CBSE Class 10th Maths Formulas: Chapter Wise Formula and Points
Last Updated :
20 Apr, 2024
Mathematics is one of the most scoring subject in CBSE Class 10th board exam. So Students are advised to prepare well for Math in order to score good marks in CBSE Class 10 board exam.
GeeksforGeeks has curated the chapter wise Math formulae for CBSE Class 10th exam. These Formulae include chapters such as, Number system, Polynomials, Trigonometry, Algebra, Mensuration, Probability, and Statistics.
CBSE Class 10th Maths Formulas
Below is the chapter wise formulae for CBSE Class 10th Exam.
Chapter 1 Real Numbers
The first chapter of mathematics for class 10th will introduce you to a variety of concepts such as natural numbers, whole numbers, and real numbers, and others.
Let's look at some concepts and formulas for Chapter 1 Real numbers for Class 10 as:
| Concepts | Description | Examples/Formula |
|---|
| Natural Numbers | Counting numbers starting from 1. | N = {1, 2, 3, 4, 5, ...} |
| Whole Numbers | Counting numbers including zero. | W = {0, 1, 2, 3, 4, 5, ...} |
| Integers | All positive numbers, zero, and negative numbers. | …, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … |
| Positive Integers | All positive whole numbers. | Z+ = 1, 2, 3, 4, 5, … |
| Negative Integers | All negative whole numbers. | Z– = -1, -2, -3, -4, -5, … |
| Rational Number | Numbers expressed as a fraction where both numerator and denominator are integers and the denominator is not zero. | Examples: 3/7, -5/4 |
| Irrational Number | Numbers that cannot be expressed as a simple fraction. | Examples: π, √5 |
| Real Numbers | All numbers that can be found on the number line, including rational and irrational numbers. | Includes Natural, Whole, Integers, Rational, Irrational |
| Euclid’s Division Algorithm | A method for finding the HCF of two numbers. | a = bq + r, where 0 ≤ r < b |
| Fundamental Theorem of Arithmetic | States that every composite number can be expressed as a product of prime numbers. | Composite Numbers = Product of Primes |
| HCF and LCM by Prime Factorization | Method to find the highest common factor and least common multiple. | HCF = Product of smallest powers of common factors;
LCM = Product of greatest powers of each prime factor; HCF(a,b) × LCM(a,b) = a × b |
Learn More
Chapter 2 Polynomials
Polynomial equations are among the most common algebraic equations involving polynomials. Learning algebra formulae in class 10 will assist you in turning diverse word problems into mathematical forms.
These algebraic formulae feature a variety of inputs and outputs that may be interpreted in a variety of ways. Here are all of the key Algebra Formulas and properties for Class 10:
| Category | Description | Formula/Identity |
|---|
| General Polynomial Formula | Standard form of a polynomial | F (x) = anxn + bxn-1 + an-2xn-2 + …….. + rx + s |
| Special Case: Natural Number n | Difference of powers formula | an – bn = (a – b)(an-1 + an-2b +…+ bn-2a + bn-1) |
| Special Case: Even n (n = 2a) | Sum of even powers formula | xn + yn = (x + y)(xn-1 – xn-2y +…+ yn-2x – yn-1) |
| Special Case: Odd Number n | Sum of odd powers formula | xn + yn = (x + y)(xn-1 – xn-2y +…- yn-2x + yn-1) |
| Division Algorithm for Polynomials | Division of one polynomial by another | p(x) = q(x) × g(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x). Here p(x) is divided, g(x) is divisor, q(x) is quotient, g(x) ≠ 0 and r(x) is remainder. |
Types of Polynomials: Here are some important concepts and properties are mentioned in the below table for each type of polynomials.
