# NCERT Solutions for Class 10 Maths

## Chapter-Wise NCERT Solutions for Class 10 Maths

**NCERT Solutions for class 10 maths** will be helpful for Class 10 students of mathematics, as we know that there is just one NCERT book for all subjects. It is a broad syllabus where students are being taught maths and science, but it is not always possible to consult the textbook during class lectures or home study.

NCERT Solutions for class 10 maths is a student-friendly compilation of problems with detailed solutions on topics such as functions, geometry, vectors, and trigonometry.

- 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

## NCERT Solutions for Class 10 Maths Updated for 2022-23 – Free PDF Download

### 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

### How to score fully Marks in Class 10 Maths with NCERT textbook and NCERT solutions for class 10 Maths

Maths is the subject of the application of formulas, and it needs very good planning and hard work to score good marks in class 10 Maths. There are 15 chapters in class 10 Maths, and you need to have good practice and understanding of all 15 chapters. A few tips to score good marks in class 10 Maths are as follows.

- Always attend the school lecture of class 10 Maths and listen carefully to what teachers explain if required ask questions and doubts you have. Note down all important points in class and try to understand the application of the concept in numerical.
- Once your school class or tuition class is over read the NCERT textbook and note down all important formulas and points, read line by line, and spend time in the chapters, while preparing the chapter don’t be in hurry give adequate time to understand the entire chapter.
- Start your practice from the questions given in the NCERT textbook exercise before attempting questions from any reference book make sure you have solved all the questions given in the NCERT exercise you can use NCERT solutions for class 10 Maths prepared and uploaded by home tution.com experts if you face problem in any question.

To score good marks in class 10 Maths one must understand the pattern of the CBSE class 10 Board. Let’s understand the pattern of questions asked in board exams.

### CBSE Class 10 Maths Unit-wise Weightage

#### Marks weightage of CBSE Class 10 Maths Term 2 Exam are as follows

- Algebra (10 Marks)

- Chapter 4: Quadratic Equations
- Chapter 5: Arithmetic Progression

- Geometry (9 Marks)

- Chapter 10: Circles
- Chapter 11: Constructions

- Trigonometry (7 Marks)

- Chapter 9: Some Applications of Trigonometry

- Mensuration (6 Marks)

Total- 40 Marks

CBSE Class 10 Maths Term 2 Exam- a total of 50 marks having 40 marks of theory paper and 10 marks of term 2 internal assessment Marks in the internal assessment will be

1. Periodic Tests (Marks-3)

2. Multiple Assessments (Marks-2)

3. Portfolio (Marks-2)

4. Student Enrichment Activities-practical work (Marks-3)

Total Marks-10

### CBSE Class 10 Maths Question Paper Pattern for Term 2 Exam are

- All the questions asked in term 2 are subjective-type questions.
- Questions are asked in three sections (Section A, B, and C) and have 14 questions.
- Section-A: It consists of 6 questions with 2 marks each.
- Section-B: It consists of 4 questions with 3 marks each. One of the questions will be provided with an internal choice.
- Section-C: This section consists of 4 questions with 4 marks each. One question will be provided with an internal choice.

### Important formulas are given in NCERT Class 10 Maths textbook

An algebraic expression f(x) of the form f(x)= a_{0} + a_{1}x +a_{2}x^{2}+…….+a_{n }x^{n }, where a_{0},a_{1,} a_{2}……. A_{n }are real numbers and all the index of x are non-negative integers is called polynomial in x and the highest index n is called the degree of the polynomial.

