**NCERT Solutions for Class 10 Maths** covers all of the questions from the textbook that are prescribed by the CBSE board. Maths is one of the important subjects in Class 10. NCERT Solutions for Class 10 provides comprehensive knowledge of all the concepts. These NCERT questions help students to study for their CBSE board exams by developing a sound understanding of all the topics. NCERT Solutions for Class 10 Maths help students to enhance their problem-solving skills. Our experts have designed the CBSE NCERT Solutions in such a way that students can gain a better understanding of all the important topics and problems mentioned in the textbook. By solving these questions, students can analyse their level of preparation and focus on developing their weak areas.

The NCERT Solutions for Class 10 Maths are curated by subject-matter experts to help students with their term-wise preparations. For all the exercises in Chapters 1 to 15, we have provided NCERT Class 10 Solutions for Maths in this article. Students can download all chapter-wise PDFs solutions from this article itself to score good marks in their exams. Continue reading to learn everything about Class 10 NCERT Solutions for Maths.

**Latest Update**

👉 The CBSE Term 2 Exams will commence in March/ April 2022. The officials will soon release the exam date sheet and other important information in February 2022.

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Here at Embibe, you can get the Free CBSE Revised MCQ Mock Test 2021 for all topics. The MCQ Test offered by Embibe is curated based on revised CBSE Class Books, paper patterns and syllabus for the year 2021. This mock test series has a comprehensive selection of relevant questions and their solutions. Candidates in CBSE Board can take these free mock tests to practise and find areas where they need to improve for their board exams.

Table of Contents

- Class 10 Maths NCERT Solutions PDF
- Class 10 Maths NCERT Solutions: Chapter-wise
- NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Term I)
- NCERT Solutions for Class 10 Maths Chapter 2 Polynomials (Term I)
- NCERT Solutions for Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables (Term I)
- NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Term II)
- NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions (Term II)
- NCERT Solutions for Class 10 Maths Chapter 6 Triangles (Term I)
- NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry (Term I)
- NCERT Solutions for Class 10 Maths Chapter 8 Introduction To Trignometry (Term I)
- NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry (Term II)
- NCERT Solutions for Class 10 Maths Chapter 10 Circles (Term II)
- NCERT Solutions for Class 10 Maths Chapter 11 Constructions (Term II)
- NCERT Solutions for Class 10 Maths Chapter 12 Areas Related To Circles (Term I)
- NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volumes (Term II)
- NCERT Solutions for Class 10 Maths Chapter 14 Statistics (Term II)
- NCERT Solutions for Class 10 Maths Chapter 15 Probability (Term I)
- Why Embibe for Class 10 Maths NCERT Solutions?
- FAQs

## Class 10 Maths NCERT Solutions PDF

You can download all the NCERT Solutions of Class 10 Maths provided on this page in the PDF format for free. Studying from the NCERT Class 10 Maths Solutions will help you in clearing your basics and improving your problem-solving skills. In this solutions, we will begin with the number system and then go to algebra. After that, we will look at coordinate geometry. We will also go over trigonometry and mensuration principles. Finally, we will look at Probability and Statistics. Click on the links provided below to download chapter-wise PDFs for Class 10 Maths.

**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**

As discussed earlier, we have provided the direct links for NCERT Maths Book Class 10 Solutions PDF Free Download. Using these links, students can download Chapter-wise NCERT Class 10 Maths Solutions for offline access.

It is crucial to note that CBSE has opted to cut the syllabus by 30% for the academic year 2021-22. However, many of the subjects that have been eliminated can serve as connectors, so students can read those topics to develop their knowledge and abilities. NCERT Solutions for Class 10 Maths will aid in the review of those areas. These topics will be covered not just in Class 10 but also in the Science stream, so you should be familiar with them. You may learn more about the CBSE Class 10 Maths Curriculum 2021-22 and the test format by looking at the complete syllabus.

