## NCERT Solutions for Class 10 Maths Updated for 2023-24 Session – Free PDF Download

**NCERT Solutions for Class 10 Maths**for all the exercises from Chapters 1 to 15 are provided here. These NCERT Solutions are curated by our expert faculty to help students in their exam preparations. Students looking for the NCERT Solutions of Class 10 Maths can download all chapter-wise PDFs to find a better approach to solving the problems.

The answers to the questions present in the NCERT books are undoubtedly the best study material a student can get hold of. These **CBSE NCERT Solutions for Class 10 **Maths 2023-24 will also help students to build a deeper understanding of concepts covered in the textbook. Practising the textbook questions will help students analyse their level of preparation and knowledge of concepts. The solutions to these questions present in the NCERT books can help students to clear their doubts quickly.

NCERT Solutions Class 10 Maths Chapters NCERT Solutions Class 10 Maths Chapter DetailsCBSE Class 10 Maths Exam Pattern 2023-2024CBSE Class 10 Chapter-wise Marks Weightage CBSE Question Paper Design

## 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 Areas Related to Circles****Chapter 12 Surface Areas and Volumes****Chapter 13 Statistics****Chapter 14 Probability**

The following are the chapters that have been removed from the NCERT Class 10 Maths textbook 2023-24.

Constructions

### NCERTSolutions for Class 10 Maths Free PDF Download

NCERT Solutions of Class 10 Maths list comprises all the chapter-wise answers to the questions present in the **NCERT Book for Class 10 Maths,** written in a very precise and lucid manner, maintaining the objective of textbooks. The students can refer to theNCERT Solutions for Class 10 as their additional references and study materials. Practising NCERT textbook exercise solutions will surely help the students in their preparation for the examination.

## NCERT Solutions Class 10 Maths Chapter Details and Exercises

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

In Chapter 1 of Class 10, students will explore real numbers and irrational numbers. The chapter starts with the Euclid’s Division Lemma, which states that “Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0≤r<b”. The Euclid’s Division algorithm is based on this lemma and is used to calculate the HCF of two positive integers. Then, the Fundamental Theorem of Arithmetic is defined, which is used to find the LCM and HCF of two positive integers. After that, the concept of an irrational number, a rational number and the decimal expansion of rational numbers are explained with the help of the theorem.

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

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

Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5

**Important Steps –**

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

**Step 1:** Apply Euclid’s division lemma to c and d. So, we find whole numbers, q and r, such that c = dq + r, 0 ≤ r < d.

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

**Step 3:** Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. This algorithm works because HCF (c, d) = HCF (d, r), where the symbol HCF (c, d) denotes the HCF of c and d, etc.

Class 10 Maths NCERT Solutions PDF Chapter 1 Exercises |
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☞NCERT Solutions Real Numbers Class 10Exercise 1.1 – 5 Questions (4 Long Answers, 1 Short Answer) |

☞NCERT Solutions Real Numbers Class 10 Exercise 1.2 – 7 Questions (4 Long Answers, 3 Short Answers) |

☞NCERT Solutions Real Numbers Class 10 Exercise 1.3 – 3 Questions (3 Short Answers) |

☞NCERT Solutions Real Numbers Class 10 Exercise 1.4 – 3 Questions (3 Short Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 1 Real Numbers, at BYJU’S:*

- NCERT Real Numbers Class 10 Notes
- Real Numbers Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10Real Numbers
- RD Sharma SolutionsReal Numbers Class 10

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

In Polynomials, the chapter begins with the definition of the degree of the polynomial, linear polynomial, quadratic polynomial and cubic polynomial. This chapter has a total of 4 exercises, including an optional exercise. Exercise 2.1 includes questions on finding the number of zeroes through a graph. It requires an understanding of the Geometrical Meaning of the Zeroes of a Polynomial. Exercise 2.2 is based on the Relationship between Zeroes and Coefficients of a Polynomial, where 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 concept of the division algorithm is defined, and students will find the questions related to it. The optional exercise, 2.4, consists of the questions from all the concepts of Chapter 2.

