# Maths Formulas for Class 6 to Class 12 PDF | All Basic Maths Formulas

Maths Formulas – Most of you might feel Maths as your biggest nightmare. But, it’s not and it can be quite interesting once you get to know the applications of it in real life. It’s all about connecting the dots and knowing which calculation to use. Maths Formulas are difficult to memorize and Learn Cram Experts have curated some of the List of Basic Mathematical Formulas that you may find useful in your way of preparation.

Students of Class 6 to 12 can utilise the Maths Formulas PDF and cover the entire syllabus. Revise these formulae thoroughly and identify your strengths and weaknesses in the subject and its formulae. Resolve your doubts while solving the problems by making use of these General Maths Formulas for Classes 6, 7, 8, 9, 10, 11, 12.

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Comprehensive list of Maths Formulas for Classes 12, 11, 10, 9 8, 7, 6 to solve problems efficiently. Download Mathematics Formula PDF to complete the syllabus and excel in your exams.

## List of Basic Maths Concepts

Sets and Relations

• Set
• Subset and Superset
• Venn Diagram
• Operations on Sets
• Ordered Pair
• Relation
• Composition of Relation

Functions and Binary Operations

• Functions
• Equal Functions
• Real-Valued and Real Functions
• Standard Real Functions and their Graphs
• Operations on Real Functions
• Compositions of Two Functions
• Even and Odd Functions
• Binary Operations

Complex Numbers

• Equality of Complex Numbers
• Algebra of Complex Numbers
• Argand Plane and Argument of a Complex Number
• Cube Roots of Unity
• nth Roots of Unity
• Geometry of Complex Numbers

• Polynomial
• Inequality
• Linear Inequality
• Solution Set

Sequences and Series

• Sequence
• Series
• Arithmetic Progression (AP)
• Geometric Progression (GP)
• Harmonic Progression (HP)
• Arithmetic-Geometric Progression

Permutations and Combinations

• Fundamental Principles of Counting
• Factorial
• Permutation Circular
• Permutation
• Combination

Binomial Theorem and Principle of Mathematical Induction

• Binomial Theorem for Positive Integer
• General Term in a Binomial Expansion
• Middle Term in a Binomial Expansion
• Greatest Term Multinomial Theorem
• R-f Factor Relations
• Binomial Theorem for Any Index
• Principle df Mathematical Induction

Matrices

• Matrix
• Algebra of Matrices
• Transpose of Matrix
• Symmetric and Skew- Symmetric Matrices
• Elementary Operations (Transfo r matio ns of a Matrix)
• Coniugate of a Matrix
• Rank of a Matrix

Determinants

• Determinant
• Minor and Cofactors
• Inve rse of a Matrix
• Homogeneous and Non- homogeneous System of Linear Equations

Probability

• Experiment
• Algebra of Events
• Bayes Theorem
• Random Variable
• Bernoulli Trials and Binomial Distribution

Trigonometric Functions, Identities and Equations

• Measurement of Angles
• Relation Between Degree and Radian
• Trigonometric Ratios For Acute Angle
• Trigonometric (or Circular) Functions
• Graph of Trigonometric Functions
• Fundamental Trigonometric Identities
• Trigonometric Functions of Compound Angles
• Transformation Formulae
• Trigonometric Functions of Multiple Angles
• Trigonometric Periodic Functions
• Trigonometric Equations

Solution of Triangles

• Basic Rules of Triangle
• Trigonometrical Ratios of Half of the Angles of Triangle
• Area of a Triangle

Heights and Distances

• Angie of Elevation
• Angle of Depression

Inverse Trigonometric Functions

• Domain and Range of Inverse Trigonometric Functions
• Graphs oflnverse Triginometric Functions
• Elementary Properties of Inverse Trigonometric Functions
• Inverse Hyperbolic Functions

Rectangular Axis

• Rectangular Axis
• Distance Formulae
• Section Formulae
• Shifting of Origin/Rot at ion of Axes
• Equation of Locus

Straight Line

• Slope (Gradient) of a Line
• Angle between Two Lines
• Equation of a Straight Line
• Distance of a Point from a Line
• Equation of the Bisectors
• Pair of Lines