### NCERT Class 10 Maths Chapter-wise Weightage

CBSE Class 10 Maths board exams will be conducted in two terms namely Term I and Term II as per the new assessment policy. The exam patterns of Term I and II are different. Students preparing for the Class 10 board exam must check the chapter-wise weightage and create study plans accordingly. The unit-wise weightage distribution as per the **CBSE Class 10 Maths Sample Paper** and **Class 10 Marking Scheme** is tabulated below:

### Chapter-wise Weightage for CBSE 10 Maths 1st Term Exam

Chapters | Marks |

Chapter 1: Real Numbers | 6 |

Chapter 2: Polynomials and Chapter 3: Pair of Linear Equations in Two Variables | 10 |

Chapter 6: Triangles | 6 |

Chapter 7: Coordinate Geometry | 6 |

Chapter 8: Introduction to Trigonometry | 5 |

Chapter 12: Area Related to Circles | 4 |

Chapter 15: Probability | 3 |

Internal Assessment | 10 |

First Term Total Marks | 50 |

### Chapter-wise Weightage for CBSE 10 Maths 2nd Term Exam

Chapters | Marks |

Chapter 4: Quadratic Equations and Chapter 5: Arithmetic Progression | 10 |

Chapter 9: Some Applications of Trigonometry | 7 |

Chapter 10: Circles and Chapter 11: Constructions | 9 |

Chapter 13: Surface Areas and Volumes | 6 |

Chapter 14: Statistics | 8 |

Internal Assessment | 10 |

Second Term Total Marks | 50 |

## Class 10 Maths NCERT Solutions: Chapter-wise

These NCERT Solutions for Class 10 Maths have been compiled by professionals in a comprehensive format that will help you clear your questions quickly. It will help you become well-versed on a range of topics and enable you to recall and shape your responses quickly. Students should also examine their main areas using previous year’s questions, practise test papers, and worksheets. Now let’s understand the topics and sub-topics you will learn with the Maths solutions provided here.

### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Term I)

**NCERT Solutions for Class 10 Maths Chapter 1:** After understanding some basic rules regarding real and irrational numbers in earlier grades, the first chapter of Class 10 Maths starts with some of the most critical concepts in Algebra. The Division Lemma (lemma – an algorithm, a proven statement used to prove other statements) by Euclid is one of the first concepts that students will learn, followed by the Fundamental Theorem of Arithmetic by Carl Freidrich Gauss.

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

Fundamental Theorem of Arithmetic – statements along with reviews of earlier works done. Illustrations and clarifications through examples. The decimal representation of rational numbers as both terminating/non-terminating recurring decimals.

**Important Steps**:

To obtain the HCF of two positive integers, say c and d, where c > d, the following steps must be taken:

**Step 1:** First apply Euclid’s division lemma, to both c and d. This helps us find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.

**Step 2:** If r = 0, then d is the HCF of c and d. However, if r ≠ 0, then apply the division lemma to d and r.

**Step 3:** Continue this process until the remainder is zero. The divisor that we obtain at this stage is the required HCF. The reason why this algorithm works is HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) indicates the HCF of c and d, etc.

**Topics Deleted for Academic Year 2021-22 Board Exams: Euclid’s division lemma **

The total number of exercises and questions are tabulated below:

Exercise No & Name | Total Questions |

Ex 1.1 Euclid’s Division Lemma | 5 Questions |

Ex 1.2 The Fundamental Theorem of Arithmetic | 7 Questions |

Ex 1.3 Revisiting Irrational Numbers | 3 Questions |

Ex 1.4 Revisiting Rational Numbers and Their Decimal Expansions | 3 Questions |

Total: Contains 4 Exercises | 18 Questions |

### NCERT Solutions for Class 10 Maths Chapter 2 Polynomials (Term I)

** NCERT Solutions for Class 10 Maths Chapter 2: **Expanding from the basics of polynomials learned in earlier classes, the chapter explains the concept of degree of polynomials and how polynomial with degree 1 is linear polynomial, degree 2 a quadratic polynomial, and degree 3 a cubic polynomial.

**Topics Covered in Class 10 Maths Chapter 2 Polynomials for Term I:**

The most important topics covered in this article are zeroes of a polynomial and the relationship between zeroes and coefficients of quadratic polynomials only.

**Important Steps**:

We first arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Recall that arranging the terms in this order is called writing the polynomials in standard form.

**Step 1:** First we must obtain the first term of the quotient by dividing the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process.

**Step 2:** Then, we can obtain the second term of the quotient by dividing the highest degree term of the new dividend by the highest degree term of the divisor. Carry out the division process again.

**Step 3:** The degree of the remainder is lesser than the degree of the divisor. So, we cannot continue the division anymore.