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

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

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

**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:** To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. Then carry out the division process.

**Step 2:** Now, to obtain the second term of the quotient, divide the highest degree term of the new dividend by the highest degree term of the divisor. Again, carry out the division process.

**Step 3:** Now, the degree of the remainder is less than the degree of the divisor. So, we cannot continue the division any further.

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

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 result is known as the Division Algorithm for polynomials.

Class 10 Maths NCERT Solutions PDF Chapter 2 Exercises |
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☞NCERT Solutions Polynomials Class 10 Exercise 2.1 – 1 Question (1 Short Answer) |

☞NCERT Solutions Polynomials Class 10 Exercise 2.2 – 2 Questions (2 Short Answers) |

☞NCERT Solutions Polynomials Class 10 Exercise 2.3 – 5 Questions (2 Short Answers, 3 Long Answers) |

☞NCERT Solutions Polynomials Class 10 Exercise 2.4 – 5 Questions (2 Short Answers, 3 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 2 Polynomials, at BYJU’S:*

- NCERTPolynomials Class 10 Notes
- Polynomials Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Polynomials
- RD Sharma Solutions Polynomials Class 10

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

This chapter explains the concept of Pair of Linear Equations in Two Variables. This chapter has a total of 7 exercises, and in these exercises, different methods of solving the pair of linear equations are described. Exercise 3.1 describes how to represent a situation algebraically and graphically. Exercise 3.2 explains the methods of solving the pair of the linear equation through the Graphical Method. Exercises 3.3, 3.4, 3.5 and 3.6 describe the Algebraic Method, Elimination Method, Cross-Multiplication Method, and Substitution Method, respectively. Exercise 3.7 is an optional exercise which contains all types of questions. Students must practise these exercises to master the method of solving linear equations.

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

*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 number of solutions. Solution of a pair of linear equations in two variables algebraically – by substitution, by elimination. Simple situational problems.

**Important Formulas –**

The general form for a pair of linear equations in two variables, x and y, is

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 all real numbers and a_{1} ^{2} + b_{1} ^{2} ≠ 0, a_{2} ^{2} + b_{2} ^{2} ≠ 0.

Class 10 Maths NCERT Solutions PDF Chapter 3 Exercises |
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☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.1 – 3 Questions (2 Short Answers, 1 Long Answer) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.2 – 7 Questions (5 Short Answers, 2 Long Answers) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.3 – 3 Questions (2 Short Answers, 1 Long Answer) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.4 – 2 Questions (2 Long Answers) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.5 – 4 Questions (4 Short Answers) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.6 – 2 Questions (2 Long Answers) |

☞NCERT Solutions Pair of Linear Equations in Two Variables Class 10 Maths Exercise 3.7 – 8 Questions (1 Short Answer, 7 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 3 Pair of Linear Equations in Two Variables, at BYJU’S:*

- NCERT Pair of Linear Equations in Two Variables Class 10 Notes
- Pair of Linear Equations in Two Variables Class 10 Important Questions,
- NCERT Exemplar SolutionsClass 10 Pair of Linear Equations in Two Variables
- RD Sharma Solutions Pair of Linear Equations in Two Variables Class 10

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

In this chapter, students will get to know the standard form of writing a Quadratic Equation. The chapter goes on to explain the method of solving the quadratic equation through the factorization method and completing the square method. The chapter ends with the topic on 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, ifb² – 4ac < 0

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

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

The standard form of a quadratic equation ax^{2} + bx + c = 0, (a ≠ 0). Solutions of quadratic equations (only real roots) by factorization and by using the quadratic formula. Relationship between discriminant and nature of roots.

Situational problems based on quadratic equations related to day-to-day activities are to be incorporated.

**Important Formulas –**

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

\(\begin{array}{l}-\frac{b}{2a}+\frac{\sqrt{b^{2}-4ac}}{2a}\: and\: -\frac{b}{2a}-\frac{\sqrt{b^{2}-4ac}}{2a}\end{array} \)

If b ^{2} – 4ac = 0, then

\(\begin{array}{l}\\x=-\frac{b}{2a}\pm 0 \\ \\x=-\frac{b}{2a}\: or-\frac{b}{2a}\end{array} \)

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.