| Types of Polynomials | General Form | Zeroes | Formation of Polynomial | Relationship Between Zeroes and Coefficients |
|---|
| Linear | ax+b | 1 | f(x)=a (x−α) | α=−b/a |
| Quadratic | ax2+bx+c | 2 | f(x)=a (x−α)(x−β) | Sum of zeroes α+β=−b/a ; Product of zeroes, αβ= c/a |
| Cubic | ax3+bx2+cx+d | 3 | f(x)=a (x−α)(x−β)(x−γ) | Sum of zeroes, α+β+γ=−b/a; Sum of product of zeroes taken two at a time, αβ+βγ+γα = c/a; Product of zeroes, αβγ= −ad |
| Quartic | ax4+bx3+cx2+dx+e | 4 | f(x)=a(x−α)(x−β)(x−γ)(x−δ) | Relationships become more complex; involves sums and products of zeroes in various combinations. |
Algebraic Polynomial Identities
- (a+b)2 = a2 + b2 + 2ab
- (a-b)2 = a2 + b2 – 2ab
- (a+b) (a-b) = a2 – b2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (x + a)(x – b) = x2 + (a – b)x – ab
- (x – a)(x + b) = x2 + (b – a)x – ab
- (x – a)(x – b) = x2 – (a + b)x + ab
- (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – b3 – 3ab(a – b)
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz -xz)
- x2 + y2 =½ [(x + y)2 + (x – y)2]
- (x + a) (x + b) (x + c) = x3 + (a + b +c)x2 + (ab + bc + ca)x + abc
- x3 + y3= (x + y) (x2 – xy + y2)
- x3 – y3 = (x – y) (x2 + xy + y2)
- x2 + y2 + z2 -xy – yz – zx = ½ [(x-y)2 + (y-z)2 + (z-x)2]
Learn More
Chapter 3 Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables is a crucial chapter that contains a range of significant Maths formulas for class 10, particularly for competitive examinations. Some of the important concepts from this chapter are included below:
- Linear Equations: An equation which can be put in the form ax + by + c = 0, where a, b and c are Pair of Linear Equations in Two Variables, and a and b are not both zero, is called a linear equation in two variables x and y.
- Solution of a system of linear equations: The solution of the above system is the value of x and y that satisfies each of the equations in the provided pair of linear equations.
- Consistent system of linear equations: If a system of linear equations has at least one solution, it is considered to be consistent.
- Inconsistent system of linear equation: If a system of linear equations has no solution, it is said to be inconsistent.
S. No.
| Types of Linear Equation
| General form
| Description
| Solutions
|
| 1. | Linear Equation in one Variable | ax + b=0 | Where a ≠ 0 and a & b are real numbers | One Solution |
| 2. | Linear Equation in Two Variables | ax + by + c = 0 | Where a ≠ 0 & b ≠ 0 and a, b & c are real numbers | Infinite Solutions possible |
| 3. | Linear Equation in Three Variables | ax + by + cz + d = 0 | Where a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c, d are real numbers | Infinite Solutions possible |
- Simultaneous Pair of Linear Equations: The pair of equations of the form:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
- Graphically represented by two straight lines on the cartesian plane as discussed below:
Read More
Chapter 4 Quadratic Equations
| Concept | Description |
|---|
| Quadratic Equation | A polynomial equation of degree two in one variable, typically written as f(x) = ax² + bx + c, where 'a,' 'b,' and 'c' are real numbers, and 'a' is not equal to zero. |
| Roots of Quadratic Equation | The values of 'x' that satisfy the quadratic equation f(x) = 0 are the roots (α, β) of the equation. Quadratic equations always have two roots. |
| Quadratic Formula | The formula to find the roots (α, β) of a quadratic equation is given by: (α, β) = [-b ± √(b² - 4ac)] / (2a), where 'a,' 'b,' and 'c' are coefficients of the equation. |
| Discriminant | The discriminant 'D' of a quadratic equation is given by D = b² - 4ac. It determines the nature of the roots of the equation. |
| Nature of Roots | Depending on the value of the discriminant 'D,' the nature of the roots can be categorized as follows: |
| - D > 0: Real and distinct roots (unequal). |
| - D = 0: Real and equal roots (coincident). |
| - D < 0: Imaginary roots (unequal, in the form of complex numbers). |
| Sum and Product of Roots | The sum of the roots (α + β) is equal to -b/a, and the product of the roots (αβ) is equal to c/a. |
| Quadratic Equation in Root Form | A quadratic equation can be expressed in the form of its roots as x² - (α + β)x + (αβ) = 0. |
| Common Roots of Quadratic Equations | Two quadratic equations have one common root if (b₁c₂ - b₂c₁) / (c₁a₂ - c₂a₁) = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁). |
| Both equations have both roots in common if a₁/a₂ = b₁/b₂ = c₁/c₂. |
| Maximum and Minimum Values | For a quadratic equation ax² + bx + c = 0: |
| Roots of Cubic Equation | - If 'a' is greater than zero (a > 0), it has a minimum value at x = -b/(2a). |
| - If 'a' is less than zero (a < 0), it has a maximum value at x = -b/(2a). |
| If α, β, γ are roots of the cubic equation ax³ + bx² + cx + d = 0, then: |
| - α + β + γ = -b/a |
| - αβ + βγ + λα = c/a |
| - αβγ = -d/a |
Learn More
Chapter 5 Arithmetic Progressions
Many things in our everyday lives have a pattern to them. Sequences are the name given to these patterns.
