NCERT Solutions for Class 10 Maths 2023-24 Download
NCERT Solutions for Class 10 Maths are prepared by our expert team to promote students in achieving excellent results in their 10th board examinations. We have included solutions for all the questions and exercises in the CBSE NCERT Books, which makes this article a comprehensive resource.
NCERT Solutions for Class 10 Maths: Are you struggling with understanding the concepts taught in your Class 10 syllabus? Are lengthy calculations and intricate diagrams making you nervous? Then worry not, as NCERT Solutions for Class 10 Maths have arrived to ease your journey and make learning Mathematics enjoyable.
These workbooks provide a depth of knowledge on topics ranging from basic algebraic equations to complex theorems, enabling learners to master foundational maths skills, which serve as a strong base for pursuing higher education. With step-by-step solutions presented through videos and illustrations, it is easier to understand tough concepts and get the best out of your studies!
Our Class 10 Maths NCERT Solutions are presented with a detailed step-by-step explanation, ensuring students can grasp the concepts easily. With our solutions, students can build a strong understanding of the subject, preparing them for their board exams. Explore the complete range of NCERT Solutions for Class 10 Mathematics.
NCERT Solutions for Class 10 Maths Overview:
NCERT Solutions for Class 10 Maths is a valuable resource to assist students in their mathematics studies. These solutions are created to align with the NCERT (National Council of Educational Research and Training) syllabus, which is commonly followed in CBSE (Central Board of Secondary Education) and other state board schools in India.
These solutions cover all chapters and exercises presented in the NCERT Class 10 Maths textbook. It eliminates confusion and ensures that students can self-study effectively.
NCERT Solutions are beneficial for exam preparation, helping students practice and understand the types of questions that can appear in their board exams. NCERT Solutions for Class 10 Maths are often free online, making them a cost-effective study tool.
Overall, NCERT Solutions for Class 10 Maths is a valuable asset for students to enhance their mathematical knowledge, prepare for exams, and improve their problem-solving skills. They serve as a reliable and comprehensive guide for the NCERT syllabus.
Check Out All Sample Papers: CBSE 15 Sample Question Papers Class 10 2024 Exam
Also Read: NCERT Solutions for Class 10 Science
NCERT Solutions for Class 10 Maths Overview
| Chapter Number | Chapter Name |
| 1 | Real Numbers |
| 2 | Polynomials |
| 3 | Pair of Linear Equations in Two Variables |
| 4 | Quadratic Equations |
| 5 | Arithmetic Progressions |
| 6 | Triangles |
| 7 | Coordinate Geometry |
| 8 | Introduction to Trigonometry |
| 9 | Some Applications of Trigonometry |
| 10 | Circles |
| 11 | Constructions |
| 12 | Areas Related to Circles |
| 13 | Surface Areas and Volumes |
| 14 | Statistics |
| 15 | Probability |
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers
NCERT Solutions for Class 10 Maths Chapter 1 are detailed answers and explanations to the questions and exercises in Chapter 1 of the Class 10 Mathematics NCERT textbook. These solutions help students understand the concepts and solve mathematical problems effectively.
Chapter 1 of Class 10 delves into real and irrational numbers. It commences with Euclid's division lemma, which establishes that "given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b".
The Euclidean division algorithm is grounded in this lemma, offering a method to compute the Highest Common Factor (HCF) of two positive integers. Subsequently, the chapter introduces the Fundamental Theorem of Arithmetic, which is pivotal in finding both the Least Common Multiple (LCM) and HCF of two positive integers.
The chapter further elaborates on concepts like irrational numbers, rational numbers, and the decimal expansion of rational numbers, employing the theorem as a fundamental tool.
Key Topics Covered in Chapter 1:
-
Fundamental Theorem of Arithmetic: The chapter reviews the fundamental theorem, both in its statement and through practical illustrations.