We can see here that Dividend = Divisor × Quotient + Remainder. What we are applying here is an algorithm that is similar to Euclid’s division algorithm that you studied in Chapter 1.

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This says that:

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

This obtained result is called the Division Algorithm for polynomials.

**Topics Deleted for Academic Year 2021-22 Board Exams: Statement and simple problems on division algorithm for polynomials with real coefficients.**

The total number of exercises and questions in polynomials are tabulated below:

Exercise No & Name | Total Questions |

Exercise 2.1 | 1 Question & Solutions (1 Short Answer) |

Exercise 2.2 | 2 Questions & Solutions (2 Short Answers) |

Exercise 2.3 | 5 Questions & Solutions (2 Short Answers, 3 Long Answers) |

Exercise 2.4 | 5 Questions & Solutions (2 Short Answers, 3 Long Answers) |

Total: 4 Exercises | 13 Questions |

### NCERT Solutions for Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables (Term I)

**NCERT Solutions for Class 10 Maths****Chapter 3: **In this chapter, we will learn several methods such as the substitution method, elimination method, and cross-multiplication method. The final concept in the chapter is the reduction of equations in two variables. A brief summary of the NCERT Solutions for Class 10 Maths Chapter 3 is given below:

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

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

This lesson covers the Pair of linear equations in two variables and the graphical method of their solution, consistency/inconsistency, algebraic conditions for the number of solutions. Solutions of a pair of linear equations in two variables algebraically – by substitution and by elimination, simple situational problems, simple problems on equations reducible to linear equations.

**Important Formulas**:

We have given the general form for a pair of linear equations in two variables x and y below:

a_{1} x + b_{1} y + c_{1} = 0

and

a_{2} x + b_{2} y + c_{2} = 0,

where a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are real numbers where a_{1}^{2} + b_{1}^{2} ≠ 0, a_{2}^{2} + b_{2}^{2} ≠ 0.

**Topics Deleted for Academic Year 2021-22 Board Exams: Cross Multiplication Method**

The total number of questions and exercises in the chapter pair of linear equations in two variables are tabulated below:

Exercise No & Name | Total Questions |

Exercise 3.1 | 3 Questions & Solutions (2 Short Answers, 1 Long Answer) |

Exercise 3.2 | 7 Questions & Solutions (5 Short Answers, 2 Long Answers) |

Exercise 3.3 | 3 Questions & Solutions (2 Short Answers, 1 Long Answer) |

Exercise 3.4 | 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) |

Total: 7 Exercises | 24 Questions |

### NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Term II)

**NCERT Solutions for Class 10 Maths****Chapter 4: **In this chapter, we will learn the standard form of quadratic equations, different methods of solving quadratic equations (by factorization, by completing the square), and the nature of roots. The Quadratic Equation Class 10 PDF solutions will be extremely helpful.

The chapter ends with the topic of finding the nature of roots which states that, a quadratic equation ax² + bx + c = 0 has

- two distinct real roots, if b² – 4ac > 0
- two equal roots, if b² – 4ac = 0
- no real roots, if b² – 4ac < 0

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

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

The relationship between discriminant and nature of roots, standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0), solutions of quadratic equations (only real roots) by factorization, and by using the quadratic formula, the relationship between discriminant and nature of roots, situational problems based on quadratic equations related to day to day activities (problems on equations reducible to quadratic equations are excluded)

**Important Formulas**:

If b ^{2} – 4ac > 0, we get two distinct real roots

If b ^{2} – 4ac = 0, then,

So, the roots of the equation ax^{2} + bx + c = 0 are both -b/2a.

Therefore, we say that the quadratic equation ax^{2} + bx + c = 0 has two equal real roots in this case.

If b ^{2} – 4ac < 0, then there is no real number whose square is b ^{2} – 4ac. Therefore, there are no real roots for the given quadratic equation in this case.

Since b ^{2} – 4ac determines whether the quadratic equation ax^{2} + bx + c = 0 has real roots or not, b ^{2} – 4ac is called the discriminant of this quadratic equation.

So, a quadratic equation ax^{2} + bx + c = 0 has:

(i) two distinct real roots, if b ^{2} – 4ac > 0,

(ii) two equal real roots, if b ^{2} – 4ac = 0,

(iii) no real roots, if b ^{2} – 4ac < 0.

