# Ncert Solutions for Class 10 Maths

**Ncert Solutions For Class 10 Maths All Chapters **

**Ncert Solutions for Class 10 Maths Updated Pdf – Free Download **

**NCERT Solutions for Class 10 Maths** are provided here for all exercises from Chapters 1 to 15. Our expert faculty has curated these **NCERT Solutions** to assist students with their exam preparations. Students looking for **NCERT Class 10 Math Solutions** can download all chapter-wise PDFs to find a better approach to solving the problems.

The answers to the questions in the **NCERT books **are without a doubt the best study material a student can obtain. These **CBSE NCERT Solutions for Class 10 Maths** 2022-23 will also assist students in developing a more in-depth understanding of concepts covered in the textbook. Practicing textbook questions will allow students to assess their level of preparation and concept knowledge. The answers to these questions provided in the books can assist students in quickly clearing their doubts.

**NCERT Solutions for Class 10 Maths** are provided here, and they cover all of the chapters and exercises in Class 10 Math. These solutions have been prepared by expert teachers with years of experience teaching Math to Class 10 students. Class 10 Maths NCERT Solutions are available in PDF format and are explained step by step. For an uninterrupted learning experience, students can now download the class 10 Maths NCERT solutions PDFs.

The **Class 10 Maths NCERT Solution PDFs** cover all of the exercises in the **NCERT books**. These Class 10 Maths solutions are written in accordance with the **CBSE **exam guidelines to help students understand the subject and score well. The **NCERT solutions for class 10 Mathematics** PDFs will not only help students prepare for final exams but will also assist them in better understanding the concept. Science and Math are two important subjects in class 10, and you can find reliable and accurate **NCERT Solutions for Class 10 Science** on Utopper.

**NCERT Solutions for Class 10 Maths | Chapter-wise List**

The **NCERT Solutions for Class 10** are provided below, chapter by chapter. The Maths experts at Utopper provide detailed solutions to these problems. Go through these chapter-by-chapter solutions to become thoroughly acquainted with the concepts.

**NCERT Solutions Class 10 Maths Chapters**

- Chapter 1 Real Numbers
- Chapter 2 Polynomials
- Chapter 3 Pair of Linear Equations in Two Variables
- Chapter 4 Quadratic Equations
- Chapter 5 Arithmetic Progressions
- Chapter 6 Triangles
- Chapter 7 Coordinate Geometry
- Chapter 8 Introduction to Trigonometry
- Chapter 9 Some Applications of Trigonometry
- Chapter 10 Circles
- Chapter 11 Constructions
- Chapter 12 Areas Related to Circles
- Chapter 13 Surface Areas and Volumes
- Chapter 14 Statistics
- Chapter 15 Probability

The Solutions to **NCERT Textbooks** also aid students in gaining a comprehensive understanding of the topic, which is crucial beyond the Class 10 examination. Greater comprehension provides a solid foundation for their future studies. Frequently, questions from textbooks are included in competitive examinations. Both in-text and exercise questions have been covered in depth.

Class 10 students can now prepare for their board exams for all subjects with the help of Utopper. **Click on these links to download the NCERT Solutions for all subjects of Class 10.**

**NCERT Solutions for Class 10 Maths****NCERT Solutions for Class 10 Science**- NCERT Solutions for Class 10 Hindi
- NCERT Solutions for Class 10 Social Science
- Ncert Solutions for Class 10 EnglishÂ

**NCERT Solutions for Class 10 Maths PDF Updated for 2022-23 Session**

**Class 10 NCERT Maths solutions** Pdfâ€™s are a reliable and accurate source of information. All questions are exhaustively explained in the study materials. These responses are written in a clear and sequential manner to maximize retention. As they examine these solutions, students will gain a deeper understanding of the concept. Students must engage in regular practice if they wish to perform well in mathematics. The **NCERT solutions** for Class 10 serve as an ideal guide. You can consult the guide to refresh your memory or for assistance if you become lost.

Utopper’s solutions are formulated and created by professionals with years of experience in the field of education. The answers are provided in simple language so that students can comprehend the concept more fully. Utopper aims to provide solutions to each question that students of varying abilities can easily comprehend. The entire curriculum and solutions have been revised in accordance with changes to the **NCERT textbook** or the** CBSE curriculum** guidelines.

Utopper is the premier online resource for answers to all the solutions for every subject and grade. Utopper’s mission is to aid students in comprehending a subject and developing a love for learning. Consequently, the comfort and convenience of the students should take precedence when developing solutions to the questions. The solutions are useful if students become stuck or need to comprehend where they went wrong. In addition to analyzing their answers and performance, students can also gain additional practice by referring to the additional questions in the solutions PDF.

**NCERT Solutions Class 10 Maths Chapter Details and Exercises**

**NCERT Solutions for Class 10 Maths – Chapter 1 Real Numbers**

In this chapter, there are 4 exercises. In the first chapter of **Class 10 Maths**, students will learn about **real numbers** and numbers that don’t make sense. Euclid’s Division Lemma, HCF, LCM, Divisibility of integers, rational number, irrational number, and decimal expansion of rational numbers are all explained in Chapter 1 of **CBSE Class 10 Maths.**

The Division of Euclid Lemma says, “Given two positive integers a and b, there exists a unique pair of positive integers q and r such that a = bq + r, where 0 â‰¤r <b .” This gives the starting point for the algorithm used to find the HCF of two positive integers by dividing them. In this chapter, you will also learn about the Fundamental Theorem of Arithmetic. It shows how to figure out the LCM and HCF of two positive numbers.