-
Proofs of Irrationality: The chapter also delves into the proofs of the irrationality of specific numbers such as √2, √3, and √5.
| NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers |
| Exercise 1.1 |
| Exercise 1.2 |
| Exercise 1.3 |
| Exercise 1.4 |
Chapter 1 provides students with a foundational understanding of real and irrational numbers, as well as the methods to explore their properties and relationships. It equips them with the tools to solve mathematical problems related to these topics, making it a crucial part of Class 10 Mathematics. You can also access other helpful resources Class 10 Maths Chapter 1 Real Numbers:
-
Important questions for class 10 Maths Chapter 1 Real Numbers
-
CBSE 15 Sample Question Paper - Class 10 Science, Mathematics, English, Hindi, Social Science by PW offers a collection of 15 sample question papers for Class 10 students, covering a wide range of subjects, including Science, Mathematics, English, Hindi, and Social Science. These sample papers are valuable resources for students as they can help them practice and understand the exam pattern.
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials
Chapter 2 of Class 10 Mathematics, titled "Polynomials," embarks on a journey to explore the world of polynomial expressions. It introduces students to the degree of polynomials, categorizing polynomials into linear, quadratic, and cubic types. This chapter comprises four exercises, including an optional one.
Exercise 2.1 delves into finding the number of zeros on a graph, urging students to grasp the geometric significance of the zeros of a polynomial. Exercise 2.2 is centered on establishing the link between the zeros and coefficients of a polynomial, challenging students to determine the zeros of a quadratic polynomial. Some questions in this exercise require students to identify a quadratic polynomial.
In Exercise 2.3, the division algorithm takes the spotlight, offering questions that assess students' understanding of this algorithm. Finally, Exercise 2.4 covers questions that encompass all the concepts presented in Chapter 2, creating a comprehensive test of students' knowledge.
Key Topics Covered in Chapter 2:
-
Zeros of a Polynomial: The chapter begins by discussing the concept of zeros of a polynomial.
-
Relationship Between Zeros and Coefficients of Quadratic Polynomials: It delves into the correlation between the zeros and coefficients of quadratic polynomials.
| NCERT Solutions for Class 10 Maths Chapter 2 Polynomials |
| Exercise 2.1 |
| Exercise 2.2 |
| Exercise 2.3 |
| Exercise 2.4 |
Chapter 2 provides students with a fundamental understanding of polynomials, their properties, and their practical applications, serving as an essential part of Class 10 Mathematics. Check below links for additional resources for Class 10 exams:
-
Important questions for class 10 Maths Chapter 2 Polynomials
-
CBSE Class 10th Past Year Paper by PW books are focused on providing past year question papers for Class 10th students. Past year question papers are an excellent resource for students to familiarize themselves with the types of questions asked in previous examinations
NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables
Chapter 3 of Class 10 Mathematics delves into the intricate concept of a "Pair of Linear Equations in Two Variables." This chapter offers comprehensive insights into solving equations with two variables. It encompasses a total of 7 exercises, each focusing on different methods to solve such pairs of linear equations.
In Exercise 3.1, students are introduced to the technique of representing situations both algebraically and graphically. Exercise 3.2 elucidates the graphical method for solving a pair of linear equations, offering students a valuable tool for understanding solutions graphically.
Subsequent exercises, including 3.3, 3.4, 3.5, and 3.6, unravel additional methods for solving linear equations, such as the algebraic method, elimination method, cross-multiplication method, and substitution method, respectively.
These exercises provide a comprehensive exploration of the various techniques to tackle linear equations. To further bolster students' grasp of the concepts presented, Exercise 3.7 offers an optional exercise containing diverse question types. It is a comprehensive practice ground to master the art of solving linear equations.
Key Topics Covered in Chapter 3:
-
Pair of Linear Equations in Two Variables and Graphical Method of Their Solution: The chapter begins with an explanation of the fundamental concepts, including graphical representations.
-
Consistency/Inconsistency: It also explores the character and inconsistency of linear equations.