Class 10 Maths NCERT Solutions PDF Chapter 4 Exercises |
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☞NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.1 – 2 Questions (1 Short Answer, 1 Long Answer) |

☞NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.2 – 6 Questions (6 Short Answers) |

☞NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.3 – 11 Questions (8 Short Answers, 3 Long Answers) |

☞NCERT Solutions Maths Quadratic Equations Class 10 Exercise 4.4 – 5 Questions (2 Short Answers, 3 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 4 Quadratic Equations, at BYJU’S:*

- NCERT Quadratic Equations Class 10 Notes
- Quadratic Equations Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Quadratic Equations
- RD Sharma Solutions Quadratic Equations Class 10

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

This chapter introduces students to a new topic, Arithmetic Progression, i.e. AP. The chapter constitutes a total of 4 exercises. In Exercise 5.1, students will find the questions related to representing a situation in the form of AP, finding the first term and difference of an AP, finding out whether a series is AP or not. Exercise 5.2 includes the questions on finding out the nth term of an AP by using the following formula:

a_{n}= a + (n-1) d

The next exercise, i.e. 5.3, contains the questions on finding the sum of the first n terms of an AP. The last exercise includes higher-level questions based on AP to enhance students’ analytical and problem-solving skills.

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

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

Motivation for studying Arithmetic Progression Derivation of the n^{th} term and sum of the first n terms of A.P. and their application in solving daily life problems.

**Important Formulas –**

If a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6},_{…}are the terms of AP and d is the common difference between each term, then 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. Now, n^{th}term for arithmetic progression is given as;

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

Sum of the first n termsin Arithmetic Progression;

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

Class 10 Maths NCERT Solutions PDF Chapter 5 Exercises |
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☞NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.1 – 4 questions (1 Short Answer, 3 Long Answers) |

☞NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.2 – 20 questions (10 Short Answers, 10 Long Answers) |

☞NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.3 – 20 Questions (7 Short Answers, 13 Long Answers) |

☞NCERT Solutions Class 10 Maths Arithmetic Progressions Exercise 5.4 – 5 Questions (5 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 5 Arithmetic Progressions, at BYJU’S:*

- NCERT Arithmetic Progressions Class 10 Notes
- Arithmetic Progressions Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Arithmetic Progressions
- RD Sharma Solutions Arithmetic Progressions Class 10

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

In Chapter 6 of Class 10 CBSE Maths, students will study those figures which have the same shape, but not necessarily the same size. The chapter Triangles starts with the concept of a similar and congruent figure. It further explains the condition for the similarity of two triangles and theorems related to the similarity of triangles. After that, the areas of similar triangles have been explained with a theorem. At the end of this chapter, the Pythagoras Theorem and the Converse of Pythagoras Theorem are described.

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

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

Definitions, examples, and counter examples of similar triangles.

1. (Prove) 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. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.

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

4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal, and the two triangles are similar.

5. (Motivate) 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.

**Important Theorems –**

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.

Class 10 Maths NCERT Solutions PDF Chapter 6 Exercises |
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☞Triangles Class 10 Exercise 6.1 – 3 Questions (3 Short Answers) |

☞Triangles Class 10 Exercise 6.2 – 10 Questions (9 Short Answers, 1 Long Answer) |

☞Triangles Class 10 Exercise 6.3 – 16 Questions (12 Short Answers, 4 Long Answers) |

☞Triangles Class 10 Exercise 6.4 – 9 Questions (7 Short Answers, 2 Long Answers) |

☞Triangles Class 10 Exercise 6.5 – 17 Questions (15 Short Answers, 2 Long Answers) |

☞Triangles Class 10 Exercise 6.6 – 10 Questions (5 Short Answers, 5 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 6 Triangles, at BYJU’S:*

- NCERT Triangles Class 10 Notes
- Triangles Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Triangles
- RD Sharma Solutions Triangles Class 10

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

In this chapter, students will learn how to find the distance between two points whose coordinates are given, and to find the area of the triangle formed by three given points. Along with this, students will also study how to find the coordinates of the point which divides a line segment joining two given points in a given ratio. For this purpose, students will get introduced to the Distance Formula, Section Formula and Area of a Triangle in this chapter on Coordinate Geometry.