Arithmetic and geometric sequences are two examples of such sequences. The terms of a sequence are the various numbers that appear in it.
| Concept | Description |
|---|
| Arithmetic Progressions (AP) | A sequence of terms where the difference between consecutive terms is constant. |
| Common Difference | The constant difference between any two consecutive terms in an AP. It is denoted as 'd'. d=a2- a1 = a3 - a2 = ... |
| nth Term of AP | an = a + (n - 1) d,, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference. |
| Sum of nth Terms of AP | Sn= n/2 [2a + (n - 1)d], where 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference. |
Learn More
Chapter 6 Triangles
Triangle is a three-side closed figure made up of three straight lines close together. In CBSE Class 10 curriculum, chapter 6 majorly discusses the similarity criteria between two triangles and some important theorems which may help to understand the problems of triangles.
The main points of the chapter triangle's summary are listed as:
Chapter 6: Triangles
|
|---|
| Concept | Description |
|---|
| Similar Triangles | Triangles with equal corresponding angles and proportional corresponding sides. |
| Equiangular Triangles | Triangles with all corresponding angles equal. The ratio of any two corresponding sides is constant. |
| Criteria for Triangle Similarity |
| Angle-Angle-Angle (AAA) Similarity | Two triangles are similar if their corresponding angles are equal. |
| Side-Angle-Side (SAS) Similarity | Two triangles are similar if two sides are in proportion and the included angles are equal. |
| Side-Side-Side (SSS) Similarity | Two triangles are similar if all three corresponding sides are in proportion. |
| Basic Proportionality Theorem | If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally. |
| Converse of Basic Proportionality Theorem | If in two triangles, corresponding angles are equal, then their corresponding sides are proportional and the triangles are similar. |
Learn More
Chapter 7 Coordinate Geometry
Coordinate geometry helps in the presentation of geometric forms on a two-dimensional plane and the learning of its properties. To gain an initial understanding of Coordinate geometry, we will learn about the coordinate plane and the coordinates of a point, as discussed in the below-mentioned points:
Formulas Related to Coordinate Geometry
|
|---|
| Description | Formula |
|---|
| Distance Formula | Distance between two points A(x1, y1) and B(x2, y2) | AB= √[(x2 − x1)2 + (y2 − y1)2] |
| Section Formula | Coordinates of a point P dividing line AB in ratio m : n | P={[(mx2 + nx1) / (m + n)] , [(my2 + ny1) / (m + n)]} |
| Midpoint Formula | Coordinates of the midpoint of line AB | P = {(x1 + x2)/ 2, (y1+y2) / 2} |
| Area of a Triangle | Area of triangle formed by points A(x1, y1), B(x2, y2) and C(x3, y3) | (∆ABC = ½ |x1(y2 − y3) + x2(y3 – y1) + x3(y1 – y2)| |
Learn More :
Chapter 8 Introduction to Trigonometry
Trigonometry is the science of relationships between the sides and angles of a right-angled triangle. Trigonometric ratios are ratios of sides of the right triangle. Here are some important trigonometric formulas related to trigonometric ratios:
| Category | Formula/Identity | Description/Equivalent |
|---|
| Arc Length in a Circle | l =r × θ | l is arc length, r is radius, θ is angle in radians |
|---|
| Radian and Degree Conversion | Radian Measure = π/180 × Degree Measure | Conversion from degrees to radians |
|---|
| | Degree Measure= 180/π × Radian Measure | Conversion from radians to degrees |
|---|
Trigonometric Ratios
| Trigonometric Ratio | Formula | Description |
|---|
| sin θ | P / H | Perpendicular (P) / Hypotenuse (H) |
| cos θ | B / H | Base (B) / Hypotenuse (H) |
| tan θ | P / B | Perpendicular (P) / Base (B) |
| cosec θ | H / P | Hypotenuse (H) / Perpendicular (P) |
| sec θ | H / B | Hypotenuse (H) / Base (B) |
| cot θ | B / P | Base (B) / Perpendicular (P) |
Read More
Reciprocal of Trigonometric Ratios
| Reciprocal Ratio | Formula | Equivalent to |
|---|
| sin θ | 1 / (cosec θ) | Reciprocal of cosecant |