-
Algebraic Conditions for the Number of Solutions: The chapter delves into algebraic conditions related to the number of solutions.
-
Pair of Linear Equations in Two Variables Solution: Students are introduced to solving these equations using techniques like substitution and elimination.
-
Simple Situational Problems: The chapter concludes with applying these techniques in solving simple situational problems.
| NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables |
| Exercise 3.1 |
| Exercise 3.2 |
| Exercise 3.3 |
| Exercise 3.4 |
| Exercise 3.5 |
| Exercise 3.6 |
| Exercise 3.7 |
Chapter 3 offers students a solid foundation in handling pairs of linear equations, whether algebraically or graphically. It equips them with diverse problem-solving tools, making it a pivotal segment of Class 10 Mathematics.
NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations
In this chapter, students will embark on a comprehensive journey through the standard form of writing a Quadratic Equation. The chapter meticulously elucidates the methods for solving quadratic equations using two fundamental approaches: the completing the square method and the factorization method.
By the end of this chapter, students will possess a clear understanding of determining the nature of roots in a quadratic equation, where ax² + bx + c = 0, taking into account the value of the discriminant:
-
When b² – 4ac > 0, the equation yields two distinct real roots.
-
When b² – 4ac = 0, the equation results in two equal roots.
-
When b² – 4ac < 0, the equation exhibits no real roots.
Key Topics Covered in Chapter 4:
-
Standard Form of a Quadratic Equation: The chapter commences by introducing students to the standard form of a quadratic equation, ax² + bx + c = 0, where "a" is not equal to zero.
-
Solutions of Quadratic Equations: Students will be well-versed in solving quadratic equations and obtaining real roots through two primary methods: factorization and the quadratic formula.
-
Relationship Between Discriminant and Nature of Roots: This chapter concludes by exploring the connection between the discriminant (b² - 4ac) and the nature of the roots, shedding light on the outcomes of distinct real roots, equal roots, or no real roots.
| NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations Exercises |
| Exercise 4.1 |
| Exercise 4.2 |
| Exercise 4.3 |
| Exercise 4.4 |
Chapter 4 empowers students with a profound understanding of quadratic equations, equipping them with the tools to solve them efficiently. It is a pivotal component of Class 10 Mathematics, expanding students' mathematical knowledge and skills.
NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions
This chapter introduces students to the concept of Arithmetic Progression (AP). In this chapter, students will learn about the definition of an AP, the derivation of the nth term, how to find the sum of the first 'n' terms of an AP, and how to apply AP to solve real-life problems.
The first exercise of this chapter focuses on teaching students how to represent real-world problems or situations in the form of an AP. It also covers topics such as finding the first term and the common difference of the AP and determining whether a given series is an AP.
In the second exercise, labeled Ex 5.2, students learn how to find the nth term of an AP using the formula: nth term = a + (n - 1)d, where a is the first term, 'n' is the position of the term, and 'd' is a common difference.
The third exercise provides insights into calculating the sum of the first 'n' terms of an AP and includes questions related to this topic. The fourth exercise contains questions that revolve around the concepts covered in the chapter, allowing students to practice and apply their knowledge.
To facilitate problem-solving in this chapter, students should remember the following essential formulas:
Finding the nth term of a progression:
Nth term = a + (n - 1)d
Where 'a' is the first term, 'n' represents the position of the term, and 'd' is the common difference.
The sum of the first 'n' terms in a sequence:
Sn = (n/2) [2a + (n - 1)d]
Here, 'n' signifies the number of terms, 'a' is the first term, and 'd' stands for the common difference.
These formulas are valuable tools for solving problems and exercises in the textbook, making understanding and applying AP more manageable for students.
| NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions |
| Exercise 5.1 |
| Exercise 5.2 |
| Exercise 5.3 |
| Exercise 5.4 |
NCERT Solutions for Class 10 Maths Chapter 6 Triangles
Chapter 6 of Class 10 CBSE Mathematics delves into the realm of triangles, focusing on figures that share the same shape but not necessarily the same size. This chapter, titled "Triangles," begins by introducing the concepts of similarity and congruence in figures.