**Topics Deleted for Academic Year 2021-22 Board Exams: Situational problems based on equations reducible to quadratic equations**

The total number of exercises and questions in Quadratic Equations for Term II are tabulated below:

Exercise No & Name | Total Questions |

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 Answer) |

Total: 4 Exercises | 24 Questions |

### NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions (Term II)

** NCERT Solutions for Class 10 Maths Chapter 5:** Progressions are found in nature as well as in Mathematics. One of the most important ones is the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21,…). However, if we want to find a particular number in the sequence such as the 30th or 40th number, we have to use arithmetic progression. By using simple formulas of arithmetic progression it is possible to find numbers at specific places in a long series.

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

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

Reasons for studying Arithmetic Progression Derivation of the nth term and sum of the first n terms of A.P. and their real-life application in solving day to day problems.

(Applications based on the sum to n terms of an A.P. are excluded)

**Important Formulas –**

If a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6},_{…} are terms of an AP and d is the common difference between each term in the AP, we can write the sequence as; a,_{ }a+d, a+2d, a+3d, a+4d, a+5d,….,nth term… where a is the first term of the series. So, n^{th} term for arithmetic progression is given as;

**n ^{th} term = a + (n-1) d**

Sum of the first n terms in the Arithmetic Progression;

**S _{n} = n/2 [2a + (n-1) d]**

**Topics Deleted for Academic Year 2021-22 Board Exams: Application in solving daily life problems based on the sum to n terms**

The total number of exercises and questions in Arithmetic Progressions are tabulated below:

Exercise No & Name | Total Questions |

Exercise 5.1 | 4 Questions & Solutions (1 Short Answer, 3 Long Answers) |

Exercise 5.2 | 20 Questions & Solutions (10 Short Answers, 10 Long Answers) |

Exercise 5.3 | 20 Questions & Solutions (7 Short Answer, 13 Long Answers) |

Exercise 5.4 | 5 Questions & Solutions (5 Long Answers) |

Total: 4 Exercises | 49 Questions |

### NCERT Solutions for Class 10 Maths Chapter 6 Triangles (Term I)

** NCERT Solutions for Class 10 Maths Chapter 6:** Coming to geometry, we start with triangles. In this chapter, we will learn how similar triangles follow similarities in sizes and angles, and how we can determine similarities to solve problems. We will learn through exercises about how to determine such similarities and further learn how to even find areas of triangles that are similar.

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*Topics Covered in Class 10 Maths Chapter 6 Triangles for Term I:*

*Topics Covered in Class 10 Maths Chapter 6 Triangles for Term I:*

The important topics covered in this chapter are definitions, examples, counterexamples of similar triangles. Moreover,

1. We need to prove that 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.

2. Also the converse of this theorem should also be proven, that is, if a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.

3. If the corresponding angles are equal in two triangles are equal, their corresponding sides are proportional and the triangles are similar.

4. The converse is also true, that is, if the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.

5. Moreover, if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.

6. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse of the triangle, the triangles on each side of the perpendicular will be right triangles which are similar to the whole triangle and to each other.

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

8. The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

9. If the square on one side of a triangle is equal to the sum of the squares on the other two sides, the angle opposite to the first side is a right angle.

**Important Theorems –**

**Theorem 6.1:** If we draw a line 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, 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.

**Topics Deleted for Academic Year 2021-22 Board Exams: Proof of the following theorems are deleted**

- The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
- In a triangle, if the square on one side is equal to sumof the squares on the other two sides, the angle opposite to the first side is a right angle.

The number of exercises and questions included in the Triangles chapter are tabulated below:

Exercise No & Name | Total Questions |

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) |

Total: Exercises 6 | 65 Questions |

### NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry (Term I)

**NCERT Solutions for Class 10 Maths****Chapter 7: **Furthering the knowledge of triangles, this chapter teaches how we can find the distance between two points through the distance formula for triangles. Similarly, we will also learn how to find areas of triangles through a sectional formula for triangles.