**Class 10 Maths NCERT Solutions PDF Chapter 1 Exercises**

- NCERT Solutions Real Numbers Class 10 Exercise 1.1 â€“ 5 Questions (4 Long Answers, 1 Short Answer)
- NCERT Solutions Real Numbers Class 10 Exercise 1.2 â€“ 7 Questions (4 Long Answers, 3 Short Answers)
- NCERT Solutions Real Numbers Class 10 Exercise 1.3 â€“ 3 Questions (3 Short Answers)
- NCERT Solutions Real Numbers Class 10 Exercise 1.4 â€“ 3 Questions (3 Short Answers)

**Topics Covered in Class 10 Maths Chapter 1 Real Numbers :**

- Fundamental Theorem of Arithmetic â€“ statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of the irrationality of âˆš2, âˆš3, âˆš5

In addition, students can download additional study materials for **NCERT Class 10 Chapter 1 Real Numbers** at Utopper.

- Real Numbers Class 10 Important Questions
- Important Formulas for Real Numbers Class 10
- Ncert Revision Notes Class 10 Real Numbers
- NCERT Exemplar Solutions Class 10 Real Numbers
- RD Sharma Solutions for Real Numbers Class 10
- RS Aggarwal Solutions for Real Numbers Class 10
- Ncert Book for Class 10 Real Numbers
- Ncert Exemplar Book For Class 10 Real Numbers

**NCERT Solution for Class 10 Mathematics Chapter 2 – Polynomials**

In **Polynomials**, the first part of the chapter explains what a polynomial’s degree, linear polynomial, quadratic polynomial, and cubic polynomial are. There are a total of 4 exercises in this chapter, one of which is optional. In Exercise 2.1, there are questions about how to use a graph to find the number of zeroes. It’s important to know what the zeroes of a polynomial mean in terms of geometry.

Exercise 2.2 is about the relationship between the zeros and coefficients of a polynomial. Students have to find the zeros of a quadratic polynomial, and in some of the questions, they have to find the quadratic polynomial. In Exercise 2.3, the idea of the division algorithm is explained, and the questions that go with it are given. The optional exercise, 2.4, is made up of questions about all of Chapter 2’s ideas.

**Class 10 Maths NCERT Solutions PDF Chapter 2 Exercises**

- NCERT Solutions Polynomials Class 10 Exercise 2.1 â€“ 1 Question (1 Short Answer)
- NCERT Solutions Polynomials Class 10 Exercise 2.2 â€“ 2 Questions (2 Short Answers)
- NCERT Solutions Polynomials Class 10 Exercise 2.3 â€“ 5 Questions (2 Short Answers, 3 Long Answers)
- NCERT Solutions Polynomials Class 10 Exercise 2.4 â€“ 5 Questions (2 Short Answers, 3 Long Answers)

**Topics Covered in Class 10 Maths Chapter 2 Polynomials :**

- Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials.

In addition, students can download additional study materials for **Chapter 2 of CBSE Class 10 Maths Solutions** provided by Utopper.

- Polynomials Class 10 Important Questions
- Important Formulas for Polynomials Class 10
- NCERT Revision Notes Class 10 Polynomials
- NCERT Exemplar Solutions Class 10 Polynomials
- RD Sharma Solutions for Polynomials Class 10
- RS Aggarwal Solutions Polynomials Class 10
- Ncert Book For Class 10 Polynomials
- Ncert Exemplar Book For Class 10 Polynomials

**NCERT Solutions of Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables**

This chapter explains the idea of **Pair of Linear Equations in Two Variables**. There are a total of 7 exercises in this chapter. In these exercises, different ways to solve a pair of linear equations are shown. Exercise 3.1 shows how to show a situation using both algebra and pictures.

Exercise 3.2 shows you how to use the Graphical Method to solve a pair of linear equations. The Algebraic Method, Elimination Method, Cross-Multiplication Method, and Substitution Method are explained in Exercises 3.3, 3.4, 3.5, and 3.6, respectively. Exercise 3.7 is optional and has many different kinds of questions. Students must do these exercises over and over again to learn how to solve linear equations.

**Class 10 Maths NCERT Solutions PDF Chapter 3 Exercises**

- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.1 â€“ 3 Questions (2 Short Answers, 1 Long Answer)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.2 â€“ 7 Questions (5 Short Answers, 2 Long Answers)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.3 â€“ 3 Questions (2 Short Answers, 1 Long Answer)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.4 â€“ 2 Questions (2 Long Answers)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.5 â€“ 4 Questions (4 Short Answers)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.6 â€“ 2 Questions (2 Long Answers)
- NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.7 â€“ 8 Questions (1 Short Answer, 7 Long Answers)

**Topics Covered in Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables:**

- Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency.
- Algebraic conditions for a number of solutions. Solution of a pair of linear equations in two variables algebraically â€“ by substitution, by elimination. Simple situational problems.

