresponding angles are equal |
| SSS | All corresponding sides are in the same ratio |
| SAS | Two sides are in the same ratio and included angles are equal |
Important Theorems
| Theorem | Formula |
| Basic Proportionality Theorem | DE/BC = AD/AB = AE/AC |
| Pythagoras Theorem | c² = a² + b² |
| Converse of Pythagoras | If c² = a² + b², then triangle is right-angled |
Areas of Similar Triangles
If triangles are similar, then ratio of their areas = (ratio of corresponding sides)²
Coordinate Geometry
Distance Formula
| Formula | Description |
| d = √[(x₂-x₁)² + (y₂-y₁)²] | Distance between two points (x₁,y₁) and (x₂,y₂) |
Section Formula
| Type | Formula |
| Internal Division | x = (mx₂ + nx₁)/(m+n), y = (my₂ + ny₁)/(m+n) |
| External Division | x = (mx₂ - nx₁)/(m-n), y = (my₂ - ny₁)/(m-n) |
Midpoint Formula
| Formula | Description |
| x = (x₁+x₂)/2, y = (y₁+y₂)/2 | Midpoint of line segment joining (x₁,y₁) and (x₂,y₂) |
Area of Triangle
| Formula | Description |
| Area = ½ | x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) |
Introduction to Trigonometry
Trigonometric Ratios
| Ratio | Formula | Reciprocal |
| sin θ | Opposite/Hypotenuse | cosec θ = 1/sin θ |
| cos θ | Adjacent/Hypotenuse | sec θ = 1/cos θ |
| tan θ | Opposite/Adjacent | cot θ = 1/tan θ |
Fundamental Identities
| Identity | Formula |
| Pythagorean Identity | sin²θ + cos²θ = 1 |
| 1 + tan²θ = sec²θ | |
| 1 + cot²θ = cosec²θ |
Trigonometric Values for Standard Angles
| Angle | sin θ | cos θ | tan θ |
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
Complementary Angle Formulas
| Formula | Value |
| sin(90° - θ) | cos θ |
| cos(90° - θ) | sin θ |
| tan(90° - θ) | cot θ |
Some Applications of Trigonometry
Height and Distance Problems
| Term | Definition |
| Angle of Elevation | Angle above horizontal |
| Angle of Depression | Angle below horizontal |
| Line of Sight | Direct line from eye to object |
Key Relationships
| Formula | Description |
| tan θ = Height/Base | For right triangle |
| Height = Base × tan θ | Finding height |
| Base = Height/tan θ | Finding base |
Circles
Basic Formulas
| Formula | Description |
| C = 2πr | Circumference of circle |
| A = πr² | Area of circle |
Theorems Related to Circles
| Theorem | Description |
| Tangent-Radius | Tangent is perpendicular to radius at point of contact |
| Two Tangents | Two tangents from external point are equal in length |
| Tangent-Secant | (Tangent)² = External segment × Whole secant |
Length of Tangent
| Formula | Description |
| L = √(d² - r²) | Length of tangent from external point, where d = distance from center, r = radius |
Areas Related to Circles
Sector and Segment
| Formula | Description |
| Area of sector = (θ/360°) × πr² | Where θ is in degrees |
| Area of sector = ½r²θ | Where θ is in radians |
| Length of arc = (θ/360°) × 2πr | Where θ is in degrees |
| Area of segment = Area of sector - Area of triangle |
Combined Figures
| Shape | Area Formula |
| Ring/Annulus | π(R² - r²) |
| Semi-circle | πr²/2 |
| Quarter-circle | πr²/4 |
Surface Areas and Volumes
Cube
| Formula | Description |
| Surface Area = 6a² | Where a is side length |
| Volume = a³ |
Cuboid
| Formula | Description |
| Surface Area = 2(lb + bh + hl) | Where l, b, h are length, breadth, height |
| Volume = l × b × h |
Cylinder
| Formula | Description |
| Curved Surface Area = 2πrh | |
| Total Surface Area = 2πr(r + h) | |
| Volume = πr²h |
Cone
| Formula | Description |
| Curved Surface Area = πrl | Where l is slant height |
| Total Surface Area = πr(r + l) | |
| Volume = ⅓πr²h | |
| Slant height, l = √(r² + h²) |
Sphere
| Formula | Description |
| Surface Area = 4πr² | |
| Volume = ⁴⁄₃πr³ |
Hemisphere
| Formula | Description |
| Curved Surface Area = 2πr² | |
| Total Surface Area = 3πr² | |
| Volume = ⅔πr³ |
Frustum of Cone
| Formula | Description |
| Volume = ⅓πh(r₁² + r₂² + r₁r₂) | Where r₁, r₂ are radii of ends |
| Curved Surface Area = π(r₁ + r₂)l | Where l is slant height |
Statistics
Measures of Central Tendency
Mean
| Type | Formula |
| Direct Method | x̄ = Σx/n |
| Assumed Mean Method | x̄ = a + Σd/n |
| Step Deviation Method | x̄ = a + h(Σu/n) |
Median
| Type | Formula |
| Individual Series | Median = ((n+1)/2)th term |
| Grouped Data | Median = l + [(n/2 - cf)/f] × h |
Mode
| Type | Formula |
| Grouped Data | Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h |
Where:
- l = lower boundary of modal class
- f₁ = frequency of modal class
- f₀ = frequency of class before modal class
- f₂ = frequency of class after modal class
- h = class width
Empirical Relationship
Mode = 3Median - 2Mean
Probability
Basic Probability
| Formula | Description |
| P(E) = Number of favorable outcomes / Total number of outcomes | Basic probability formula |
| 0 ≤ P(E) ≤ 1 | Range of probability |
| P(E) + P(Ē) = 1 | Complementary events |
Properties
| Property | Description |
| P(Sure event) = 1 | Probability of certain event |
| P(Impossible event) = 0 | Probability of impossible event |
Important Constants and Values
Mathematical Constants
| Constant | Value |
| π (pi) | 3.14159... or 22/7 |
| e | 2.71828... |
Square Roots
| Number | Square Root |
| √2 | 1.414 |
| √3 | 1.732 |
| √5 | 2.236 |
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