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

LINES (In two dimensions)

Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. Section formula (internal division)

**Important Formulas**:

Distance Formula

Section Formula

**Topics Deleted for Academic Year 2021-22 Board Exams: Area of a triangle**

The number of exercises and questions included in the coordinate geometry chapter are tabulated below:

Exercise No & Name | Total Questions |

Exercise 7.1 | 10 Questions & Solutions (3 Short Answers, 7 Long Answer) |

Exercise 7.2 | 10 Questions & Solutions (2 Short Answers, 8 Long Answer) |

Exercise 7.3 | 5 Questions & Solutions (2 Short Answers, 3 Long Answer) |

Exercise 7.4 | 8 Questions & Solutions (3 Short Answers, 5 Long Answer) |

Total Exercises 4 | 23 Questions |

### NCERT Solutions for Class 10 Maths Chapter 8 Introduction To Trignometry (Term I)

**NCERT Solutions for Class 10 Maths****Chapter 8:** In Class 10, we finally get to trigonometry – one of the most interesting and practical concepts in the book. Trigonometry is the measurement (metric) of triangles (trigono). Through limited information on angles/distances, formulas in trigonometry help us find areas and distances.

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

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

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined). Values of the trigonometric ratios of 30^{0}, 45^{0} and 60^{0}. Relationships between the ratios.**Trigonometric Identities**Proof and applications of the identity sin

^{2}A + cos

^{2}A = 1.

**Important Formulas:**

Trigonometry Maths formulas for Class 10 cover three major functions Sine, Cosine and Tangent for a right-angle triangle. Let a right-angled triangle ABC is right-angled at point B and have ∠θ.

**Trigonometry Table**

Angle | 0° | 30° | 45° | 60° | 90° |

Sinθ | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cosθ | 1 | √3/2 | 1/√2 | ½ | 0 |

Tanθ | 0 | 1/√3 | 1 | √3 | Undefined |

Cotθ | Undefined | √3 | 1 | 1/√3 | 0 |

Secθ | 1 | 2/√3 | √2 | 2 | Undefined |

Cosecθ | Undefined | 2 | √2 | 2/√3 | 1 |

**Trigonometric Ratios of Complementary Angles**

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

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°

**Topics Deleted for Academic Year 2021-22 Board Exams: Motivate the ratios whichever are defined at 0o and 90o**

The number of exercises and questions included in the introduction to the trigonometry chapter are tabulated below:

Exercise No & Name | Total Questions |

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) |

Total Exercises 4 | 27 Questions |

### NCERT Solutions for Class 10 Maths **Chapter 9 Some Applications of Trigonometry** (Term II)

** NCERT Solutions for Class 10 Maths Chapter 9:** After learning the basics of trigonometry, we will delve into some practical applications of trigonometry. Through the use of trigonometry, we will try to solve practical problems by using concepts such as the line of sight, angle of depression, and angle of elevation to determine height or distance.

HEIGHTS AND DISTANCES-Angle of elevation, Angle of Depression.

Simple problems with heights and distances. Problems should not involve more than two right triangles. Angles of elevation/depression should be only 30°, 45°, 60°.

**Important Points**:

The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object.

The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed.

You would need to know the following:

(i) the distance DE at which the student is standing from the foot of the minar

(ii) the angle of elevation, ∠ BAC, of the top of the minar

(iii) the height AE of the student.

Assuming that the above three conditions are known, how can we determine the height of the minar?

In the figure, CD = CB + BD. Here, BD = AE, which is the height of the student.

To find BC, we will use trigonometric ratios of ∠ BAC or ∠ A.

In ∆ ABC, the side BC is the opposite side in relation to the known ∠ A. Our search narrows down to using either tan A or cot A, as these ratios involve AB and BC.

Therefore, tan A = BC/AB or cot A = AB/BC, which on solving would give us BC.

By adding AE to BC, you will get the height of the minar.

The number of exercises and questions included in some applications of trigonometry chapter are tabulated below:

Exercise No & Name | Total Questions |

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

Total Exercises: 1 | 16 Questions |

### NCERT Solutions for Class 10 Maths Chapter 10 Circles (Term II)

** NCERT Solutions for Class 10 Maths Chapter 10**: Students learn about elements related to a circle such as a chord, arc, etc. in Class 9. In this chapter, students will learn about tangents and the different scenarios when lines touch or bisect circles on a given plane. How a line interacts with a circle opens up different possibilities and problems and students will learn how to solve them.

*Topics Covered in Class 10 Maths Chapter 10 Circles for Term II:*

*Topics Covered in Class 10 Maths Chapter 10 Circles for Term II:*

Tangent to a circle at, point of contact

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.

2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.