It further elucidates the conditions for the similarity of two triangles and the theorems associated with triangle similarity. The chapter explores areas of similar triangles with the help of a theorem. Towards the end, it covers the Pythagoras Theorem and the Converse of the Pythagoras Theorem.
| NCERT Solutions for Class 10 Maths Chapter 6 Triangles |
| Exercise 6.1 |
| Exercise 6.2 |
| Exercise 6.3 |
| Exercise 6.4 |
| Exercise 6.5 |
NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry
In Chapter 7, students delve into Coordinate Geometry, where they will acquire skills to calculate the distance between two points with given coordinates and determine the triangle area formed by three specified points.
Additionally, students will master the art of finding the coordinates of a point that bisects a line joining two given points at a fixed ratio. Towards the culmination of the chapter, students will be introduced to vital concepts, including the distance formula, section formula, and the computation of the area of a triangle in the realm of Coordinate Geometry.
| NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry |
| Exercise 7.1 |
| Exercise 7.2 |
| Exercise 7.3 |
| Exercise 7.4 |
NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry
Chapter 8 of NCERT Solutions serves as an introduction to the fascinating world of trigonometry for students. Trigonometry is the study of the relationships and ratios that exist within right triangles, primarily focusing on the trigonometric ratios of angles. This chapter elucidates the trigonometric ratios for the angles of 0° and 90°.
As students progress, they will learn to compute these ratios for various given angles, along with gaining insight into a few fundamental trigonometric identities.
Furthermore, this chapter equips students with foundational knowledge in trigonometry, which proves to be invaluable in subsequent classes, such as Class 11 and 12. Students are introduced to essential trigonometric formulas to ensure they possess a solid understanding of the basics, enabling them to navigate more advanced topics in the future.
The chapter not only presents the formulas but also elucidates the connections between these ratios and the corresponding angles.
Crucial trigonometric ratios for complementary angles include:
-
sin (90° – A) = cos A,
-
cos (90° – A) = sin A,
-
tan (90° – A) = cot A,
-
cot (90° – A) = tan A,
-
sec (90° – A) = cosec A,
-
cosec (90° – A) = sec A
Furthermore, the chapter establishes fundamental trigonometric identities, such as:
-
sin^2 A + cos^2 A = 1,
-
sec^2 A – tan^2 A = 1 for 0° ≤ A < 90°,
-
cosec^2 A = 1 + cot^2 A for 0° < A ≤ 90°
| NCERT Solutions for Class 10 Maths Chapter 8 Trigonometry |
| Exercise 8.1 |
| Exercise 8.2 |
| Exercise 8.3 |
| Exercise 8.4 |
This foundational understanding of trigonometry prepares students for more complex concepts and applications in higher classes. Check below links for addition sources:
-
Important questions for class 10 Maths Chapter 8 Introduction to Trigonometry
-
CBSE 15 Sample Question Paper - Class 10th by PW books sample papers are likely to encompass a broad range of subjects, which is typical for Class 10 board examinations.
NCERT Solutions for Class 10 Maths Chapter 9 Applications of Trigonometry
Chapter 9 expands on the principles covered in Chapter 8 and delves into the real-life utility of trigonometry. Within this chapter, we witness trigonometry in action, precisely computing the height and distance of objects in practical scenarios.
It underscores the indispensable role of trigonometry in navigation, ensuring accurate direction-finding and positioning. Furthermore, the chapter showcases trigonometry's integral part in construction endeavors, facilitating the determination of angles and distances for architectural projects.
| NCERT Solutions for Class 10 Maths Chapter 9 Applications of Trigonometry |
| Exercise 9.1 |
Moreover, trigonometry finds extensive application in geography and cartography, enabling precise location identification based on latitude and longitude coordinates.