Circles

• Standard Equation of a Circle
• Equation of Circle Passing Through Three Points »
• Parametric Equation of Circle
• Equation of Tangent
• Equation of Normal
• Pair of Tangents
• Common Tangents of Two Circles
• Family of Circles
• Limiting Points
• Diameter of Circle

Parabola

• Conicsection
• general Equation of Conic
• Standard Forms of a Parabola and Related Terms
• Equation of Tangent
• Point of I ntersection of Two Tangents
• Equation of Normal
• Length of Tangent and Normal
• Equation of Diameter
• Pair of Tangents
• Chor of Contact

Ellipse

• Parametric Equation
• Equation of Tangent
• Equation of Normal

Hyperbola

• Hyperbola of the Form
• Conjugate Hyperbola
• Equation of Hyperbola in Different Forms
• Tangent Equation of Hyperbola
• Normal Equation of Hyperbola
• Asymptote
• Rectangular Hyperbola

limits, Continuity & Differentiability

• Limit
• Methods of Evaluating Limits
• Sandwich Theorem
• Continuity
• Differentiability
• Fundamental Theorems of Differentiability

Derivatives

• Derivatives of Standard Functions
• Fundamental Rules for Derivatives
• Derivatives of Different Types of Function
• Differentiation of a Determinant
• Successive Differentiations
• Partial Differentiations

Application of Derivatives

• Derivatives as the Rate of Change
• Tangents and Normals
• Rolle’s Theorem
• Lagrange’s Mean Value Theorem
• Approximations and Errors
• Increasing Function
• Maxima and Minima of Functions

Indefinite Integrals

• Some Standard Integral Formulae
• Properties of Integration
• Intergation by Substitution
• Integration by Parts
• Integration by Partial Fractions
• Integration of Irrational Algebraic Function

Definite Integrals

• Fundamental Theorem of Calculus
• Properties of Definite Integral
• Integral Function

Applications of Integrals

• Area of Curves Given by Polar Equations
• Area of Curves Given by Parametric Curves
• Curve Sketching
• Volume and Surface Area

Differential

• Equations
• Order and Degree of a Differential Equation
• Linear and Non-Linear Differential Equations
• Solution of Differential Equations
• Formation of Differential Equations

Vectors

• Types of Vectors
• Addition of Vectors Differences (Subtraction) of Vectors
• Multiplication of a Vector by a Scalar
• Components of a Vector
• Vector joining Two Points
• Section Formulae
• Scalar or Dot Product of Two Vectors
• Vector or Cross Product of Two Vectors
• Scalar Triple Product Vector Triple Product

Three Dimensional Geometry

• Coordinate System
• Direction Cosines
• Line in Space
• Plane
• Angle Between Two Planes
• Parallelism and Perpendicularity of Two Planes

Statistics

• Graphical Representation of Frequency Distributions
• Measures of Central Tendency
• Arithmetic Mean
• Geometric Mean
• Harmonic Mean
• Median
• Mode
• Covariance
• Rank Correlation (Spearman’s)
• Regression

Mathematical Reasoning

• Statement (Proposition)
• Elementary Logical Connectives or Logical Operators
• Truth Value and Truth Table
• Quantifiers
• Validity of Statements

Linear Programming Problem (LPP)

• Objective Function
• Constraints
• Non-negative Restrictions
• Optimal Value
• Solution of Simultaneous Linear Inequations
• Graphical Method of Solving a Linear Programming Problem
• Different Types of Linear Programming Problems

Elementary Arithmetic-I

• Types of Number System
• ClassificationofNumbcrsin Decimal Number System
• Test of Divisibility of a Natural Number
• Rule of Determine the Digit at Unit Place
• Rational Numbers
• Irrational Number
• Real Number
• Complex Numbers
• Fraction
• Ascending/Descending Orders in Fraction
• Power of Index
• Surds
• HCF and LCM
• Simplification
• Average
• Ratio and Proportion
• Proportion

Elementary Arithmetic-II

• Fundamental Formula
• Speed, Time and Distance
• Problem Based on Trains
• Boats and Streams
• Pipes and Cisterns
• Clock
• Calendar

Elementary Arithmetic- lll

• Percentage
• Profit, Loss and Discount
• Simple Interest
• Compound Interest
• Growth and Depriciation
• Partnership
• Share and Debenture
• Alligation or Mixture