Students can also download extra study materials from Utopper for **NCERT Class 10 Chapter 3 Pair of Linear Equations in Two Variables.**

- Pair of Linear Equations in Two Variables Class 10 Important Questions
- Important Formulas for Pair of Linear Equations in Two Variables Class 10
- NCERT Revision Notes Class 10 Pair of Linear Equations in Two Variables
- NCERT Exemplar Solutions Class 10 Pair of Linear Equations in Two Variables
- RD Sharma Solutions for Pair of Linear Equations in Two Variables Class 10
- RS Aggarwal Solutions Pair of Linear Equations in Two Variables Class 10
- Ncert Books For Class 10 Pair of Linear Equations in Two Variables
- Ncert Exemplar Book For Class 10 Pair of Linear Equations in Two Variables

**NCERT Solutions of Class 10 Maths Chapter 4 Quadratic Equations**

**Quadratic equations** are covered in Chapter 4 of the **Class 10 Maths NCERT book PDF solution**. In this chapter, you’ll learn what a quadratic equation is and how to solve it. It also shows how to use the square method and the factorization method to solve quadratic equations. The chapter talks about how discriminants relate to each other and what their roots are. In this chapter, real-life problems have been used to show how to solve them.

- Two distinct real roots, if bÂ² â€“ 4ac > 0
- Two equal roots, if bÂ² â€“ 4ac = 0
- No real roots, if bÂ² â€“ 4ac < 0

**Class 10 Maths NCERT Solutions Chapter 4 Exercises**

- NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.1 â€“ 2 Questions (1 Short Answer, 1 Long Answer)
- NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.2 â€“ 6 Questions (6 Short Answers)
- NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.3 â€“ 11 Questions (8 Short Answers, 3 Long Answers)
- NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.4 â€“ 5 Questions (2 Short Answers, 3 Long Answers)

**Topics Covered in Class 10 Maths Chapter 4 Quadratic Equations :**

- The standard form of a quadratic equation ax2 + bx + c = 0, (a â‰ 0). Solutions of quadratic equations (only real roots) by factorization and by using the quadratic formula. Relationship between discriminant and nature of roots.

Students can also download extra study materials from Utopper for Chapter 4 of **NCERT Class 10 Chapter 4 Quadratic Equations** â€“ Utopper.

- Quadratic Equations Class 10 Important Questions
- Important Formulas for Quadratic Equations Class 10
- NCERT Revision Notes Class 10 Quadratic Equations
- NCERT Exemplar Solutions Class 10 Quadratic Equations
- RD Sharma Solutions for Quadratic Equations Class 10
- RS Aggarwal Solutions Quadratic Equations Class 10
- Ncert Book For Class 10 Quadratic Equations
- Ncert Exemplar Book For Class 10 Quadratic Equations

**NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions**

This chapter teaches the students about a new subject called Arithmetic Progression, or AP for short. In this chapter, students will learn what AP is, how to find the nth term, find the sum of the first n terms of AP, and use AP to solve real-world problems.

**Class 10 Maths NCERT Solutions Chapter 5 Exercises**

- NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.1 â€“ 4 questions (1 Short Answer, 3 Long Answers)
- NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.2 â€“ 20 questions (10 Short Answers, 10 Long Answers)
- NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.3 â€“ 20 Questions (7 Short Answers, 13 Long Answers)
- NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.4 â€“ 5 Questions (5 Long Answers)

**Topics Covered in Class 10 Maths Chapter 5 Arithmetic Progressions:**

If a1, a2, a3, a4, a5, a6,… are the terms of AP and d is the common difference between each term, then the sequence can be written as a, a+d, a+2d, a+3d, a+4d, a+5d,…., nth term… where an is the first term. Now, the nth term in the arithmetic progression is given as;

- nth term = a + (n-1) d
- Sum of the first n terms in Arithmetic Progression: Sn = n/2 [2a + (n-1) d]

Students can also download extra study materials from Utopper for Chapter 5 of **NCERT Class 10 Chapter 5 Arithmetic Progressions** â€“ Utopper.

- Arithmetic Progressions Class 10 Important Questions
- Important Formulas for Arithmetic Progressions Class 10
- NCERT Revision Notes Class 10 Arithmetic Progressions
- NCERT Exemplar Solutions Class 10 Arithmetic Progressions
- RD Sharma Solutions for Arithmetic Progressions Class 10
- RS Aggarwal Solutions Arithmetic Progressions Class 10
- Ncert Textbook For Class 10 Arithmetic Progressions
- Ncert Exemplar Book For Class 10 Arithmetic Progressions

**NCERT Solutions for Class 10 Maths Chapter 6 Triangles**

In Chapter 6 of Class 10 CBSE Mathematics, students will examine figures with identical shapes but not necessarily identical sizes. The Triangles chapter begins with the idea of a similar and congruent figure. It elaborates on the condition for the similarity of two triangles and theorems associated with triangle similarity. After that, a theorem was used to explain the areas of triangles with the same angles. At the conclusion of this chapter, the Pythagorean Theorem and its converse are described.