**Important Theorems**:

**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.

The total number of exercises and questions included in the Circles chapter are given below:

Exercise No & Name | Total Questions |

Exercise 10.1 | 4 Questions & Solutions (2 Short Answer, 2 Long Answers) |

Exercise 10.2 | 13 Questions & Solutions (2 Short Answers, 14 Long Answers) |

Total Exercises 2 | 17 Questions |

### NCERT Solutions for Class 10 Maths Chapter 11 Constructions (Term II)

**NCERT Solutions for Class 10 Maths****Solutions:** In this chapter on Construction, we will apply the rules of geometry learned so far, and construct shapes from dimensions and measurements provided. Using tools such as a ruler, pencil, and protractor, students will form shapes according to different problems in the exercises. Referring to Construction Class 10 PDF solutions is essential for this chapter to check if the structure created by students is as intended in the question.

*Topics Covered in Class 10 Maths Chapter 11 Constructions for Term II:*

*Topics Covered in Class 10 Maths Chapter 11 Constructions for Term II:*

1. Division of a line segment in a given ratio (internally).

2. Tangents to a circle from a point outside it.

**Important Points**:

**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.

**Topics Deleted for Academic Year 2021-22 Board Exams: Construction of a triangle similar to a given triangle.**

The total number of exercises and questions included in the constructions chapter are given below:

Exercise No & Name | Total Questions |

Exercise 11.1 | 7 Questions & Solutions (7 Long Answers) |

Exercise 11.2 | 7 Questions & Solutions (7 Long Answers) |

Total Exercises 2 | 14 Questions |

### NCERT Solutions for Class 10 Maths Chapter 12 Areas Related To Circles (Term I)

**NCERT Solutions for Class 10 Maths****Chapter 12:** Circles are some of the commonly found shapes in our surroundings. By merging lines and other shapes within circles, students will learn how to measure the areas of different segments using theorems and the number Pi.

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

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

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above-said plane figures. (In calculating the area of a segment of a circle, problems should be restricted to the central angle of 60° and 90° only. Plane figures involving triangles, simple quadrilaterals and circles should be taken.)

**Important Formulas**:

circumference = 2πr

area of the circle = πr ^{2}

Area of the sector of angle θ = (θ/360) × π r^{2}

Length of an arc of a sector of angle θ = (θ/360) × 2 π r where r is the radius of the circle

**Topics Deleted for Academic Year 2021-22 Board Exams: Problems on the central angle of 120°**

The total number of questions and exercises under the chapter Areas Related To Circles are tabulated below:

Exercise No & Name | Total Questions |

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) |

Total Exercises: 3 | 35 Questions |

### NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas And Volumes (Term II)

**NCERT Solutions for Class 10 Maths****Chapter 13:** After looking at shapes on a plane, we move on to a 3-dimensional world and look at shapes like squares or rectangles as cubes or cuboids, triangles as cones, and circles as cylinders or spheres. We will learn in this chapter how to view objects as shapes and solve practical problems such as finding surface area and volume as mentioned in the exercises.

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

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

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

2. Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken).

**Important Formulas –**

TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere

Diameter of sphere = 2r

Surface area of sphere = 4 π r^{2}

Volume of Sphere = 4/3 π r^{3}

Curved surface area of Cylinder = 2 πrh

Area of two circular bases = 2 πr^{2}

Total surface area of Cylinder = Circumference of Cylinder + Curved surface area of Cylinder = 2 πrh + 2 πr^{2}

Volume of Cylinder = π r^{2 }h

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

Curved surface area of cone = πrl

Total surface area of cone = πr (l + r)

Volume of cone = ⅓ π r^{2 }h

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

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

Total surface area of cuboid = 2(l×b + b×h + l×h)

Volume of Cuboid = l × b × h

**Topics Deleted for Academic Year 2021-22 Board Exams: Frustum of a cone**

The total number of questions and exercises in the chapter surface areas and volumes are given below:

Exercise No & Name | Total Questions |

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) |

Total Exercises 5 | 38 Questions |

### NCERT Solutions for Class 10 Maths Chapter 14 Statistics (Term II)

** NCERT Solutions for Class 10 Maths Chapter 14:** Used holistically in business, economics, and many other areas, we will learn how to find mean (average), median, and mode from a grouped dataset. After learning that, we will also study how to portray data in a graphical manner to understand trends and correlations within data.