- (x+a)
^{2}= x^{2 }+2ax+a^{2} - (x-a)
^{2}= x^{2}; 2ax +a^{2 } - x
^{2}- a^{2}=(x+a)(x-a) - x
^{3 }+ a^{3}= (x+a) (x^{2}-ax+a^{2}) = (x+a)^{3}3xa (x+a) - x
^{3}-a^{3}= (x-a) (x^{2}+ax+a^{2}) = (x+a)^{3}+ 3xa(x-a) - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ac - (a + b)
^{3}= a^{3}+ b^{3}+ 3ab (a + b) - (a-b)
^{3}= a^{3}-b^{3}-3ab (a-b) - a
^{3}+ b^{3}+ c^{3}- 3abc = (a + b + c) (a^{2}+ b^{2}+ c^{2}- ab - bc - ca)

**SPECIAL CASE:** If a + b + c = 0 then a^{3}+ b^{3}+ c^{3}= 3abc

**ZEROS OR ROOTS OF A POLYNOMIAL**

A real number ‘a’ is a zero of a polynomial f(x), if f(a) = 0. Here ‘a’ is called a root of the equation f(x) = 0

**FACTOR THEOREM**

Let p(x) be a polynomial of degree greater than or equal to ‘a' be a real number such that p(a) = 0, then (x - a) is a factor of p(x), then p(a) = 0.

**REMAINDER THEOREM**

Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x)is a dividend by (x-a), then remainder is equal to p(a). Let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x-a), then, dividend = Divisor x Quotient + Remainder

### Chapter-wise introduction to NCERT Solutions for class 10 maths

Here we provide complete **NCERT maths solutions for class 10 ** so, with our great tutors, you can learn in both languages: Hindi and English.

### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Chapter-1 Real Numbers in the NCERT textbook consist of real numbers and irrational numbers apart from these students will learn about the Divisibility of integers, rational number, irrational number, Euclid’s Division Lemma, HCF, LCM, and decimal expansion of rational numbers with the help of theorem. There are 4 Exercises present in this chapter.

**Chapter-1 Real Numbers Exercise and Questions**

- Exercise 1.1 consist of 5 Questions Solutions (4 Long Answers, 1 Short Answer
- Exercise 1.2 consist of 7 Questions Solutions (4 Long Answers, 3 Short Answers)
- Exercise 1.3 consist of 3 Questions Solutions (3 Short Answers)
- Exercise 1.4 consist of 3 Questions Solutions (3 Short Answers)

### NCERT Solutions for Class 10 Maths Chapter-2 Polynomials

Chapter-2 Polynomials in the NCERT textbook consist of 4 Exercises including an optional Exercise. Find the concept used in this chapter and questions are asked about which part of the concept.

- Exercise 2.1 Consist of questions based on finding zeros of polynomials p(x) - 1 Question & Solutions (1 Short Answer)
- Exercise 2.2 Consist of questions based on quadratic polynomial and relationship between 0’s and coefficients of a polynomial 2 Questions & Solutions (2 Short Answers)
- Exercise 2.3 Consist of questions based on concepts in entire chapter 5 Questions & Solutions (2 Short Answers, 3 Long Answers)
- Exercise 2.4 Consist of questions based on Division Algorithm for polynomials 5 Questions & Solutions (2 Short Answers, 3 Long Answers)

### NCERT Solutions for Class 10 Maths Chapter-3 Pair of Linear Equations in Two Variables

Chapter 3 Pair of Linear Equations in Two Variables has concepts about linear equations and their application in questions. NCERT textbook consists of lots of questions based on these chapters and has 7 exercises. Find details about the exercise and the types of questions and concepts used to solve these exercises

- Exercise 3.1 It consists of problems that required the use of algebraically and graphically explanation. Total questions in this exercise are 3 Questions Solutions (2 Short Answers, 1 Long Answer)
- Exercise 3.2 This exercise required solving linear equations by graphical method. The total questions in this exercise are 7 Questions Solutions (5 Short Answers, 2 Long Answers)
- Exercise 3.3 - 3 Questions Solutions (2 Short Answers, 1 Long Answer)
- Exercise 3.4 questions in this exercise required different types of methods of solving the pairs of linear equations. Total questions in this exercise are 2 Questions Solutions (2 Long Answers)
- Exercise 3.5 - 4 Questions Solutions (4 Short Answers)
- Exercise 3.6 - 2 Questions Solutions (2 Long Answers)
- Exercise 3.7 - 8 Questions & Solutions (1 Short Answer, 7 Long Answers)

### NCERT Solutions for Class 10 Maths Chapter - 4 - Quadratic Equations

An equation *ax*^{2}+*bx*+*c* = 0 (where *a* ≠ 0, and *a*, *b*, *c* ∊ *R*), is called a quadratic equation. Here *a*, *b,* and *c* are called coefficients of the equation. This equation always has two roots. Let the roots be alpha and beta.