**Class 10 Maths NCERT Solutions Chapter 6 Exercises**

- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.1 â€“ 3 Questions (3 Short Answers)
- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.2 â€“ 10 Questions (9 Short Answers, 1 Long Answer)
- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.3 â€“ 16 Questions (12 Short Answers, 4 Long Answers)
- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.4 â€“ 9 Questions (7 Short Answers, 2 Long Answers)
- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.5 â€“ 17 Questions (15 Short Answers, 2 Long Answers)
- NCERT Solutions Class 10 Maths Triangles Class 10 Exercise 6.6 â€“ 10 Questions (5 Short Answers, 5 Long Answers)

**Topics Covered in Class 10 Maths Chapter 6 Triangles :**

The following are essential theorems that students must know thoroughly while preparing for the exam:

**Theorem 6.1: **If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

**Theorem 6.2:** If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

**Theorem 6.3:** If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion), and hence the two triangles are similar.

**Theorem 6.4:** If in two triangles, the sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal, and hence the two triangles are similar.

**Theorem 6.5: **If one angle of a triangle is equal to one angle of the other triangle and the sides, including these angles, are proportional, then the two triangles are similar.

**Theorem 6.6: **The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

**Theorem 6.7:** If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

**Theorem 6.8:** In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

**Theorem 6.9: **In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Students can also download extra study materials from Utopper for Chapter 6 of **NCERT Class 10 Chapter 6 Triangles **â€“ Utopper.

- Triangles Class 10 Important Questions
- Important Formulas for Triangles Class 10
- NCERT Revision Notes Class 10 Triangles
- NCERT Exemplar Solutions Class 10 Triangles
- RD Sharma Solutions for Triangles Class 10
- RS Aggarwal Solutions Triangles Class 10
- Ncert Textbook For Class 10 Triangles
- Ncert Exemplar Book For Class 10 Triangles

**NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry**

In this chapter, students will learn how to calculate the distance between two points with given coordinates and the area of a triangle formed by three points. In addition, students will learn how to determine the coordinates of the point that divides a line segment connecting two given points in a given ratio. This chapter on Coordinate Geometry will introduce students to the Distance Formula, Section Formula, and Triangle Area for this purpose.

**Class 10 Maths NCERT Solutions Chapter 7 Exercises**

- NCERT Solutions Class 10 Coordinate Geometry Class 10 Exercise 7.1 â€“ 10 Questions (3 Short Answers, 7 Long Answers)
- NCERT Solutions Class 10 Coordinate Geometry Class 10 Exercise 7.2 â€“ 10 Questions (2 Short Answers, 8 Long Answers)
- NCERT Solutions Class 10 Coordinate Geometry Class 10 Exercise 7.3 â€“ 5 Questions (2 Short Answers, 3 Long Answers)
- NCERT Solutions Class 10 Coordinate Geometry Class 10 Exercise 7.4 â€“ 8 Questions (3 Short Answers, 5 Long Answers)

**Topics Covered in Class 10 Maths Chapter 7 Coordinate Geometry:**

**Distance Formula =**

PQ =\(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

**Section Formula =**

m1 : m2 = ( \(\frac{m_1x_2+m_2x_1}{m_1+m_2}\) , \(\frac{m_1y_2+m_2y_1}{m_1+m_2}\) )

**Area of Triangle =**

** **\(\frac{1}{2}\) [ \( x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right) \) ]

Students can also download extra study materials from Utopper for Chapter 7 of **NCERT Class 10 Chapter 7 Coordinate Geometry **â€“ Utopper.

- Coordinate Geometry Class 10 Important Questions
- Important Formulas for Coordinate Geometry Class 10
- NCERT Revision Notes Class 10 Coordinate Geometry
- NCERT Exemplar Solutions Class 10 Coordinate Geometry
- RD Sharma Solutions for Coordinate Geometry Class 10
- RS Aggarwal Solutions Coordinate Geometry Class 10
- Ncert Textbook For Class 10 Coordinate Geometry
- Ncert Exemplar Book For Class 10 Coordinate Geometry

**NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry**

In Chapter 8 of NCERT Solutions, students are taught about trigonometry. Trigonometry is the study of the relationship between right triangles and their acute angles. These relationships are called trigonometric ratios of the angles. Also, the ratios for angles 0Â° and 90Â° in trigonometry. Students will also learn how to figure out trigonometric ratios and identities for given angles.

This chapter gives students in Class 11 and 12 a basic introduction to trigonometry that they can use. They learn the basic formulas of trigonometry so they can understand the basics and use them in higher classes and in the future. In addition to the formulas, the chapter explains the relationship between ratios and angles.

**Class 10 Maths NCERT Solutions Chapter 8 Exercises**

- NCERT Solutions Class 10 Introduction to Trigonometry Class 10 Exercise 8.1 â€“ 11 Questions (8 Short Answers, 3 Long Answers)
- NCERT Solutions Class 10 Introduction to Trigonometry Class 10 Exercise 8.2 â€“ 4 Questions (2 Short Answers, 2 Long Answers)
- NCERT Solutions Class 10 Introduction to Trigonometry Class 10 Exercise 8.3 â€“ 7 Questions (5 Short Answers, 2 Long Answers)
- NCERT Solutions Class 10 Introduction to Trigonometry Class 10 Exercise 8.4 â€“ 5 Questions (3 Short Answers, 2 Long Answers)

