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Class 10 Board Exams - Top Maths Formulae You Should Know

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Class 10 Board Exams - Top Maths Formulae You Should Know

Edited By Ramraj Saini | Updated on Jan 10, 2023 03:29 PM IST

While every component of the Mathematics curriculum is important for building an interest in and understanding of the discipline as a whole, they don’t all carry equal weight in board exams. For the final revision for the Central Board of Secondary Education (CBSE) Class 10 Maths exam, knowing which ones carry the most marks and the most critical formulae associated with them can help build confidence.

Class 10 Board Exams - Top Maths Formulae You Should Know
Class 10 Board Exams - Top Maths Formulae You Should Know

The NCERT Class 10 Mathematics textbook contains the following units

  • Algebra

  • Geometry

  • Trigonometry

  • Statistics and Probability

  • Mensuration

  • Coordinate Geometry

  • Number Systems

Given below is a list of the most important Maths formulae for CBSE Class 10 board exams, in the decreasing order of the marks they carry.

Algebra: 20 Marks

This unit contains the maximum marks in the Class 10 CBSE Maths board exam and includes chapters like Polynomials, Pair of Linear Equations in two Variables, Quadratic Equations, and Arithmetic Progression. Here are the important formulae for the Class 10 board exam from each chapter.

Polynomials

This chapter mainly discusses the definition of a polynomial, degree of a polynomial, the zeros of a polynomial, the fundamental theorem of algebra, the division algorithm for polynomials, synthetic division, and the zeros theorem.

These are the most important formulae:

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

(a + b)(a - b) = a2 - b2

(x + a)(x + b) = x2 + (a + b)x + ab

a3 - b3 = (a - b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 - ab + b2)

(a + b)3 = a3 + b3 + 3ab(a + b)

(a - b)3 = a3 - 3ab(a - b) + b3

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

If a quadratic equation is given as ax2 + bx + c = 0 then

Sum of zeros = α + β = -b/a = -(coefficient of x)/coefficient of x2

Product of zeros = αβ = c/a = -(coefficient of x)/coefficient of x2

If a cubic equation is given as ax3 + bx2 + cx + d = 0 then

Sum of zeros = α + β + γ = -b/a = -(coefficient of x2)/coefficient of x3

Sum of multiple of alternate zeros = αβ + βγ + γα = c/a = coefficient of x/coefficient of x3

Product of zeros = αβγ = -d/a =-(constant term)/coefficient of x3

If p(x) and g(x) are two polynomials such that g(x) is not equal to 0, then we can find polynomials q(x) and r(x) such that the following equation holds:

p(x) = g(x) * q(x) + r(x)

Pair Of Linear Equations In Two Variables

There are three possible ways to solve a pair of linear equations in two variables: graphing, substitution method, and elimination method.

  • The graphing method involves plotting the two equations on the same coordinate plane and finding the point(s) of intersection.

  • The substitution method involves solving one of the equations for one variable in terms of the other and substituting this expression into the other equation to solve for the remaining variable.

  • The elimination method involves adding or subtracting the two equations to eliminate one of the variables.

If a pair of linear equations in two variables has exactly one solution, it is called a consistent pair of equations. If it has no solutions, it is called an inconsistent pair. If it has infinite solutions, it is called a dependent pair. The following are important formulae from this chapter.

If a pair of linear equations is given as

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Comparison Of Ratios

Graphical Interpretation

Algebraic Interpretation

(a1/a2) ≠ (b1/b2)

Intersecting lines

Exactly one or unique solution

a1/a2 = b1/b2 = c1/c2

Coincident lines

Infinitely many solutions

a1/a2 = b1/b2 ≠ c1/c2

Parallel lines

No solutions


When there is a unique solution then

x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)

This formula can be remembered as the diagram given below

4

Quadratic Equations

This chapter contains concepts of quadratic equations, solution of quadratic equation with factorisation method, completing the square method, and lastly the nature of roots.

The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are constants and x is the variable. The solutions to a quadratic equation are also known as the roots of the equation and can be found using the quadratic formula:

x = {-b ± √(b2 - 4ac)}/2a

If the value of the discriminant b2-4ac is positive, the equation has two distinct real roots. If the value of the discriminant is zero, the equation has one repeated real root. If the value of the discriminant is negative, the equation has no real roots.