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

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

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

**Important Formulas –**

Distance Formula

\(\begin{array}{l}PQ=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\end{array} \)

Section Formula

\(\begin{array}{l}m_{1}:m_{2}=(\frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},\frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}})\end{array} \)

\(\begin{array}{l}Area\: of\: Triangle=\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]\end{array} \)

Class 10 Maths NCERT Solutions PDF Chapter 7 Exercises |
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☞Coordinate Geometry Class 10 Exercise 7.1 – 10 Questions (3 Short Answers, 7 Long Answers) |

☞Coordinate Geometry Class 10 Exercise 7.2 – 10 Questions (2 Short Answers, 8 Long Answers) |

☞Coordinate Geometry Class 10 Exercise 7.3 – 5 Questions (2 Short Answers, 3 Long Answers) |

☞Coordinate Geometry Class 10 Exercise 7.4 – 8 Questions (3 Short Answers, 5 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 7 Coordinate Geometry, at BYJU’S:*

- NCERT Coordinate Geometry Class 10 Notes
- Coordinate Geometry Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Coordinate Geometry
- RD Sharma Solutions Coordinate Geometry Class 10

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

This chapter will introduce students to Trigonometry. They will study some ratios of a right triangle with respect to its acute angles, called trigonometric ratios of the angles. The chapter also defines the trigonometric ratios for angles of 0^{0} and 90^{0}. Further, students will also know how to calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.

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

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

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0^{o} and 90^{o}. 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. Only simple identities to be given

**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.

Sin θ=

\(\begin{array}{l}\frac{Side\, opposite\, to\, angle\, \theta}{Hypotenuse}\end{array} \)

=\(\begin{array}{l}\frac{Perpendicular}{Hypotenuse}\end{array} \)

= P/HCos θ =

\(\begin{array}{l}\frac{Adjacent\, side\, to\, angle\, \theta}{Hypotenuse}\end{array} \)

=\(\begin{array}{l}\frac{Base}{Hypotenuse}\end{array} \)

= B/HTan θ =

\(\begin{array}{l}\frac{Side\, opposite\, to\, angle\, \theta}{Adjacent\, side\, to\, angle\, \theta}\end{array} \)

= P/BSec θ =

\(\begin{array}{l}\frac{1}{cos\, \theta }\end{array} \)

Cot θ =

\(\begin{array}{l}\frac{1}{tan\, \theta }\end{array} \)

Cosec θ =

\(\begin{array}{l}\frac{1}{sin\, \theta }\end{array} \)

Tan θ =

\(\begin{array}{l}\frac{Sin\, \theta }{Cos\, \theta }\end{array} \)

**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°

Class 10 Maths NCERT Solutions PDF Chapter 8 Exercises |
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☞Introduction to Trigonometry Class 10 Exercise 8.1 – 11 Questions (8 Short Answers, 3 Long Answers) |

☞Introduction to Trigonometry Class 10 Exercise 8.2 – 4 Questions (2 Short Answers, 2 Long Answers) |

☞Introduction to Trigonometry Class 10 Exercise 8.3 – 7 Questions (5 Short Answers, 2 Long Answers) |

☞Introduction to Trigonometry Class 10 Exercise 8.4 – 5 Questions (3 Short Answers, 2 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 8 Trigonometry, at BYJU’S:*

- NCERT Introduction to Trigonometry Class 10 Notes
- Introduction to Trigonometry Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Introduction to Trigonometry
- RD Sharma Solutions Introduction to Trigonometry Class 10

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

This chapter is the continuation of the previous chapter; here, the students will study the applications of trigonometry. It is used in geography, navigation, construction of maps, and determining the position of an island in relation to the longitudes and latitudes. In this chapter, students will see how trigonometry is used for finding the heights and distances of various objects without actually measuring them. They will get introduced to the term line of sight, angle of elevation, and angle of depression.