*Topics Covered in Class 10 Maths Chapter 14 Statistics for Term II:*

*Topics Covered in Class 10 Maths Chapter 14 Statistics for Term II:*

Mean, median and mode of grouped data (bimodal situation to be avoided). Mean by Direct Method and Assumed Mean Method only.

**Important Formulas –**

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

1. **Direct Method: x̅** =

, where ∑f_{i }x_{i }is the sum of observations from value i = 1 to n And ∑f_{i }is the number of observations from value i = 1 to n

2. **Assumed mean method**: **x̅** =

3. **Step deviation method: x̅** =

**The mode of grouped data:**

Mode =

**The median for grouped data:**

Median =

**Topics Deleted for Academic Year 2021-22 Board Exams: Step deviation Method for finding the mean, Cumulative Frequency graph**

Exercise No & Name | Total Questions |

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) |

Total Exercises 4 | 25 Questions |

### NCERT Solutions for Class 10 Maths Chapter 15 Probability (Term I)

**NCERT Solutions for Class 10 Maths****Chapter 15:** From weather, sports, politics, to insurance, the probability is used in a multitude of areas. Students will learn some theoretical aspects of probability and apply them to determine impossible events, certain and uncertain events. Conclusively, the concepts of probability in the chapter will allow students to learn how to determine the outcomes of events through formulas.

**Topics Covered in Class 10 Maths Chapter 15 Probability for Term I:**

- The classical definition of probability
- Simple problems on finding the probability of an event

**Important Formulas**:

- The theoretical probability (also called classical probability) of an event E, written as P(E), is defined as:

where we assume that the outcomes of the experiment are equally likely.

- The probability of a sure event (or certain event) is 1.
- The probability of an impossible event is 0.
- The probability of an event E is a number P(E) such that 0 ≤ P (E) ≤ 1
- An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

Exercise No & Name | Total Questions |

Exercise 15.1 | 25 Questions & Solutions (22 Short Answers, 3 Long Answers) |

Exercise 15.2 | 5 Questions & Solutions (5 Short Answers) |

Total Exercises 2 | 30 Questions |

*Also, Download:*

**NCERT Solutions Class 10 Science**- NCERT Solutions Class 10 Social Science
- NCERT Solutions Class 10 English

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**Download** – **Algebra Formulas for Class 10**

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### FAQs

*Q1.* Which is the best website for NCERT Maths Solution Class 10?** Ans:** Embibe is the best website for detailed NCERT Maths Solutions for Class 10. Our experts at Embibe have created a step by step solutions in simple and easy language. Students searching for NCERT Solutions for Class 10 Maths can read this article and download Free NCERT Maths Book Class 10 Solutions from this article.

**Q2. Where can I download NCERT Solutions for Class 10 Maths PDF?**** Ans:** NCERT Solution for Class 10 Maths is available on this article free of cost. You can download it for later use.

**Q3. How can I save these Class 10 NCERT Maths Solutions for reference?**** Ans:** You can download the PDF files through the respective chapter links and keep them on your computer or mobile phone for future reference. You can also print these files if you prefer that.

**Q4. How many chapters does the NCERT Maths textbook have?**** Ans: **The NCERT Maths textbook for Class 10 features 15 chapters.

**Q5. How can I easily learn formulas for all chapters in NCERT Class 10 textbook?**** Ans:** You can easily refer to this article on

**Maths formulas for Class 10**. It covers the entire syllabus and contains all the formulas that are important for the exam.

**Q6. Why is it important to refer to NCERT 10 Maths Solutions?*** Ans:* Maths NCERT Class 10 Solutions for Class 10 exercise questions by Embibe provide a detailed and step-by-step explanation. Keeping the concepts in mind, they are easy to understand and handy to practice.

**Q7. How can I score maximum marks in the 10th Maths exam?**** Ans: **The most important thing is to practice sums and problems every day. Pay special attention to units like Algebra, Trigonometry, Statistics, and Probability as these topics have the highest weightage and can help you score maximum marks in the exam.

**Q8. What are the benefits of CBSE Class 10 Maths solutions?**** Ans:** The benefits of having a Class 10 Maths NCERT solution are that you can refer to them later in offline mode. You don’t have to wait for your tutors to get your doubts cleared.

**Q9. Is NCERT Maths textbook enough for Class 10 Board Exams?**** Ans:** Yes, for the Class 10 board exam, the NCERT maths textbook is enough. However, you must also solve previous year papers and sample papers.

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