This chapter illuminates the pragmatic use of trigonometry for resolving tangible challenges and augmenting our comprehension of the surrounding world. The competencies acquired from this chapter are adaptable across numerous domains, underscoring trigonometry's versatility and its significance in education.
NCERT Solutions for Class 10 Maths Chapter 10 Circles
Chapter 10 of the Class 10 Maths NCERT Solutions builds upon the foundation of Circle studies from prior classes. This chapter introduces students to tangents and the count of tangents originating from a point on a circle.
Key Theorems Covered:
-
Theorem 1: The tangent at any point on a circle is perpendicular to the radius passing through the point of contact.
-
Theorem 2: The lengths of tangents drawn from an external point to a circle are identical.
| NCERT Solutions for Class 10 Maths Chapter 10 Circles |
| Exercise 10.1 |
| Exercise 10.2 |
NCERT Solutions for Class 10 Maths Chapter 11 Constructions
Chapter 11 offers an engaging and hands-on exploration of constructing various geometric figures. In this module, students will acquire the skills needed to bisect line segments and create tangents to circles. The chapter delivers thorough and lucid explanations of the construction techniques, facilitating students' comprehension of these concepts.
Each construction method is presented in a step-by-step format, accompanied by numerous illustrative examples before each tangent construction exercise. These examples are thoughtfully designed to assist students in developing a deeper understanding of the construction principles and the associated theorems.
| NCERT Solutions for Class 10 Maths Chapter 11 Constructions |
| Exercise 11.1 |
| Exercise 11.2 |
| Exercise 11.3 |
By diligently following these examples and completing the exercises, students can refine their proficiency in geometric constructions and grow more self-assured in resolving construction-related problems. This chapter not only imparts valuable hands-on experience but also sets a solid groundwork for tackling more intricate constructions in advanced grade levels.
NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles
In Chapter 12, we delve into the fascinating realm of "Areas Related to Circles." This module expands upon the knowledge acquired in prior chapters and explores the intricate relationship between circles and their associated regions.
This chapter covers various essential topics, including the calculation of the areas of circles, sectors, and segments. Students will learn to compute the area of the sector and segment of a circle, further enhancing their geometric skills.
| NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles |
| Exercise 12.1 |
| Exercise 12.2 |
| Exercise 12.3 |
"Areas Related to Circles" is a pivotal chapter that strengthens students' understanding of geometry and equips them with valuable tools for solving mathematical problems related to circular regions. This knowledge will prove invaluable in various mathematical and real-life situations.
NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes
In this chapter, we delve into the fundamental concepts of circle perimeter and area. Our exploration extends to the calculation of sector areas and circle segments.
Additionally, the chapter provides insights into determining the total area of shapes that encompass either a complete circle or a portion of it. Here are some essential formulas that students will encounter and practice in this chapter:
-
Circumference = 2πr (where 'r' represents the circle's radius)
-
Area of the circle = πr^2 (where 'r' signifies the circle's radius)
-
Area of a sector with an angle θ = (θ/360) × πr^2 (where 'θ' denotes the sector's angle, and 'r' stands for the circle's radius)
-
Length of an arc within a sector with an angle θ = (θ/360) × 2πr (where 'θ' represents the sector's angle, and 'r' signifies the circle's radius)
| NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes |
| Exercise 13.1 |
| Exercise 13.2 |
| Exercise 13.3 |
| Exercise 13.4 |
| Exercise 13.5 |
Mastering these formulas empowers students to compute circle perimeters and areas confidently. Furthermore, they learn to determine regions in diverse scenarios involving circles and circular segments. This chapter equips students with valuable skills to tackle circle-related challenges across various practical applications.
NCERT Solutions for Class 10 Maths Chapter 14 Statistics
In this chapter, students embark on a captivating journey into numerical data representation, whether in grouped or ungrouped formats. They acquire valuable skills for computing the mean, median, and mode of a given dataset, which are essential in revealing the data's central tendencies.
