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.
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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 |
Exam Tips and Study Strategy
Formula Memorization Techniques
- Practice regularly – Write formulas daily
- Understand derivations – Know how formulas are derived
- Create formula sheets – Organize by chapters
- Use mnemonics – Create memory aids for complex formulas
- Apply in problems – Regular problem-solving reinforces memory
Common Mistakes to Avoid
- Confusing similar formulas
- Sign errors in calculations
- Unit conversion mistakes
- Forgetting to check reasonableness of answers
Frequently Asked Questions (FAQs) on Class 10 Mathematics Formulas
1. How to memorize Class 10 maths formulas quickly and effectively?
Memorizing Class 10 maths formulas requires a strategic approach combining understanding and practice:
Understanding-Based Techniques:
- Learn the derivation: Understanding how a formula is derived helps in long-term retention. For example, knowing why the quadratic formula x = [-b ± √(b² – 4ac)] / 2a works makes it easier to remember.
- Visual learning: Create colorful formula charts and stick them where you study. Visual memory is powerful for mathematical formulas.
- Connect related formulas: Group similar formulas together. For instance, all trigonometric identities (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ) follow a similar pattern.
Practice-Based Methods:
- Daily writing: Write each formula at least 10 times daily for one week
- Solve problems: Regular problem-solving automatically reinforces formula memory
- Create flashcards: Make digital or physical flashcards with formula on one side and application on the other
- Teach others: Explaining formulas to classmates solidifies your own understanding
2. Which are the most important formulas for Class 10 board exams?
Based on previous years’ board exam patterns and question paper analysis, here are the most frequently tested formulas:
High Priority Formulas (Must Know):
- Quadratic Equations:
- Quadratic formula: x = [-b ± √(b² – 4ac)] / 2a
- Discriminant: Δ = b² – 4ac
- Sum and product of roots
- Arithmetic Progressions:
- nth term: aₙ = a + (n-1)d
- Sum of n terms: Sₙ = n/2[2a + (n-1)d]
- Trigonometry:
- All three Pythagorean identities
- Trigonometric ratios for 0°, 30°, 45°, 60°, 90°
- Complementary angle formulas
- Coordinate Geometry:
- Distance formula
- Section formula (internal division)
- Area of triangle
- Surface Areas and Volumes:
- Cone: Volume = ⅓πr²h, CSA = πrl
- Sphere: Volume = ⁴⁄₃πr³, SA = 4πr²
- Cylinder: Volume = πr²h, TSA = 2πr(r + h)
- Circles:
- Area = πr²
- Circumference = 2πr
- Area of sector = (θ/360°) × πr²
Medium Priority:
- Polynomial zero-coefficient relationships
- Statistics formulas (mean, median, mode)
- Probability basic formula
Focus 70% of your preparation time on high-priority formulas as they carry maximum weightage in board exams.
Q. What is the discriminant formula and how to use it in Class 10?
The discriminant is one of the most important concepts in Class 10 quadratic equations chapter.
Discriminant Formula:Δ = b² – 4ac
Where a, b, and c are coefficients in the standard quadratic equation: ax² + bx + c = 0
Understanding the Discriminant:
The discriminant tells you the nature of roots without actually solving the equation:
| Discriminant Value | Nature of Roots | Example |
|---|---|---|
| Δ > 0 (Positive) | Two distinct real roots | x² – 5x + 6 = 0 (Δ = 1) |
| Δ = 0 (Zero) | Two equal real roots | x² – 4x + 4 = 0 (Δ = 0) |
| Δ < 0 (Negative) | No real roots (imaginary) | x² + x + 1 = 0 (Δ = -3) |
Practical Applications:
- Finding Nature of Roots: Given 2x² + 3x + 5 = 0
- Δ = (3)² – 4(2)(5) = 9 – 40 = -31
- Since Δ < 0, the equation has no real roots
- Determining Values: Find k if x² + kx + 4 = 0 has equal roots
- For equal roots: Δ = 0
- k² – 4(1)(4) = 0
- k² = 16, therefore k = ±4
- Real-World Problems: Used in physics for projectile motion, economics for profit/loss calculations, and engineering for optimization problems