| cosec θ | 1 / (sin θ) | Reciprocal of sine |
| cos θ | 1 / (sec θ) | Reciprocal of secant |
| sec θ | 1 / (cos θ) | Reciprocal of cosine |
| tan θ | 1 / (cot θ) | Reciprocal of cotangent |
| cot θ | 1 / (tan θ) | Reciprocal of tangent |
Read More
Trigonometric Identities
| Identity | Formula |
|---|
| Pythagorean Identity | sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 - cos2 θ ⇒ cos2 θ = 1 - sin2 θ |
| Cosecant-Cotangent Identity | cosec2 θ - cot2 θ = 1 ⇒ sin2 θ = 1 - cos2 θ ⇒ cos2 θ = 1 - sin2 θ |
| Secant-Tangent Identity | sec2 θ - tan2 θ = 1 ⇒ sec2 θ = 1 + tan2 θ ⇒ tan2 θ = sec2 θ - 1 |
Read More
Chapter 9 Some Applications of Trigonometry
Trigonometry can be used in many ways in the things around us like we can use it for calculating the height and distance of some objects without calculating them actually. Below mentioned is the chapter summary of Some Applications of Trigonometry as:
Important Concepts in Chapter 9 Trigonometry
|
|---|
| Line of Sight | The line formed by our vision as it passes through an item when we look at it. |
| Horizontal Line | A line representing the distance between the observer and the object, parallel to the horizon. |
| Angle of Elevation | The angle formed above the horizontal line by the line of sight when an observer looks up at an object. |
| Angle of Depression | The angle formed below the horizontal line by the line of sight when an observer looks down at an object. |
Learn More
Chapter 10 Circles
A circle is a collection of all points in a plane that are at a constant distance from a fixed point. The fixed point is called the centre of the circle and the constant distance from the centre is called the radius.
Let's learn some important concepts discussed in Chapter 10 Circles of your NCERT textbook.
| Concept | Description |
|---|
| Circle | A circle is a closed figure consisting of all points in a plane that are equidistant from a fixed point called the center. |
| Radius | The radius of a circle is the distance from the center to any point on the circle's circumference. |
| Diameter | The diameter of a circle is a line segment that passes through the center and has endpoints on the circle's circumference. It is twice the length of the radius. |
| Chord | A chord is a line segment with both endpoints on the circle's circumference. A diameter is a special type of chord that passes through the center. |
| Arc | An arc is a part of the circumference of a circle, typically measured in degrees. A semicircle is an arc that measures 180 degrees. |
| Sector | A sector is a region enclosed by two radii of a circle and an arc between them. Sectors can be measured in degrees or radians. |
| Segment | A segment is a region enclosed by a chord and the arc subtended by the chord. |
| Circumference | The circumference of a circle is the total length around its boundary. It is calculated using the formula: Circumference = 2πr, where 'r' is the radius. |
| Area of a Circle | The area of a circle is the total space enclosed by its boundary. It is calculated using the formula: Area = πr², where 'r' is the radius. |
| Central Angle | A central angle is an angle whose vertex is at the center of the circle, and its sides pass through two points on the circle's circumference. |
| Inscribed Angle | An inscribed angle is an angle formed by two chords in a circle with its vertex on the circle's circumference. |
| Tangent Line | A tangent line to a circle is a straight line that touches the circle at only one point, known as the point of tangency. |
| Secant Line | A secant line is a straight line that intersects a circle at two distinct points. |
| Concentric Circles | Concentric circles are circles that share the same center but have different radii. |
| Circumcircle and Incircle | The circumcircle is a circle that passes through all the vertices of a polygon, while the incircle is a circle that is inscribed inside the polygon. |
Chapter 11 Constructions
Construction helps to understand the approach to construct different types of triangles for different given conditions using a ruler and compass of required measurements.