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

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

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

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

Class 10 Maths NCERT Solutions PDF Chapter 9 Exercise |
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☞Some Applications of Trigonometry Class 10 Exercise 9.1 – 16 Questions (16 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 9, Some Applications of Trigonometry, at BYJU’S:*

- NCERT Some Applications of Trigonometry Class 10 Notes
- Some Applications of Trigonometry Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Some Applications of Trigonometry
- RD Sharma Solutions Some Applications of Trigonometry Class 10

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

In earlier classes, students have studied about a circle and various terms related to a circle, such as a chord, segment, arc, etc. In this chapter, students will study the different situations that arise when a circle and a line are given in a plane. So, they will get thorough with the concept of Tangent to a Circle and Number of Tangents from a Point on a Circle.

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

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

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.

Class 10 Maths NCERT Solutions PDF Chapter 10 Exercises |
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☞Circles Class 10 Exercise 10.1 – 4 Questions (2 Short Answer, 2 Long Answers) |

☞Circles Class 10 Exercise 10.2 – 13 Questions (2 Short Answers, 14 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 10 Circles at BYJU’S:*

- NCERT Circles Class 10 Notes
- Circles Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Circles
- RD Sharma Solutions Circles Class 10

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

This chapter consists of a total of 2 exercises. Whatever students have learned about construction in earlier classes will also help them. In Exercise 11.1, students will study how to divide a line segment, whereas in Exercise 11.2, they will study the construction of tangents to a circle. Methods and steps for construction are explained, and also some examples are additionally given to make it clearer to the students.

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

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

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

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

3. Construction of a triangle similar to a given triangle.

**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.

Class 10 Maths NCERT Solutions PDF Chapter 11 Exercises |
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☞Constructions Class 10 Exercise 11.1 – 7 Questions (7 Long Answers) |

☞Constructions Class 10 Exercise 11.2 – 7 Questions (7 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 11 Constructions, at BYJU’S:*

- NCERT Constructions Class 10 Notes
- Constructions Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Constructions
- RD Sharma Solutions Constructions Class 10

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

This chapter begins with the concepts of the perimeter and area of a circle. Using this concept, the chapter further explains how to find the area of sector and segment of a circular region. Moreover, students will get clarity on findingthe areas of some combinations of plane figures involving circles or their parts.

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

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

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 segment of a circle, problems should be restricted to the central angle of 60°, 90° and 120° only.

**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

Class 10 Maths NCERT Solutions PDF Chapter 12 Exercises |
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☞Areas Related to Circles Class 10 Exercise 12.1 – 5 Questions (5 Short Answers) |

☞Areas Related to Circles Class 10 Exercise 12.2 – 14 Questions (9 Short Answers, 5 Long Answers) |

☞Areas Related to Circles Class 10 Exercise 12.3 – 16 Questions (9 Short Answers, 7 Long Answers) |

*Also access the following resources for NCERT Class 10 Chapter 12 Areas Related to Circles at BYJU’S:*

- NCERT Areas Related to Circles Class 10 Notes
- Areas Related to Circles Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Areas Related to Circles
- RD Sharma Solutions Areas Related to Circles Class 10

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

In Chapter 13, there are a total of 5 exercises. The first exercise consists of questions based on finding the surface area of an object formed by combining any two of the basic solids, i.e. cuboid, cone, cylinder, sphere and hemisphere. In exercise 13.2, questions are based on finding the volume of objects formed by combining any two of a cuboid, cone, cylinder, sphere and hemisphere. Exercise 13.3 deals with the questions in which a solid is converted from one shape to another. Exercise 13.4 is based on finding the volume, curved surface area and total surface area of a frustum of a cone. The last exercise is optional and has high-level questions based on all the topics of this chapter.