Then, α + β = -b/a (sum of roots)

αβ = c/a (product of the roots)

The quantity D = b^{2}-4ac is called the discriminant of the equation.

Roots of the equation are given by *x* = - b + - root D/2a

- If D > 0, the roots of the equation are real and distinct.
- If D = 0, the roots of the equation are real and equal.
- If D < 0, the roots of the equation are imaginary and conjugates of each other.
- If D is the square of a rational number, the roots of the equation are rational. (Provided a, b and c are rational numbers)

- Exercise 4.1-2 Questions Solutions (1 Short Answer, 1 Long Answer)
- Exercise 4.2-6 Questions Solutions (6 Short Answers)
- Exercise 4.3-11 Questions & Solutions (8 Short Answers, 3 Long Answers)
- Exercise 4.4-5 Questions & Solutions (2 Short Answers, 3 Long Answers)

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

Chapter-5 Arithmetic Progressions introduces Arithmetic Progression, commonly called AP. You are going to learn about the basics of AP.

- Exercise 5.1 This exercise consists of questions based on how to represent a problem or situation in the form of AP- Total questions in this exercise are 4 Questions Solutions (1 Short Answer, 3 Long Answers)
- Exercise 5.2 This exercise consists of questions based on how to find the first term and difference of the AP and understand how to find whether the given series is AP or something else. Also, you are going to get the questions about finding the nth term of the AP or not. The total questions in this exercise are 20 Questions Solutions (10 Short Answers, 10 Long Answers)
- Exercise 5.3 This exercise consists of questions based on how to find the sum of the first n terms of AP Total questions in this exercise are 20 Questions Solutions (7 Short Answers, 13 Long Answers)
- Exercise 5.4 This exercise consists of questions based on entire chapters. The total questions in this exercise are 5 Questions Solutions (5 Long Answers)

### NCERT Solutions for Class 10 Maths Chapter-6-Triangles

Chapter-6 Triangles consist of basic information about triangles and types of triangles and theorems used to find the different data.

**Types of Triangles based on sides:**

- A triangle with no sides equal is called a scalene triangle.
- A triangle with two sides equal is called an isosceles triangle.
- A triangle with all sides equal is called an equilateral triangle.

**Based on Angles:**

- A triangle with all angles acute is called an acute-angled triangle.
- A triangle with one angle and a right angle is called a right-angled triangle.
- A triangle with one angle and an obtuse angle is called an obtuse-angled triangle.

**Based on sides:**

- A triangle with no sides equal is called a scalene triangle.
- A triangle with two sides equal is called an isosceles triangle.
- A triangle with all sides equal is called an equilateral triangle.

**Based on Angles:**

- A triangle with all angles acute is called an acute-angled triangle.
- A triangle with one angle and a right angle is called a right-angled triangle.
- A triangle with one angle and an obtuse angle is called an obtuse-angled triangle.

**The similarity of triangles: **Two figures are similar if they are of the same shape but not necessarily of the same size.

Whereas two congruent figures have the ‘same shape’ and the ‘same size’. Hence two congruent figures are similar but the converse is not necessarily true.

All regular polygons of the same number of sides such as equilateral triangles, squares, hexagons, etc. are similar. In particular, all circles are also similar. Two triangles are said to be similar, if

- Their corresponding angles are equal
- Their corresponding sides are in the same ratio

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

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

The study of geometry employing algebra is called coordinate geometry. It is also called “analytic geometry”. This system of coordinates after the name Rene Descartes is called the Cartesian coordinate system.

In the Cartesian coordinate system position of a point is described in terms of its distance from fixed lines that are perpendicular to each other.