Here the list of important constructions learned in this chapter of class 10 are :
- Determination of a Point Dividing a given Line Segment, Internally in the given Ratio M : N
- Construction of a Tangent at a Point on a Circle to the Circle when its Centre is Known
- Construction of a Tangent at a Point on a Circle to the Circle when its Centre is not Known
- Construction of a Tangents from an External Point to a Circle when its Centre is Known
- Construction of a Tangents from an External Point to a Circle when its Centre is not Known
- Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m<n.
- Construction of a Triangle Similar to a given Triangle as per given Scale Factor m/n, m > n.
Read More
Chapter 12 Areas Related to Circles
The fundamentals of area, circumference, segment, sector, angle and length of a circle, and area for a circle's sector are all covered here. This section also covers the visualization of several planes and solid figure areas.
Below mentioned are the major points from the chapter summary of Areas Related to Circles.
Formulas of Areas Related to Circles
|
|---|
| Concept | Description | Formula |
|---|
| Area of a Circle | The space enclosed by the circle's circumference | Area=πr2 |
| Circumference of a Circle | The perimeter or boundary line of a circle | Circumference=2πr or πd |
| Area of a Sector | The area of a ‘pie-slice’ part of a circle | Area of Sector= (θ/360) × πr2 (θ in degrees) |
| Length of an Arc | The length of the curved line forming the sector | Length of Arc= (θ/360) × 2πr (θ in degrees) |
| Area of a Segment | Area of a sector minus the area of the triangle formed by the sector | Area of Segment = Area of Sector - Area of Triangle |
- r is the radius of the circle.
- d is the diameter of the circle.
- θ is the angle of the sector or segment in degrees.
Read More
Chapter 13 Surface Areas and Volumes
This page explains the concepts of surface area and volume for Class 10. The surface area and volume of several solid shapes such as the cube, cuboid, cone, cylinder, and so on will be discussed in this article. Lateral Surface Area (LSA), Total Surface Area (TSA), and Curved Surface Area are the three types of surface area (CSA).
Formulas Related to Surface Areas and Volumes
|
|---|
| Geometrical Figure | Total Surface Area (TSA) | Lateral/Curved Surface Area (CSA/LSA) | Volume |
|---|
| Cuboid | 2(lb + bh + hl) | 2h(l + b) | l × b × h |
| Cube | 6a² | 4a² | a³ |
| Right Circular Cylinder | 2πr(h + r) | 2πrh | πr²h |
| Right Circular Cone | πr(l + r) | πrl | 1/3πr²h |
| Sphere | 4πr² | 2πr² | 4/3πr³ |
| Right Pyramid | LSA + Area of the base | ½ × p × l | 1/3 × Area of the base × h |
| Prism | LSA × 2B | p × h | B × h |
| Hemisphere | 3πr² | 2πr² | 2/3πr³ |
- l = length, b = breadth, h = height, r = radius, a = side, p = perimeter of the base, B = area of the base.
- TSA includes all surfaces of the figure, CSA/LSA includes only the curved or lateral surfaces, and
- Volume measures the space occupied by the figure.
Learn More
Chapter 14 Statistics
Statistics in Class 10 mainly consist of the study of given data b evaluating its mean, mode, median. The statistic formulas are given below:
| Statistical Measure | Method/Description | Formula |
|---|
| Mean | Direct method | X = ∑fi xi / ∑fi |
| Assumed Mean Method | X = a + ∑fi di / ∑fi
,(where di = xi - a) |
| Step Deviation Method: | X = a + ∑fi ui / ∑fi × h |
| Median | Middlemost Term | For even number of observations: Middle term
For odd number of observations: (n+1/2) th term |
| Mode | Frequency Distribution | \text{Mode} = 1 + \left[\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right]\times h
where l = lower limit of the modal class,
f1 =frequency of the modal class,
f0 = frequency of the preceding class of the modal class,
f2 = frequency of the succeeding class of the modal class,
h is the size of the class interval.
|
Chapter 15 Probability
Probability denotes the likelihood of something happening. Its value is expressed from 0 to 1.