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

*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 –**

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

Class 10 Maths NCERT Solutions PDF Chapter 13 Exercises |
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☞Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.1 – 9 Questions (2 Short Answers, 7 Long Answers) |

☞Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.2 – 8 Questions (1 Short Answer, 7 Long Answers) |

☞Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.3 – 9 Questions (9 Long Answers) |

☞Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.4 – 5 Questions (5 Long Answers) |

☞Solutions of NCERT Maths Class 10 Surface Areas and Volumes Exercise 13.5 – 7 Questions (7 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 13 Surface Areas and Volumes, at BYJU’S:*

- NCERT Surface Areas and Volumes Class 10 Notes
- Surface Areas and Volumes Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Surface Areas and Volumes
- RD Sharma Solutions Surface Areas and Volumes Class 10

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

Here, students will learn the numerical representation of ungrouped data to grouped data and find the Mean, Mode and Median. Also, the concept of cumulative frequency, cumulative frequency distribution and how to draw cumulative frequency curves will be explained.

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

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

Mean, median and mode of grouped data (bimodal situation to be avoided).

**Important Formulas –**

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

**Direct Method: x̅**=\(\begin{array}{l}\frac{\sum_{i=1}^{n}f_i x_i}{\sum_{i=1}^{n}f_i}\end{array} \)

, 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**Assumed mean method**:**x̅**=\(\begin{array}{l}a+\frac{\sum_{i=1}^{n}f_i d_i}{\sum_{i=1}^{n}f_i}\end{array} \)

**Step deviation method : x̅**=\(\begin{array}{l}a+\frac{\sum_{i=1}^{n}f_i u_i}{\sum_{i=1}^{n}f_i}\times h\end{array} \)

**The mode of grouped data:**

Mode =

\(\begin{array}{l}l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2} \times h\end{array} \)

**The median for a grouped data:**

Median =

\(\begin{array}{l}l+\frac{\frac{n}{2} – cf}{f} \times h\end{array} \)

Class 10 Maths NCERT Solutions PDF Chapter 14 Exercises |
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☞NCERT Solutions of Maths Class 10 Statistics Exercise 14.1 – 9 Questions (9 Long Answers) |

☞NCERT Solutions of Maths Class 10 Statistics Exercise 14.2 – 6 Questions (6 Long Answers) |

☞NCERT Solutions of Maths Class 10 Statistics Exercise 14.3 – 7 Questions (7 Long Answers) |

☞NCERT Solutions of Maths Class 10 Statistics Exercise 14.4 – 3 Questions (3 Long Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 14 Statistics, at BYJU’S:*

- NCERT Statistics Class 10 Notes
- Statistics Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Statistics
- RD Sharma Solutions Statistics Class 10

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

The last chapter deals with Probability. The chapter starts with the theoretical approach of probability. Subsequently, the chapter explains the difference betweenexperimental probability and theoretical probability. There are various examples given toexplain it in an effective way. So, before going through the exercise problems, students must solve the examples of CBSE Maths first.

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

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

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

\(\begin{array}{l}P(E)= \frac{Number\: of\: outcomes\: favourable\: to\: E}{Number\: of\: all\: possible\: outcomes\: of\: the\: experiment}\end{array} \)

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.

Class 10 Maths NCERT Solutions PDF Chapter 15 Exercises |
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☞NCERT Solutions of Maths Class 10 Probability Exercise 15.1 – 25 Questions (22 Short Answers, 3 Long Answers) |

☞NCERT Solutions of Maths Class 10 Probability Exercise 15.2 – 5 Questions (5 Short Answers) |

*Also, access the following resources for NCERT Class 10 Chapter 15 Probability, at BYJU’S:*

- NCERT Probability Class 10 Notes
- Probability Class 10 Important Questions
- NCERT Exemplar SolutionsClass 10 Probability
- RD Sharma Solutions Probability Class 10

** NCERT Solutions for Class 10 Maths PDF** in

*English Medium, as well as Hindi Medium (हिंदी मीडियम) for the academic year 2023-24,*are followed by not only CBSE but also UP Board, Uttarakhand board and all other boards following NCERT Textbooks.