Arithmetic Progression

In an arithmetic progression (AP), the difference between consecutive terms is always constant. Some important formulae for arithmetic progressions are:

If an AP is given as a1, a2, a3,..................an

Common difference = d = a2 - a1 = a3 - a2 = a4 - a3 = an-an-1

Nth term of AP = Tn = a + (n - 1)d

Sum of n term of AP = Sn = (n/2)[2a + (n - 1)d] = (n/2)[T1 + Tn]

Geometry: 15 Marks

This unit contains three chapters – Triangles, Circles, and Constructions. Below, students can find important formulae from each chapter.

Triangles

This chapter contains concepts of similar triangles, criteria of similarity of triangles, calculation of area of similar triangles, and Pythagoras’ theorem. Here are some important formulae from this chapter.

  • If a line is drawn parallel to one side of a triangle and intersects the other two sides at two different points, the lengths of the segments into which the other two sides are divided will be in the same ratio and vice versa.

ADDB=AEEC

1%20(1)

  • Similarity Of Triangles

  1. AA (Angle-Angle) Similarity: If two triangles have two pairs of corresponding angles that are congruent (equal in measure), then the triangles are similar.

  2. SAS (Side-Angle-Side) Similarity: If two triangles have two pairs of corresponding sides that are in the same ratio, and the included angles are congruent, then the triangles are similar.

  3. SSS (Side-Side-Side) Similarity: If two triangles have three pairs of corresponding sides that are in the same ratio, then the triangles are similar.

  4. HL (Hypotenuse-Leg) Similarity: If two right triangles have one pair of corresponding legs that are in the same ratio, and the hypotenuses of the triangles are in the same ratio, then the triangles are similar.

  • The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

ar(ABC)/ar(PQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2

2%20(1)

  • In a right triangle, the length of the hypotenuse is equal to the square root of the sum of the squares of the lengths of the other two sides.

AC2=AB2+BC2

3

Circles

This chapter describes concepts of tangent and tangent to a circle. Following are some important concepts from this chapter.

  • At any point of contact, the tangent to a circle is perpendicular to the radius drawn from the point of contact.

  • It is not possible to draw a tangent from a point inside a circle to the circle.

  • A single tangent can be drawn from a point on the circle to the circle.

  • Two tangents can be drawn from a point outside the circle to the circle.

  • The lengths of tangents drawn from an external point to a circle are equal.

Constructions

This chapter discusses concepts of division of a line segment and construction of a tangent to a circle. Particularly, it contains the concepts of dividing a line segment in a given ratio, constructing a triangle similar to a given triangle as per the given scale factor, and constructing the tangents to a circle from a point outside it.

Trigonometry: 12 Marks

This unit includes two chapters Introduction to Trigonometry and Some Applications of Trigonometry. Here students can find important formulae for the board examination from each chapter.

Trigonometry And Its Applications

In A right angle Δ PQR, the ratios are defined as:


1673239197108

sin Q = (opposite side to ∠Q)/(hypotenuse) = PR/PQ

cos Q = (adjacent side to ∠Q)/(hypotenuse) = RQ/PQ

tan Q = (opposite side to ∠Q)/(adjacent side to ∠Q) = PR/RQ

= sin ∠Q/cos ∠Q

cosec Q= 1/sin Q = (hypotenuse)/ (side opposite to ∠Q) = PQ/PR

sec Q = 1/cos Q = (hypotenuse)/ (side opposite to ∠Q) = PQ/RQ

cot Q = 1/tan Q = (side adjacent to ∠Q)/(side opposite to ∠Q)= RQ/PR

Trigonometric Ratios For Some Specific Angles

Trigonometric Ratios

Specific Angles

∠θ

30°

45°

60°

90°

sin θ

0

1/2

1/√2

√3/2

1

cos θ

1

√3/2

1/√2

1/2

0

tan θ

0

1/√3

1

√3

Not defined

cosec θ

Not defined

2

√2

2/√3

1

sec θ

1

2/√3

√2

2

Not defined

cot θ

Not defined

√3

1

1/√3

0

In general form, for angle θ,

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,

Following are three trigonometric identities

cos2 θ + sin2 θ = 1

1 + tan2 θ = sec2 θ

cot2θ + 1 = cosec2 θ , True for all θ such that 0° < θ ≤ 90°.

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Statistics And Probability: 11 Marks

This unit contains two chapters – Statistics and Probability. Statistics further discusses the mean, mode, and median of grouped data as well as a graphical representation of cumulative frequency distribution. Probability discusses the concepts required to calculate the probability of events. Here are some important formulae from these chapters.