Let's discuss some important Probability formulas in the Class 10 curriculum:
| Type of Probability | Description | Formula |
|---|
| Empirical Probability | Probability based on actual experiments or observations. | Empirical Probability = Number of Trials with expected outcome / Total Number of Trials |
| Theoretical Probability | Probability based on theoretical reasoning rather than actual experiments. | Theoretical Probability = Number of favorable outcomes to E / Total Number of possible outcomes |
Related :
Similar Reads
CBSE Class 10 Maths Notes PDF: Chapter Wise Notes 2024
Math is an important subject in CBSE Class 10th Board exam. So students are advised to prepare accordingly to score well in Mathematics. Mathematics sometimes seems complex but at the same time, It is easy to score well in Math. So, We have curated the complete CBSE Class 10 Math Notes for you to pr
15+ min read
Chapter 1: Real Numbers
Real Numbers
Real Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line
10 min read
Euclid Division Lemma
Euclid's Division Lemma which is one of the fundamental theorems proposed by the ancient Greek mathematician Euclid which was used to prove various properties of integers. The Euclid's Division Lemma also serves as a base for Euclid's Division Algorithm which is used to find the GCD of any two numbe
5 min read
Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic is a useful method to understand the prime factorization of any number. The factorization of any composite number can be uniquely written as a multiplication of prime numbers, regardless of the order in which the prime factors appear. The figures above represent
7 min read
HCF / GCD and LCM - Definition, Formula, Full Form, Examples
The full form of HCF is the Highest Common Factor while the full form of LCM is the Least Common Multiple. HCF is the largest number that divides two or more numbers without leaving a remainder, whereas LCM is the smallest multiple that is divisible by two or more integers. Table of Content HCF or G
10 min read
Irrational Numbers- Definition, Examples, Symbol, Properties
Irrational numbers are real numbers that cannot be expressed as fractions. Irrational Numbers can not be expressed in the form of p/q, where p and q are integers and q ≠0. They are non-recurring, non-terminating, and non-repeating decimals. Irrational numbers are real numbers but are different from
12 min read
Decimal Expansions of Rational Numbers
Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. So in this article let's discuss some rational and irrational numbers an
6 min read
Rational Numbers
Rational numbers are a fundamental concept in mathematics, defined as numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Represented in the form p/q​ (with p and q being integers), rational numbers include fractions, whole numbers, and terminating or repea
15+ min read
Chapter 2: Polynomials
Algebraic Expressions in Math: Definition, Example and Equation
Algebraic Expression is a mathematical expression that is made of numbers, and variables connected with any arithmetical operation between them. Algebraic forms are used to define unknown conditions in real life or situations that include unknown variables. An algebraic expression is made up of term
8 min read
Polynomial Formula
The polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. The General Formula of a Polynomial: f(x) = an​xn + an−1​xn−1 + ⋯ + a1​x + a0​ Where, an​, an−1​, …, a1​, a0
6 min read
Types of Polynomials (Based on Terms and Degrees)
Types of Polynomials: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. There are mainly four types of polynomials based on degree-constant polynomial (zero degree), linear polynomial ( 1st degree), quadratic polynomial (2n
9 min read
Zeros of Polynomial
Zeros of a Polynomial are those real, imaginary, or complex values when put in the polynomial instead of a variable, the result becomes zero (as the name suggests zero as well). Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing situation
14 min read
Geometrical meaning of the Zeroes of a Polynomial
An algebraic identity is an equality that holds for any value of its variables. They are generally used in the factorization of polynomials or simplification of algebraic calculations. A polynomial is just a bunch of algebraic terms added together, for example, p(x) = 4x + 1 is a degree-1 polynomial
8 min read
Factorization of Polynomial
Factorization in mathematics refers to the process of expressing a number or an algebraic expression as a product of simpler factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and we can express 12 as 12 = 1 × 12, 2 × 6, or 4 × 3. Similarly, factorization of polynomials involves expr
10 min read
Division Algorithm for Polynomials