The subsequent section delves into the concept of cumulative frequency distribution, enabling students to construct cumulative frequency curves. These graphical representations depict the cumulative frequencies of the data, offering a more comprehensive insight into the dataset's distribution.
| NCERT Solutions for Class 10 Maths Chapter 14 Statistics |
| Exercise 14.1 |
| Exercise 14.2 |
| Exercise 14.3 |
| Exercise 14.4 |
By mastering these techniques, students gain the ability to make well-informed decisions rooted in data and extract valuable insights from it.
NCERT Solutions for Class 10 Maths Chapter 15 Probability
Students gain insights into how probability is applied in real-life situations, making decisions based on uncertain outcomes. They learn about the concepts of experimental probability and theoretical probability, along with practical examples.
The chapter equips students with the skills to calculate probabilities, analyze uncertainty, and make informed decisions, making it a crucial aspect of mathematics with applications in various fields. By mastering these probability concepts, students can approach problems and situations involving uncertainty with confidence.
| NCERT Solutions for Class 10 Maths Chapter 15 Probability |
| Exercise 15.1 |
| Exercise 15.2 |
NCERT Solutions for Class 10 Maths PDF Download
The NCERT Solutions for Class 10 Mathematics provides comprehensive chapter-wise answers to the questions in the Class 10 Maths NCERT Book. These solutions are presented clearly and concisely, aligning with the objectives of the textbooks. Students can use these NCERT Solutions for Class 10 as supplementary references and study materials. Engaging with the solutions to the exercises in the NCERT textbook will benefit students in their exam preparation.
On this platform, you can access comprehensive NCERT solutions for Class 10 Mathematics, covering all the chapters and exercises in the Class 10 Math syllabus. Experienced educators meticulously craft these solutions with a strong background in teaching mathematics to Class 10 students.
They are presented in a step-by-step format and are easily downloadable in PDF, ensuring a seamless learning experience. The PDFs containing the Class 10 Mathematics NCERT solutions encompass exercises from Chapters 1 to 15, mirroring the structure of the NCERT textbooks. These solutions adhere to the CBSE exam guidelines, enhancing students' understanding of the subject and facilitating the achievement of high scores.
By utilizing these NCERT solutions in PDF format, students not only prepare effectively for their final exams but also gain a deeper comprehension of the subject.
Check out: CBSE Class 10th Question Banks
NCERT Solutions for Class 10 Maths FAQs
Q1: What are NCERT Solutions for Class 10 Maths, and why are they important?
NCERT Solutions for Class 10 Maths are comprehensive, step-by-step explanations and solutions to all the exercises and problems in the NCERT Class 10 Mathematics textbook. They are essential for students as they provide clear and accurate guidance to solve mathematical problems, aiding in conceptual clarity and exam preparation.
Q2: Can I solely rely on NCERT Solutions for Class 10 Maths for board exam preparation?
While NCERT Solutions is an excellent resource, it's advisable to supplement your preparation with additional study materials, reference books, and practice papers. A well-rounded approach to study will ensure a more comprehensive understanding and better performance in board exams.
Q3: Are NCERT Solutions for Class 10 Maths beneficial for competitive exams?
NCERT Solutions provides a strong foundation in mathematics, which can be helpful for various competitive exams. However, for specialized exams, students may need to study additional topics and practice with specific competitive exam materials.
Q4: How should I use NCERT Solutions for Class 10 Maths effectively?
To use NCERT Solutions effectively, start by thoroughly understanding the concepts in each chapter. Then, practice solving the exercises and problems using the step-by-step solutions provided. Additionally, attempt previous years' question papers to enhance your problem-solving skills.
Q5: Are NCERT Solutions for Class 10 Maths available for free?
Yes, NCERT Solutions for Class 10 Maths are free on various educational websites, making them accessible to all students. They can be downloaded in PDF format for self-study and exam preparation.