**Topics Covered in Class 10 Maths Chapter 8 Introduction to Trigonometry:**

Trigonometric Ratios of Complementary Angles

- sin (90Â° â€“ A) = cos A,
- cos (90Â° â€“ A) = sin A,
- tan (90Â° â€“ A) = cot A,
- cot (90Â° â€“ A) = tan A,
- sec (90Â° â€“ A) = cosec A,
- cosec (90Â° â€“ A) = sec A
- sin
^{2}A + cos^{2}A = 1, - sec
^{2}A â€“ tan^{2}A = 1 for 0Â° â‰¤ A < 90Â°, - cosec
^{2}A = 1 + cot^{2}A for 0Â° < A â‰¤ 90Â°

Trigonometry Table

Angle | 0Â° | 30Â° | 45Â° | 60Â° | 90Â° |

SinÎ¸ | 0 | 1/2 | 1/âˆš2 | âˆš3/2 | 1 |

CosÎ¸ | 1 | âˆš3/2 | 1/âˆš2 | 1/2 | 0 |

TanÎ¸ | 0 | 1/âˆš3 | 1 | âˆš3 | Undefined |

CotÎ¸ | Undefined | âˆš3 | 1 | 1/âˆš3 | 0 |

SecÎ¸ | 1 | 2/âˆš3 | âˆš2 | 2 | Undefined |

CosecÎ¸ | Undefined | 2 | âˆš2 | 2/âˆš3 | 1 |

Students can also download extra study materials from Utopper for Chapter 8 of **NCERT Class 10 Chapter 8 Introduction to Trigonometry **â€“ Utopper.

- Introduction to Trigonometry Class 10 Important Questions
- Important Formulas for Introduction to Trigonometry Class 10
- NCERT Revision Notes Class 10 Introduction to Trigonometry
- NCERT Exemplar Solutions Class 10 Introduction to Trigonometry
- RD Sharma Solutions for Introduction to Trigonometry Class 10
- RS Aggarwal Solutions Introduction to Trigonometry Class 10
- Ncert Textbook For Class 10 Introduction to Trigonometry
- Ncert Exemplar Book For Class 10 Introduction to Trigonometry

**NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry**

Chapter 9 is a continuation of chapter 8, where students will learn how to use the trigonometry they learned in the previous chapter. This chapter helps you understand how trigonometry is used in everyday life to figure out the height and distance of things. They will also learn how trigonometry is used in navigation, building, and finding the location of any piece of land based on the total latitudes and longitudes.

**Class 10 Maths NCERT Solutions Chapter 9 Exercises**

- NCERT Solutions Class 10 Some Applications of Trigonometry Class 10 Exercise 9.1 â€“ 16 Questions (16 Long Answers)

**Topics Covered in Class 10 Maths Chapter 9 Some Applications of Trigonometry:**

HEIGHTS AND DISTANCES: Angle of elevation and Angle of depression.

Simple problems about how high and far things are. There shouldn’t be more than two right triangles in a problem. Angles of rise and fall should only be 30, 45, and 60 degrees.

**Important Points â€“**

- The
**line of sight**is the line that goes from an observer’s eye to the point on an object that the observer can see. - The
**angle of elevation**of the point being looked at is the angle made by the line of sight and the horizontal when the point being looked at is above the horizontal level, like when we raise our head to look at something. - The
**angle of depression**of a point on the object being viewed is the angle made by the line of sight and the horizontal when the point is below the horizontal level, which is the case when we lower our head to look at the point being viewed.

Students can also download extra study materials from Utopper for Chapter 9 of **NCERT Class 10 Chapter 9 Some Applications of Trigonometry **â€“ Utopper.

- Some Applications of Trigonometry Class 10 Important Questions
- Important Formulas for Some Applications of Trigonometry Class 10
- NCERT Revision Notes Class 10 Some Applications of Trigonometry
- NCERT Exemplar Solutions Class 10 Some Applications of Trigonometry
- RD Sharma Solutions for Some Applications of Trigonometry Class 10
- RS Aggarwal Solutions Some Applications of Trigonometry Class 10
- Ncert Textbook For Class 10 Some Applications of Trigonometry
- Ncert Exemplar Book For Class 10 Some Applications of Trigonometry

**NCERT Solutions for Class 10 Maths Chapter 10 Circles**

In Chapter 10 of the Class 10 Maths NCERT Solutions, you’ll learn more about Circles, which you’ve already learned about in previous classes. Students have learned about circles and related terms like chord, segment, arc, etc. in previous classes. In this chapter, students will learn about the different ways a circle and a line can interact in a plane. So, they will fully understand what a tangent is and how many tangents a point on a circle has.

**Class 10 Maths NCERT Solutions Chapter 10 Exercises**

- NCERT Solutions Class 10 Circles Class 10 Exercise 10.1 â€“ 4 Questions (2 Short Answer, 2 Long Answers)
- NCERT Solutions Class 10 Circles Class 10 Exercise 10.2 â€“ 13 Questions (2 Short Answers, 14 Long Answers)

**Topics Covered in Class 10 Maths Chapter 10 Circles:**

**Important Theorems â€“**

**Theorem 10.1: **The tangent at any point of a circle is perpendicular to the radius through the point of contact.

**Theorem 10.2: **The lengths of tangents drawn from an external point to a circle are equal.

**Number of Tangents from a Point on a Circle**

- Case 1: There is no tangent to a circle passing through a point lying inside the circle.
- Case 2: There is one and only one tangent to a circle passing through a point lying on the circle.
- Case 3: There are exactly two tangents to a circle through a point lying outside the circle.

Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 10 Circles **â€“ Utopper.

- Circles Class 10 Important Questions
- Important Formulas for Circles Class 10
- NCERT Revision Notes Class 10 Circles
- NCERT Exemplar Solutions Class 10 Circles
- RD Sharma Solutions for Circles Class 10
- RS Aggarwal Solutions Circles Class 10
- Ncert Textbook For Class 10 Circles
- Ncert Exemplar Book For Class 10 Circles

**NCERT Solutions for Class 10 Maths Chapter 11 Constructions**

Chapter 11 is one of the chapters that are worth reading. This chapter is all about making different shapes out of shapes. The students will learn how to divide a line segment and how to draw tangents to circles. In this chapter, the methods of construction are well explained so that the ideas are clear.

In this chapter, there are enough examples before each exercise to show how the tangents to the circle are made. There are examples before each exercise to help the students understand the methods and theorems.

**Class 10 Maths NCERT Solutions Chapter 11 Exercises**

- NCERT Solutions Class 10 Constructions Class 10 Exercise 11.1 â€“ 7 Questions (7 Long Answers)
- NCERT Solutions Class 10 Constructions Class 10 Exercise 11.2 â€“ 7 Questions (7 Long Answers)

**Topics Covered in Class 10 Maths Chapter 11 Constructions:**

- Division of a line segment in a given ratio (internally).
- Tangent to a circle from a point outside it.
- Construction of a triangle similar to a given triangle.

Important Points â€“

- Construction 11.1: To divide a line segment in a given ratio.
- Construction 11.2: To construct a triangle similar to a given triangle as per the given scale factor.
- Construction 11.3: To construct the tangents to a circle from a point outside it.

Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 11 Constructions **â€“ Utopper.

- Constructions Class 10 Important Questions
- Important Formulas for Constructions Class 10
- NCERT Revision Notes Class 10 Constructions
- NCERT Exemplar Solutions Class 10 Constructions
- RD Sharma Solutions for Constructions Class 10
- RS Aggarwal Solutions Constructions Class 10
- Ncert Textbook For Class 10 Constructions
- Ncert Exemplar Book For Class 10 Constructions

**NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles**

The first part of this chapter is about the area and perimeter of a circle. Using this idea, the chapter goes on to show how to find the area of a circle’s sector and segment. Students will also understand how to find the areas of some plane shapes that are made up of circles or parts of circles.

**Class 10 Maths NCERT Solutions Chapter 12 Exercises**

- NCERT Solutions Class 10 Areas Related to Circles Class 10 Exercise 12.1 â€“ 5 Questions (5 Short Answers)
- NCERT Solutions Class 10 Areas Related to Circles Class 10 Exercise 12.2 â€“ 14 Questions (9 Short Answers, 5 Long Answers)
- NCERT Solutions Class 10 Areas Related to Circles Class 10 Exercise 12.3 â€“ 16 Questions (9 Short Answers, 7 Long Answers)

**Topics Covered in Class 10 Maths Chapter 12 Areas Related to Circles:**

The area of a circle’s sectors and segments. Problems based on the areas and perimeters/circumferences of the figures listed above. (When figuring out the area of a circle segment, problems should only involve a central angle of 60Â°, 90Â°, or 120Â°.

**Important Formulas â€“**

- Circumference = 2Ï€r
- Area of the circle = Ï€r 2
- Area of the sector of angle Î¸ = (Î¸/360) Ã— Ï€ r2
- Length of an arc of a sector of angle Î¸ = (Î¸/360) Ã— 2 Ï€ r

**where r is the radius of the circle**

Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 12 Areas Related to Circles **â€“ Utopper.

- Areas Related to Circles Class 10 Important Questions
- Important Formulas for Areas Related to Circles Class 10
- NCERT Revision Notes Class 10 Areas Related to Circles
- NCERT Exemplar Solutions Class 10 Areas Related to Circles
- RD Sharma Solutions for Areas Related to Circles Class 10
- RS Aggarwal Solutions Areas Related to Circles Class 10
- Ncert Textbook For Class 10 Areas Related to Circles
- Ncert Exemplar Book For Class 10 Areas Related to Circles

**NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes**

There are a total of 5 exercises in Chapter 13. The first set of questions is about finding the surface area of an object made by combining any two of the basic solids (cube, cone, cylinder, sphere, and hemisphere). In exercise 13.2, you have to find the volume of objects made by combining any two of a cuboid, cones, cylinders, spheres, or hemispheres. Exercise 13.3 has questions about changing the shape of a solid from one shape to another. Find the volume, curved surface area, and total surface area of a cone’s frustum for Exercise 13.4. The last exercise is optional and has high-level questions about all the topics in this chapter.