Polynomials are those algebraic expressions that contain variables, coefficients, and constants. For Instance, in the polynomial 8x2 + 3z - 7, in this polynomial, 8,3 are the coefficients, x and z are the variables, and 7 is the constant. Just as simple Mathematical operations are applied on numbers
5 min read
Algebraic Identities
Algebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Relationship between Zeroes and Coefficients of a Polynomial
Polynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero. While the coef
9 min read
Division Algorithm Problems and Solutions
Polynomials are made up of algebraic expressions with different degrees. Degree-one polynomials are called linear polynomials, degree-two are called quadratic and degree-three are called cubic polynomials. Zeros of these polynomials are the points where these polynomials become zero. Sometimes it ha
6 min read
Chapter 3: Pair of Linear equations in two variables
Linear Equation in Two Variables
Linear Equation in Two Variables: A Linear equation is defined as an equation with the maximum degree of one only, for example, ax = b can be referred to as a linear equation, and when a Linear equation in two variables comes into the picture, it means that the entire equation has 2 variables presen
9 min read
Pair of Linear Equations in Two Variables
Linear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship betwe
11 min read
Graphical Methods of Solving Pair of Linear Equations in Two Variables
A system of linear equations is just a pair of two lines that may or may not intersect. The graph of a linear equation is a line. There are various methods that can be used to solve two linear equations, for example, Substitution Method, Elimination Method, etc. An easy-to-understand and beginner-fr
8 min read
Solve the Linear Equation using Substitution Method
A linear equation is an equation where the highest power of the variable is always 1. Its graph is always a straight line. A linear equation in one variable has only one unknown with a degree of 1, such as: 3x + 4 = 02y = 8m + n = 54a – 3b + c = 7x/2 = 8There are mainly two methods for solving simul
11 min read
Solving Linear Equations Using the Elimination Method
If an equation is written in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero, it is said to be a linear equation in two variables. For example, 3x + y = 4 is a linear equation in two variables- x and y. The numbers th
10 min read
Chapter 4: Quadratic Equations
Quadratic Equations
A Quadratic equation is a second-degree polynomial equation that can be represented as ax2 + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. The solutions of a quadratic equation are known as its roots. These roots can be found using method
11 min read
Roots of Quadratic Equation
The roots of a quadratic equation are the values of x that satisfy the equation. The roots of a quadratic equation are also called zeros of a quadratic equation. A quadratic equation is generally in the form: ax2 + bx + c = 0 Where: a, b, and c are constants (with a ≠0).x represents the variable. T
13 min read
Solving Quadratic Equations
A quadratic equation, typically in the form ax² + bx + c = 0, can be solved using different methods including factoring, completing the square, quadratic formula, and the graph method. While Solving Quadratic Equations we try to find a solution that represent the points where this the condition Q(x)
8 min read
How to find the Discriminant of a Quadratic Equation?
Algebra can be defined as the branch of mathematics that deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for th
4 min read
Chapter 5: Arithmetic Progressions
Arithmetic Progressions Class 10- NCERT Notes
Arithmetic Progressions (AP) are fundamental sequences in mathematics where each term after the first is obtained by adding a constant difference to the previous term. Understanding APs is crucial for solving problems related to sequences and series in Class 10 Mathematics. These notes cover the ess
7 min read
Sequences and Series
A sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence. For example: 2, 5, 8, 11, 14
7 min read
Arithmetic Progression in Maths
Arithmetic Progression (AP) or Arithmetic Sequence is simply a sequence of numbers such that the difference between any two consecutive terms is constant. Some Real World Examples of AP Natural Numbers: 1, 2, 3, 4, 5, . . . with a common difference 1Even Numbers: 2, 4, 6, 8, 10, . . . with a common
3 min read
Arithmetic Progression - Common difference and Nth term | Class 10 Maths
Arithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9....... is in a series which has a common difference (3 - 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1,
5 min read
How to find the nth term of an Arithmetic Sequence?