Elementary Algebra

• Polynomial
• Synthetic Division Method (Horners Method)
• Remainder Theorem
• Linear Equations Rational Expression

logarithms

• Types of Logarithms
• Anti Logarithm

Geometry

• Triangles
• Congruency of Triangles
• Polygon

Mensuration

• Perimeter and Area of Plane Figure
• Surface Area and Volume of Solid Figure

A polynomial of degree two of the form ax2 + bx + c (a ≠ 0) is called a quadratic expression in x.

ax2 + bx + c = 0 (a ≠ 0) has two roots, given by
α = $$\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}$$ and β = $$\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$

3. Nature of roots
The term b2 – 4ac is called discriminant of the equation. It is denoted by ∆ or D.
(A) Suppose a, b, c ∈ R and a ≠ 0 then

1. If D > 0 ⇒ roots are real and unequal
2. If D = 0 ⇒ roots are real and equal and each equal to -b/2a
3. If D < 0 ⇒ roots are imaginary and unequal or complex conjugate.

(B) Suppose a, b, c ∈ Q, a ≠ 0 then

1. If D > 0 & D is perfect square ⇒ roots are unequal & rational
2. If D > 0 & D is not perfect square ⇒ roots are irrational & unequal

4. Conjugate roots
1. If D < 0 →
One root:
α + iβ
Other root:
α – iβ
then
2. D > 0 →
One root:
α + $$\sqrt{\beta}$$
Other root:
α – $$\sqrt{\beta}$$

5. Sum of roots
S = α + β = $$\frac{-b}{a}=-\frac{\text { Coefficient of } x}{\text { cofficient of } x^{2}}$$

6. Product of roots
P = αβ = $$\frac{c}{a}=\frac{\text { Constant term }}{\text { coefficient of } x^{2}}$$

7. Formation of an equation with given roots
x2 – Sx + P = 0

8. Relation between roots-and coefficients
If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β then

1. (α – β) = $$\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}$$ = ± $$\frac{\sqrt{b^{2}-4 a c}}{a}=\frac{\pm \sqrt{D}}{a}$$
2. α2 + β2 = (α + β)2 – 2αβ = $$\frac{b^{2}-2 a c}{a^{2}}$$
3. α2 – β2 = (α + β)$$\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}$$ = – $$\frac{b \sqrt{b^{2}-4 a c}}{a^{2}}=\frac{\pm \sqrt{D}}{a}$$
4. α3 + β3 = (α + β)3 – 3αβ(α + β) = – $$\frac{b\left(b^{2}-3 a c\right)}{a^{3}}$$
5. α3 – β3 = (α – β)3 – 3αβ(α – β)$$\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}$$ {(α + β)2 – αβ} = $$\frac{\left(b^{2}-a c\right) \sqrt{b^{2}-4 a c}}{a^{3}}$$
6. α4 + β4 = {(α + β)2 – 2αβ}2 – 2α2β2 = $$\left(\frac{b^{2}-2 a c}{a^{2}}\right)^{2}-2 \frac{c^{2}}{a^{2}}$$
7. α4 – β4 = (α2 – β2)(α2 + β2) $$=\frac{\pm b\left(b^{2}-2 a c\right) \sqrt{b^{2}-4 a c}}{a^{4}}$$
8. α2 + αβ + β2 = (α + β)2 – αβ
9. $$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}=\frac{(\alpha+\beta)^{2}-2 \alpha \beta}{\alpha \beta}$$
10. α2β + β2α = αβ(α + β)
11. $$\left(\frac{\alpha}{\beta}\right)^{2}+\left(\frac{\beta}{\alpha}\right)^{2}=\frac{\alpha^{4}+\beta^{4}}{\alpha^{2} \beta^{2}}=\frac{\left(\alpha^{2}+\beta^{2}\right)^{2}-2 \alpha^{2} \beta^{2}}{\alpha^{2} \beta^{2}}$$
(xii) nb2 = ac(1 + n)2 when one root is n times of another