These lines are respectively called as x-axis, y-axis, and z-axis in space. But in a plane (just like a board or notebook) there are only two axes respectively called as x-axis and y-axis. The intersection point of these axes is called the origin.

These axes divide the plane into four parts called Quadrants namely I quadrant, II quadrant, III quadrant, and IV quadrants.

- Exercise 7.1-10 Questions Solutions (3 Short Answers, 7 Long Answers)
- Exercise 7.2-10 Questions Solutions (2 Short Answers, 8 Long Answers)
- Exercise 7.3-5 Questions Solutions (2 Short Answers, 3 Long Answers)
- Exercise 7.4-8 Questions Solutions (3 Short Answers, 5 Long Answers)

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

You will learn about the Definition of Angle If we rotate a line segment concerning any of its endpoints, then the amount of rotation is called angle. If rotation exists in an anti-clockwise direction, the angle is said to be positive, otherwise negative.

The angle of Elevation: Angle between the horizontal line and line of sight, if the object is above the horizontal level of eyes. The angle of Depression: Angle between the horizontal line and line of sight, if the object is below the horizontal level of eyes.

- Exercise 8.1-11 Questions Solutions (8 Short Answers, 3 Long Answers)
- Exercise 8.2-4 Questions Solutions (2 Short Answers, 2 Long Answers)
- Exercise 8.3-7 Questions & Solutions (5 Short Answers, 2 Long Answers)
- Exercise 8.4-5 Questions & Solutions (3 Short Answers, 2 Long Answers)

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

Chapter 9 Some Applications of Trigonometry is an extension of the Trigonometry chapter, and it is important for higher classes. Students will learn the basic formula and its use in different types of questions

- Exercise 9.1-16 Questions & Solutions (16 Long Answers)

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

The circle is a collection of points whose distance from a fixed point always remains constant. The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle. The length of the complete circle is called its circumference. Tangents to a circle. The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent.

**Number of tangents from a point**

- From a point on a circle, there is one and only one tangent.
- From an external point of a circle, two tangents can be drawn to the circle.
- From a point inside a circle, no tangent can be drawn.

A Tangent to a circle is a line that intersects the circle at exactly one point.

- Exercise 10.1-4 Questions & Solutions(2 Short Answers, 2 Long Answers)
- Exercise 10.2-13 Questions & Solutions (2 Short Answers, 14 Long Answers)

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

For drawing geometrical figures such as a triangle, a circle, a quadrilateral, a polygon, etc. with given measurements some basic geometrical instruments are needed

- A graduated scale, on one side of which centimetres and millimetres are marked off, and on the other side, inches are marked off.
- A pair of set squares, one with angles 90°, 60°, and 30° and the other with angles 90°, 45°, and 45°
- Dividers
- A compass with the provision of fitting a pencil at one end.
- Protractors

- Exercise 11.1-7 Questions & Solutions (7 Long Answers)
- Exercise 11.2-7 Questions & Solutions (7 Long Answers)

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

Chapter 12 Areas Related to Circles will teach you how to find the area and the of a circle. The concept of finding the area of a sector and segment of a circle is very simple by the use of the right formula which is explained on this page.

- Circumference of the Circles = 2πr
- Area of the circle = πr2
- Area of the sector of angle θ = (θ/360) × πr2
- Length of an arc of a sector of angle θ = (θ/360) × 2 π r where r is the radius of the circle

- Exercise 12.1-5 Questions Solutions (5 Short Answers)
- Exercise 12.2-14 Questions Solutions (9 Short Answers, 5 Long Answers)
- Exercise 12.3-16 Questions & Solutions (9 Short Answers, 7 Long Answers)

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

The branch of mathematics that concerns with measurement of lengths, areas, and volume of plane figures and solid figures is called Mensuration.

**Area:** The magnitude of a region occupied by a figure in a plane is called its area. The area is measured in m2.

**Volume: **The magnitude of the region occupied by a solid in space is called its volume. Volume is measured in m3 (meter cube) or l (litres).