Answer - Use the formula: an = a1 + (n - 1)dWhere:an = nth term,a = first term,d = common difference,n = term number. Substitute the values of a, d, and n into the formula to calculate an. Steps to find the nth Term of an Arithmetic SequenceStep 1: Identify the First and Second Term: 1st and 2nd ter
3 min read
Arithmetic Progression | Sum of First n Terms | Class 10 Maths
Arithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2,
8 min read
Arithmetic Mean
Arithmetic Mean, commonly known as the average, is a fundamental measure of central tendency in statistics. It is defined as the ratio of all the values or observations to the total number of values or observations. Arithmetic Mean is one of the fundamental formulas used in mathematics and it is hig
13 min read
Arithmetic Progression | Sum of First n Terms | Class 10 Maths
Arithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2,
8 min read
Chapter 6: Triangles
Triangles in Geometry
A triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re
13 min read
Similar Triangles
Similar Triangles are triangles with the same shape but can have variable sizes. Similar triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are different from congruent triangles. Two congruent figures are always similar, bu
15+ min read
Criteria for Similarity of Triangles
Things are often referred similar when the physical structure or patterns they show have similar properties, Sometimes two objects may vary in size but because of their physical similarities, they are called similar objects. For example, a bigger Square will always be similar to a smaller square. In
9 min read
Basic Proportionality Theorem (BPT) Class 10 | Proof and Examples
Basic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT. BPT states that if a line is parallel to a side of a triangl
8 min read
Pythagoras Theorem | Formula, Proof and Examples
Pythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem. It states that in a right-angled triangle, the square of the hypotenuse is equ
10 min read
Chapter 7: Coordinate Geometry
Coordinate Geometry
Coordinate geometry is a branch of mathematics that combines algebra and geometry using a coordinate plane. It helps us represent points, lines, and shapes with numbers and equations, making it easier to analyze their positions, distances, and relationships. From plotting points to finding the short
3 min read
Distance Formula Class 10 Maths
The distance formula is one of the important concepts in coordinate geometry which is used widely. By using the distance formula we can find the shortest distance i.e drawing a straight line between points. There are two ways to find the distance between points: Pythagorean theoremDistance formulaTa
9 min read
Distance Between Two Points
Distance Between Two Points is the length of line segment that connects any two points in a coordinate plane in coordinate geometry. It can be calculated using a distance formula for 2D or 3D. It represents the shortest path between two locations in a given space. In this article, we will learn how
6 min read
Section Formula
Section Formula is a useful tool in coordinate geometry, which helps us find the coordinate of any point on a line which is dividing the line into some known ratio. Suppose a point divides a line segment into two parts which may be equal or not, with the help of the section formula we can find the c
14 min read
How to find the ratio in which a point divides a line?
Answer: To find the ratio in which a point divides a line we use the following formula [Tex]x = \frac{m_1x_2+m_2x_1}{m_1+m_2}Â Â [/Tex][Tex]y = \frac{m_1y_2+m_2y_1}{m_1+m_2}[/Tex]Geo means Earth and metry means measurement. Geometry is a branch of mathematics that deals with distance, shapes, sizes, r
4 min read
How to find the Trisection Points of a Line?
To find the trisection points of a line segment, you need to divide the segment into three equal parts. This involves finding the points that divide the segment into three equal lengths. In this article, we will answer "How to find the Trisection Points of a Line?" in detail including section formul
4 min read
How to find the Centroid of a Triangle?
Answer: The Centroid for the triangle is calculated using the formula[Tex]\left (\frac{[x1+x2+x3]}{3}, \frac{[y1+y2+y3]}{3}\right)[/Tex]A triangle consists of three sides and three interior angles. Centroid refers to the center of an object. Coming to the centroid of the triangle, is defined as the
4 min read
Area of a Triangle in Coordinate Geometry
There are various methods to find the area of the triangle according to the parameters given, like the base and height of the triangle, coordinates of vertices, length of sides, etc. In this article, we will discuss the method of finding area of any triangle when its coordinates are given. Area of T
6 min read
Chapter 8: Introduction to Trigonometry
Trigonometric Ratios
There are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). As giv
4 min read
Unit Circle: Definition, Formula, Diagram and Solved Examples
Unit Circle is a Circle whose radius is 1. The center of unit circle is at origin(0,0) on the axis. The circumference of Unit Circle is 2π units, whereas area of Unit Circle is π units2. It carries all the properties of Circle. Unit Circle has the equation x2 + y2 = 1. This Unit Circle helps in defi
7 min read
Trigonometric Ratios of Some Specific Angles
Trigonometry is all about triangles or to be more precise the relationship between the angles and sides of a triangle (right-angled triangle). In this article, we will be discussing the ratio of sides of a right-angled triangle concerning its acute angle called trigonometric ratios of the angle and
6 min read
Trigonometric Identities
Trigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which involve relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. In this article,
10 min read
Chapter 9: Some Applications of Trigonometry