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
NEET 2026 crash course designed for focused NCERT-based revision and exam readiness.
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
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