Mean Of Grouped Data

If x1, x2, .........xn are observations with respective frequencies f1, f2......fn Then

Mean = x=(f1x1 + f2x2 + ..........fnxn)/(f1 + f2 + .........fn) = ∑fixi/∑fi

Class Mark = (Upper class limit+lower class limit)/2

There are different methods of calculating the mean of grouped data – direct method, assumed mean method, and step derivation method.

Direct Method:

x = ∑fixi/∑fi

where ∑fixi is the sum of observations

∑fi is the number of observations

Assumed Mean Method:

x = a + ∑fidi/∑fi

where

a is assumed mean

di is the deviation of mean value from xi.

And, di = xi – a

Step Deviation Method:

x = a + (∑fiui/∑fi)h

where

ui = (xi − a)/h

h = class size

Mode Of Grouped Data

Mode = l + {(f1 - f0)/(2f1 - f0 - f2)}h

Where

l = lower limit of model class

h = size of class interval

f1 = frequency of model class

f0 = frequency of the class preceding the model class

f2 = frequency of the class succeeding the model class

Median Of Grouped Data

Median of a data can be calculated by arranging data in ascending order.

If the number of records (n) is odd then the median is n+12th observation.

If n is even the median will be the average of n2 th and n+12th observation

Median for grouped data is given by the following formula

Median = l+{(n/2-cf)/f}h

Where

l = lower limit of model class

n = the number of observations

h = size of class

cf = cumulative frequency of class preceding the median class

f = frequency of median class

The relation among the mode, median and mean is given by the following formula:

3 Median = Mode + 2 Mean

Probability of an event(E)

P(E) = Number of trials in which the event happened / the Total number of trials

P(E) = 1-P(E)

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Mensuration: 10 Marks

This unit includes two chapters – Area Related to Circle and Surface Area and Volume. The first discusses the perimeter and area of the circle as well as the area of sectors and segments of a circle. The other chapter discusses the surface area of a combination of solids, the volume of a combination of solids, and the conversion of volume from one solid to another solid. Here are some important formulae from these chapters.

  • Area of a rectangle = Length x Breadth

  • Area of a square = Side x Side

  • Area of a triangle = (Base x Height) / 2

  • Area of a parallelogram = Base x Height

  • Area of a trapezium = (Sum of parallel sides x Height) / 2

  • Area of a circle = π x (Radius x Radius)

  • Circumference = 2 π x Radius

  • Area of a sector of angle = (θ/360)πr2

  • Length of the arc of a sector of an angle = (θ/360)2πr

  • Area of the segment = area of the sector - an area of triangle

  • =(θ/360)πr2 - 1/2r2θ

  • Volume of a cube = a3

  • Volume of a cuboid = lbh

  • Volume of a sphere = (4/3)πr3

  • Volume of a cone = (1/3)πr2h

  • Surface area of cone = πrl

  • Formula involving frustum of cone

  1. Volume of a frustum of cone = (1/3)π(r12 + r22 + r1r2)h

  2. Curved surface area of a frustum of a cone = πl(r1 + r2) where l = √{h2 + (r1 - r2)2}

  3. Total surface area of a frustum of a cone = πl(r1 + r2) + π(r12 + r22)

Where h = height, l = slant height of frustum, r1 and r2 are the radius of the base.

  • The volume of a right circular cylinder = πr2h

Coordinate Geometry: 6 Marks

The distance formula to find the distance between two points (x1,y1) and (x2,y2) is

D = √{(x2 - x1)2 + (y2 - y1)2}

section formula: ((m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2))

area of triangle = 1/2 [ x1 (y2 - y3) + x2 (y3 - y1) + x3(y1 - y2) ]

Number Systems: 6 Marks

This chapter discusses Euclid’s division lemma, the fundamental theorem of arithmetic, revisiting irrational numbers, revisiting rational numbers and their decimal expansions.

Here are some important formulae from this chapter.

LCM(a,b).HCF(a,b) = ab

Rational number can be written as p/(2n5m)

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An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story. 

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available
Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

5 Jobs Available
Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

3 Jobs Available
Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.  

2 Jobs Available
Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications. 

2 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
QA Manager
4 Jobs Available
Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
Production Manager
3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
QA Lead

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

2 Jobs Available
Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software. 

2 Jobs Available
Process Development Engineer

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

2 Jobs Available
QA Manager
4 Jobs Available
AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

4 Jobs Available
Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

4 Jobs Available
Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

3 Jobs Available
ITSM Manager
3 Jobs Available
Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process. 

2 Jobs Available
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