**Perimeter:** Total length of the boundary of a plane figure is called its perimeter. The perimeter of the circle is called the circumference.

**Surface area:** The area of the surface of the solid figure is called it's surface area.

- Exercise 13.1-9 Questions & Solutions (2 Short Answers, 7 Long Answers)
- Exercise 13.2-8 Questions & Solutions (1 Short Answer, 7 Long Answers)
- Exercise 13.3-9 Questions & Solutions (9 Long Answers)
- Exercise 13.4-5 Questions & Solutions (5 Long Answers)
- Exercise 13.5-7 Questions & Solutions (7 Long Answers)

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

The branch of mathematics that deals with data to extract meaningful information is called statistics.

Data Facts or figures collected with a definite purpose are called data. They are mainly of two types:

Data collected directly is called primary data. Data collected from other sources is called secondary data.

Frequency– The number of times Data repeats is called its frequency.

Cumulative frequency– If the frequency is added for all data, it is called cumulative frequency.

- Primary data
- Secondary data

- Exercise 14.1-9 Questions & Solutions (9 Long Answers)
- Exercise 14.2-6 Questions & Solutions (6 Long Answers)
- Exercise 14.3-7 Questions & Solutions (7 Long Answers)
- Exercise 14.4-3 Questions & Solutions (3 Long Answers)

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

Chapter 15 Probability is an important chapter for board exams and higher class maths too. It consists of two exercises.

- Exercise 15.1-25 Questions Solutions (22 Short Answers, 3 Long Answers)
- Exercise 15.2-5 Questions & Solutions (5 Short Answers)

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### Why are NCERT Solutions very important for your 10th maths?

NCERT Solutions for class 10 maths are specially designed for students to help them get a solid foundation. These solutions are designed by our expert team of teachers. It will act as an ideal companion for students while studying their classes

NCERT solutions for class 10 maths provide students with the basics required to understand and learn the concepts of mathematics. It helps students to prepare well before they appear in competitive exams where they need to understand the concept well. This book is particularly useful for those who are preparing for entrance exams like JEE, CAT, NEET, etc.

### CBSE Class 10 Maths Academic Resources 2023-24

Class 10 Maths NCERT Solutions | Questions | MCQs |
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### Chapter Wise Solutions - Download Free PDF

#### Related Links

- 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

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

There are 15 chapters that cover, the syllabus of class 10 Maths covers topics ranging from number systems and arithmetic to algebra, geometry & measurement, trigonometry, and statistics.

Here at Home-Tuitions, we have provided the complete NCERT Solution for Class 10 Maths in PDF form, including a Chapter-wise maths solution for class 10 and a worksheet for the students to exercise these questions. You can download these solutions in your language.

NCERT Solutions for Class 10 Maths contains a variety of questions from every chapter in the class. These problems have been solved by our team of highly qualified teachers and subject experts to help you develop your problem-solving skills. Additionally, detailed solutions are also provided for each question so that you can improve your comprehension of the content.

If you analyze the previous year’s question paper for the class 10 board, you will get the maximum questions asked from the NCERT exercise. So the first thing you must remember is that NCERT must solve textbooks to score good marks in the class 10 final exam. To score full marks hard work and the right planning is key, start your preparation from the month of March and plan for the exam very carefully. Always attend the school or tuition class of maths and make sure you are going to make the right maths notes with mentioning all-important formulas in them. Three things are very much required to score good marks in class 10 Maths one is concepts understanding- to achieve this read the NCERT theory ask your teachers if you can’t understand any points and make notes of the concepts. The second task is practice- do solve all the questions given in the NCERT exercise with the help of NCERT solutions for class 10 Maths apart from this do solve a few reference books. Last is practice important maths formulas.

Yes, the NCERT textbook and NCERT solutions for class 10 Maths are sufficient for the class 10 final exam. All the questions and concepts covered in the NCERT textbook are more than sufficient for CBSE final exams. You just prepared the notes and solve all the questions given in the NCERT textbook exercise.