Q. How to prepare for Class 10 maths exam using formulas effectively?
Strategic formula-based preparation can significantly improve your Class 10 maths performance:
Month-wise Preparation Strategy:
3 Months Before Exam:
- Week 1-2: Complete all chapters, list all formulas topic-wise
- Week 3-4: Create a master formula booklet with derivations
- Week 5-8: Solve NCERT examples and exercises using formulas
- Week 9-12: Practice previous years’ question papers
1 Month Before Exam:
- Daily Formula Revision: Spend 30 minutes reviewing 2-3 chapters
- Formula-Problem Connection: For each formula, solve 5 different problem types
- Mock Tests: Take weekly mock tests focusing on formula application
- Identify Weak Areas: Mark formulas you frequently forget and revise them extra
Last Week Strategy:
- Quick revision of all formulas (2 hours daily)
- Focus on high-weightage formulas
- Solve sample papers under timed conditions
- Create a one-page quick reference sheet
Chapter-Wise Time Allocation:
| Chapter | Importance | Practice Time |
|---|---|---|
| Trigonometry | Very High | 20% |
| Quadratic Equations | High | 15% |
| Coordinate Geometry | High | 15% |
| Surface Areas & Volumes | High | 15% |
| Statistics | Medium | 10% |
| Others | Medium | 25% |
Smart Study Techniques:
- Formula Application Practice: Don’t just memorize – solve 10+ problems per formula
- Error Analysis: Maintain a mistake diary for formula-related errors
- Group Study: Quiz each other on formulas and their applications
- Digital Tools: Use math apps for interactive formula practice
Q. What are all the basic trigonometry formulas needed for Class 10?
Trigonometry is one of the highest-scoring chapters in Class 10 if you master the basic formulas:
1. Basic Trigonometric Ratios:
- sin θ = Opposite side / Hypotenuse
- cos θ = Adjacent side / Hypotenuse
- tan θ = Opposite side / Adjacent side
- cot θ = Adjacent side / Opposite side
- sec θ = Hypotenuse / Adjacent side
- cosec θ = Hypotenuse / Opposite side
2. Reciprocal Identities:
- sin θ = 1/cosec θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
3. Pythagorean Identities (Most Important):
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
4. Standard Angle Values (Learn by Heart):
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
Memory Trick for sin values: Think of √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1
For cos values, reverse the sin values: 1, √3/2, 1/√2, 1/2, 0
5. Complementary Angle Formulas:
- sin(90° – θ) = cos θ
- cos(90° – θ) = sin θ
- tan(90° – θ) = cot θ
- cot(90° – θ) = tan θ
- sec(90° – θ) = cosec θ
- cosec(90° – θ) = sec θ
Application in Height and Distance:
- Angle of Elevation: tan θ = Height / Distance
- Angle of Depression: tan θ = Depth / Distance
Q. Where can I get Class 10 maths formulas PDF for free download?
While there are various sources for downloading Class 10 maths formulas PDF, here’s a comprehensive guide on reliable and authorized sources:
Official and Trusted Sources:
- NCERT Official Website (ncert.nic.in)
- Free access to complete NCERT textbooks
- Authentic and curriculum-aligned content
- Available in multiple languages
- Mobile-friendly format
- State Board Education Portals
- CBSE official website (cbse.gov.in)
- State-specific education department websites
- Examination reform documents
- Sample papers with formula lists
- Educational Government Initiatives
- DIKSHA Platform (diksha.gov.in)
- SWAYAM Portal
- National Digital Library of India
- Free and quality-verified content
Creating Your Own Formula PDF:
Instead of searching for PDFs online, consider creating a personalized formula sheet:
Benefits of Self-Made Formula Sheets:
- Customized to your learning style
- Better retention through active creation
- Include your own notes and tricks
- Can be updated as you learn
How to Create Your Formula PDF:
- Use free tools: Google Docs, Microsoft Word, Canva
- Organize by chapters: Follow NCERT sequence
- Add color coding: Different colors for different topics
- Include examples: One solved example per formula
- Make it scannable: Use tables, bullet points, highlighting
Digital Study Resources (Free):
- YouTube Channels: Shiksha Nation
- Mobile Apps: Shiksha Nation
- Online Platforms: Shiksha Nation
What to Include in Your Formula Compilation:
- All chapter-wise formulas (as provided in this guide)
- Derivations for complex formulas
- Standard values (π, √2, √3, etc.)
- Quick revision tables
- Common mistakes and corrections
- Previous year important questions
Recommended Format for Your Formula Sheet:
- One formula per row in tables
- Clear headings and subheadings
- Print-friendly format (A4 size)
- Leave space for personal notes
- Include page numbers for easy reference