9. Roots under particular cases
For the quadratic equation ax2 + bx + c = 0

1. If b = 0 ⇒ roots are of equal magnitude but of opposite sign
2. If c = 0 ⇒ one root is zero other is – b/a
3. If b = c = 0 ⇒ both roots are zero
4. If a = c ⇒ roots are reciprocal to each other
5. If $$\left.\begin{array}{ll} a>0 & c<0 \\ a<0 & c>0 \end{array}\right\}$$ ⇒ both roots are of opposite signs
6. If $$\left.\begin{array}{l} \mathrm{a}>0, \mathrm{b}>0, \mathrm{c}>0 \\ \mathrm{a}<0, \mathrm{b}<0, \mathrm{c}<0 \end{array}\right\}$$ ⇒ both roots are negative
7. If $$\left.\begin{array}{l} \mathrm{a}>0, \mathrm{b}<0, \mathrm{c}>0 \\ \mathrm{a}<0, \mathrm{b}>0, \mathrm{c}<0 \end{array}\right\}$$ ⇒ both roots are positive

10. Condition for common roots
Let quadratic equations are a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0
(i) If only one root is common:
$$\frac{\alpha^{2}}{\mathrm{b}_{1} \mathrm{c}_{2}-\mathrm{b}_{2} \mathrm{c}_{1}}=\frac{\alpha}{\mathrm{a}_{2} \mathrm{c}_{1}-\mathrm{a}_{1} \mathrm{c}_{2}}=\frac{1}{\mathrm{a}_{1} \mathrm{b}_{2}-\mathrm{a}_{2} \mathrm{b}_{1}}$$
(ii) If both roots are common: $$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$$

11. Nature of the factors of the Quadrate Expression

1. Real and different, if b2 – 4ac > 0.
2. Rational and different, if b2 – 4ac is a perfect square.
3. Real and equal, if b2 – 4ac = 0.

12. Position for roots of a quadratic equation ax2 + bx + c = 0
(A) Condition for both the roots will be greater than k.
(i) D ≥ 0 (ii) k < –$$\frac{b}{2 a}$$ (iii) af(k) > 0

(B) Condition for both the roots will be less than k
(i) D ≥ 0 (ii) k > –$$\frac{b}{2 a}$$ (iii) af(k) > 0

(C) Condition for k lie between the roots
(i) D > 0 (ii) af(k) < 0

(D) Condition for exactly one root lie in the interval (k1, k2) where k1 < k2
(i) f(k1) f(k2) < 0 (ii) D > 0

(E) When both roots lie in the interval (k1, k2) where k1 < k2
(i) D > 0 (ii) f(k1) . f(k2) > 0

(F) Any algebraic expression f(x) = 0 in interval [a, b] if
(i) sign of f(a) and f(b) are of same then either no roots or even no. of roots exist.
(ii) sing of f(a) and f(b) are opposite then f(x) = 0 has at least one real root or odd no. of roots.

13. Maximum & Minimum value of Quadratic Expression
In a quadratic expression ax2 + bx + c

1. If a > 0, quadratic expression has least value at x = –$$\frac{b}{2 a}$$. This least value is given by $$\frac{4 a c-b^{2}}{4 a}=-\frac{D}{4 a}$$
2. If a < 0, quadratic expression has greatest value at x = –$$\frac{b}{2 a}$$. This
greatest value is given by $$\frac{4 a c-b^{2}}{4 a}=-\frac{D}{4 a}$$

14. Quadratic expression in two variables
The general form of a quadratic expression in two variables x & y is ax2 + 2hxy + by2 + 2gx + 2fy + c. The condition that this expression may be resolved into two linear rational factors is
∆ = $$\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|$$ = 0 ⇒ abc + 2 fgh – af2 – bg2 – ch2 = 0 and h2 – ab > 0
This expression is called discriminant of the above quadratic expression.

### FAQs on Maths Formulas

1. Where can I get all Mathematical Formulas?

You can get all Mathematical Formulas arranged in an organized manner as per the Chapters for various classes from here.

2. What are the types of mathematical formulas?

There are many types in maths as far as formulas are concerned. Have a glance at some of the types of Mathematical Formulas.

• Linear equation
• Cubic equation
• First-order Differential equations
• Integral equations
• Trigonometric equations, Matrix equations, 2nd order differentials, Fourier transforms, Laplace transforms, Hamiltonians and much more.

3. Where can I find Maths Formulas for Class 6 to Class 12 in PDF Format?

You can find Maths Formulas for Classes 12, 11, 10, 9, 8, 7, 6 in PDF Format for various concepts in a structured way by referring to our page. Make the most out of these and score better grades in the exam.