**Class 10 Maths NCERT Solutions Chapter 13 Exercises**

- Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.1 â€“ 9 Questions (2 Short Answers, 7 Long Answers)
- Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.2 â€“ 8 Questions (1 Short Answer, 7 Long Answers)
- Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.3 â€“ 9 Questions (9 Long Answers)
- Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.4 â€“ 5 Questions (5 Long Answers)
- Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.5 â€“ 7 Questions (7 Long Answers)

**Topics Covered in Class 10 Maths Chapter 13 Surface Areas and Volumes :**

**Surface areas** and **volumes **of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones.

**Important Formulas â€“**

**TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of another hemisphere**

**Sphere :**

Diameter of sphere = 2r

Surface area of sphere = 4 Ï€ r^{2}

Volume of Sphere = 4/3 Ï€ r^{3}

**Cylinder :**

Curved surface area of Cylinder = 2 Ï€rh

Area of two circular bases = 2 Ï€r^{2}

Total surface area of Cylinder = Circumference of Cylinder + Curved surface area of Cylinder = 2 Ï€rh + 2 Ï€r2

Volume of Cylinder = Ï€ r^{2} h

**Cone :**

Slant height of cone = l = âˆš(r^{2} + h^{2})

Curved surface area of cone = Ï€rl

Total surface area of cone = Ï€r (l + r)

Volume of cone = â…“ Ï€ r^{2} h

**Cuboid:**

Perimeter of cuboid = 4(l + b +h)

Length of the longest diagonal of a cuboid = âˆš(l^{2} + b^{2} + h^{2})

Total surface area of cuboid = 2(lÃ—b + bÃ—h + lÃ—h)

Volume of Cuboid = l Ã— b Ã— h

Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 13 Surface Areas and Volumes **â€“ Utopper.

- Surface Areas and Volumes Class 10 Important Questions
- Important Formulas for Surface Areas and Volumes Class 10
- NCERT Revision Notes Class 10 Surface Areas and Volumes
- NCERT Exemplar Solutions Class 10 Surface Areas and Volumes
- RD Sharma Solutions for Surface Areas and Volumes Class 10
- RS Aggarwal Solutions Surface Areas and Volumes Class 10
- Ncert Textbook For Class 10 Surface Areas and Volumes
- Ncert Exemplar Book For Class 10 Surface Areas and Volumes

**NCERT Solutions for Class 10 Maths Chapter 14 Statistics**

In this interesting chapter, students learn how to show data in numbers, whether the data are grouped or not. They also learn how to find the mean, the middle, and the most common value in a set of data. In the next activity, they will learn about the cumulative frequency distribution and draw cumulative frequency curves.

**Class 10 Maths NCERT Solutions Chapter 14 Exercises**

- NCERT Solutions of Maths Class 10 Statistics Exercise 14.1 â€“ 9 Questions (9 Long Answers)
- NCERT Solutions of Maths Class 10 Statistics Exercise 14.2 â€“ 6 Questions (6 Long Answers)
- NCERT Solutions of Maths Class 10 Statistics Exercise 14.3 â€“ 7 Questions (7 Long Answers)
- NCERT Solutions of Maths Class 10 Statistics Exercise 14.4 â€“ 3 Questions (3 Long Answers)

**Topics Covered in Class 10 Maths Chapter 14 Statistics:**

Mean, median and mode of grouped data.

**Important Formulas â€“**

**The mean of the grouped data can be found by 3 methods.**

**Direct Method: xÌ…****Assumed mean method****Step deviation method**

Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 14 Statistics **â€“ Utopper.

- Statistics Class 10 Important Questions
- Important Formulas for Statistics Class 10
- NCERT Revision Notes Class 10 Statistics
- NCERT Exemplar Solutions Class 10 Statistics
- RD Sharma Solutions for Statistics Class 10
- RS Aggarwal Solutions Statistics Class 10
- Ncert Textbook For Class 10 Statistics
- Ncert Exemplar Book For Class 10 Statistics

**NCERT Solutions for Class 10 Maths Chapter 15 Probability**

The last chapter is about probability. The chapter starts with a look at probability from a theoretical point of view. After that, the chapter talks about how experimental probability is different from theoretical probability. There are a lot of examples to help make the point clear. So, students must first solve the examples of CBSE Maths before moving on to the practice problems.

**Class 10 Maths NCERT Solutions Chapter 15 Probability**

- NCERT Solutions of Maths Class 10 Probability Exercise 15.1 â€“ 25 Questions (22 Short Answers, 3 Long Answers)
- NCERT Solutions of Maths Class 10 Probability Exercise 15.2 â€“ 5 Questions (5 Short Answers)

**Topics Covered in Class 10 Maths Chapter 15 Probability:**

Classical definition of probability. Simple problems in finding the probability of an event.

**Important Formulas â€“**

P(E)= \(\frac{Number of outcomes favourable to E}{Number of all possible outcomes of the experiment}\)

â€‹Students can also download extra study materials from Utopper for Chapter 10 of **NCERT Class 10 Chapter 15 Probability **â€“ Utopper.

- Probability Class 10 Important Questions
- Important Formulas for Probability Class 10
- NCERT Revision Notes Class 10 Probability
- NCERT Exemplar Solutions Class 10 Probability
- RD Sharma Solutions for Probability Class 10
- RS Aggarwal Solutions Probability Class 10
- Ncert Textbook For Class 10 Probability
- Ncert Exemplar Book For Class 10 Probability

**CBSE Class 10 Maths Chapter-wise Marks Weightage for 2022-23**

Sr.no | Chapter Name | Marks |

1 | Number System | 6 |

2 | Algebra | 20 |

3 | Coordinate Geometry | 6 |

4 | Geometry | 15 |

5 | Trigonometry | 12 |

6 | Mensuration | 10 |

7 | Statistics and Probability | 11 |

Total | 80 | |

Internal Assessment | 20 | |

Total | 100 |

**Internal Assessment for CBSE Class 10 Maths**

Internal Assessment | Marks |

Pen Paper Test and Multiple Assessment (5+5) | 10 Marks |

Portfolio | 05 Marks |

Lab Practical (Lab activities to be done from the prescribed books) | 05 Marks |

Total Marks | 20 Marks |

**Benefits of NCERT Solution for Class 10 Maths**

One of the most popular online study tools is the NCERT 10 Maths solutions PDF that was made at Utopper. Students read through the PDF and learn more about the topics, which helps them prepare for their final exams and for further studies. Some of the most important things about NCERT Maths solution Class 10 PDF are:

- Step-by-step calculations for each question will help you understand the subject better.
- A full explanation of the problems to promote learning based on concepts.
- Up-to-date on the changes that have been made to the curriculum.
- It is set up by chapter to make it easier to use.
- Great for reviewing and getting a refresher before tests.
- More questions to help you keep track of how far you’ve come.
- If you want to do well on tests, you should follow a test pattern.
- Great for studying because the whole curriculum is put into one PDF solution. This makes it easier for students to look things up before the test.
- The language used to explain the solutions is simple and easy to understand.
- Extra questions that have already been answered are also given, so that students can look at them as they work on more questions.
- Adapted to any changes the CBSE board makes to the curriculum or the way tests are set up.

**How are CBSE Class 10 Maths Solutions of NCERT Helpful for the Board Exams?**

Class 10 CBSE Mathematics is a crucial subject for students. Here, we have provided students with comprehensive assistance for their preparation. Class 10 is the first benchmark that will be reflected in future achievement records for any student. NCERT books are always recommended by the CBSE for exam preparation.

Exam preparation necessitates a comprehensive understanding of each chapter. This procedure necessitates arduous effort toward studies and an efficient method for navigating the solutions. NCERT 10th Grade Math Solutions play a significant role in preparing students for competitive entrance exams. The NCERT textbooks are well-known for presenting concepts in a straightforward manner to facilitate comprehension. The NCERT Class 10 Mathematics Textbooks are written in the most lucid and clear manner, which aids in breaking down complex problems in the most efficient manner.

Keep visiting to obtain free PDF downloads of complete NCERT Solutions for Class 10 Maths, chapter-by-chapter, for all classes. Class 10 Science NCERT Solutions are also available on Utopper for students.

**Frequently Asked Questions on NCERT Solutions for Class 10 Maths**

**Frequently Asked Questions on NCERT Solutions for Class 10 Maths**

**Q.1 How to understand the essential concepts covered by the NCERT Solutions for Class 10 Mathematics?**

Downloading the NCERT Solutions from Utopper is recommended for Class 10 students who wish to perform well on their exams. The solutions are curated with the utmost care by a group of faculty with extensive knowledge of the respective subject. To make learning easier for the students, each and every minute detail is explained in an interactive manner. The step-by-step solutions are developed with the mark weightings specified by the CBSE in mind.

**Q.2 Will the NCERT Solutions for Class 10 Mathematics enable problem solving?**

**Q.2 Will the NCERT Solutions for Class 10 Mathematics enable problem solving?**

The Class 10 examination is one of the most significant turning points in a student’s life. Mathematics is a subject that consists primarily of numbers, and understanding concepts helps to earn higher grades. For improved performance, it is recommended that students first comprehend the curriculum for the academic year and then learn the concepts based on it. Students’ logical reasoning and analytical abilities will be enhanced if they solve textbook problems using the PDF solutions.

**Q.3 Where can I obtain a PDF of the NCERT Solutions for Class 10 Math?**

**Q.3 Where can I obtain a PDF of the NCERT Solutions for Class 10 Math?**

UTOPPER provides PDFs of the NCERT Solutions for Class 10 Maths. The solutions are prepared by subject matter experts with consideration for the students’ level of comprehension. The primary purpose of developing solutions is to facilitate students’ comprehension of difficult concepts. Students can also cross-check their answers while solving textbook problems to gain an understanding of alternative methods for efficiently answering the questions.

**Q.4 Are the NCERT Solutions for Class 10 Mathematics helpful for CBSE students?**

**Q.4 Are the NCERT Solutions for Class 10 Mathematics helpful for CBSE students?**

Listed below are some applications of the NCERT Maths Solutions for Class 10 â€“ Mathematics.

It establishes a solid foundation of fundamental concepts and enhances exam-taking confidence.

The method for solving complex problems is simple to learn.

It is the ideal study material for students to complete their assignments on time and score higher on their exams.

**Q.5 How can I get a sense of the key concepts in the NCERT Solutions for Class 10 Mathematics?**

Before the start of the academic year, students should download the CBSE curriculum and review it to gain a thorough understanding of the concepts that will be tested. Students must solve the problems in the NCERT textbooks by selecting the appropriate study material that meets their needs. Students’ problem-solving skills are enhanced through regular problem-solving practice using the NCERT Solutions for Class 10 Maths.