*CBSE Class 10 Maths Chapter-wise Marks Weightage for 2023-24*

Unit | Unit Name | Marks |

I | Number Systems | 06 |

II | Algebra | 20 |

III | Coordinate Geometry | 06 |

IV | Geometry | 15 |

V | Trigonometry | 12 |

VI | Mensuration | 10 |

VII | Statistics and Probability | 11 |

Total | 80 | |

Internal Assessment | 20 | |

Total | 100 |

### Internal Assessment Class 10

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 ofNCERT Solutions Class 10 Maths:

The Class 10 **NCERT Solutions** of Maths given here in PDFs have several benefits, which include:

- The solutions given here are very easy-to-understand.
- Solutions are provided in steps for better understanding.
- Diagrams are given to help the students to visualise the solutions.
- All the questions from each chapter are covered.

Students are advised to prepare all the chapters covered in the solution modules, which will eventually help them to gain a deeper knowledge of concepts. For effective preparations, it is essential for students to understand all the steps provided in the solutions.

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

CBSE Class 10 Maths is an important subject for students. Here, we have provided complete assistance to students for their preparation. Class 10 is the first benchmark for any student that will reflect in future records of achievement. CBSE always prescribes the NCERT books for exam preparation.

Exam preparation is a rigorous process that requires an overall understanding of individual chapters. This process demands hard work towards studies and an effective approach to getting through the solutions. This significant role is played by **NCERT 10** Class Maths Solutions to equip students in preparation for competitive entrance exams. NCERT books are best known for putting forth the concepts in a simple way for better understanding. NCERT Class 10 Books for Mathematics are written in the most lucid and clear manner that helps to break the complex problems in the most efficient way.

Keep visiting to get complete chapter-wise **NCERT Solutions for Class 10 Maths PDF free download** for all the classes. Students can also find the NCERT Solutions for Class 10 Science, at BYJU’S. Students can download BYJU’S App to get a personalised learning experience and prepare for the exams more effectively.

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

Q1

### How to grasp the important concepts covered in the **NCERT Solutions for Class 10 Maths**?

Students who aspire to score good marks in the Class 10 exams are advised to download the NCERT Solutions from BYJU’S. The solutions are curated with utmost care by a set of faculty having vast experience in the respective subject. Each and every minute detail is explained in an interactive manner to make it easier for the students while learning. The step-wise solutions are designed by keeping in mind the marks weightage allotted as per the CBSE guidelines.

Q2

### Will the **NCERT Solutions for Class 10 Maths** help me to solve problems easily?

The Class 10 exam is one of the major turning points in the life of students. Mathematics is a subject which mainly contains numericals, and understanding of concepts helps to score more marks. Students are recommended to first understand the syllabus for the academic year and learn the concepts based on them for better performance. Solving the textbook problems by referring to the solutions PDF will improve the logical thinking and analytical skills among students.

Q3

### Where will I get the **NCERT Solutions for Class 10 Maths** PDF?

Students can get the NCERT Solutions for Class 10 Maths PDF at BYJU’S. The solutions are prepared by the subject matter experts keeping in mind the understanding capacity of students. The main objective of creating solutions is to help students learn complex concepts without any difficulty. Students can also cross-check their answers while solving the textbook problems to get an idea about the other methods of answering the questions efficiently.

Q4

### Are the **NCERT Solutions for Class 10 Maths** useful for the students under the CBSE board?

Given below are some of the uses of the **NCERT Maths** Solutions for Class 10 –

1. It builds a strong foundation of fundamental concepts and improves confidence to face exams.

2. The method of solving complex problems can be learnt easily.

3. It is a perfect study material for the students to finish their assignments on time and to score more marks in the exams.

Q5

### How to get an idea about the important concepts in the **NCERT Solutions for Class 10 Maths**?

Before the start of an academic year, students should download the CBSE syllabus and go through it for a clear idea about the concepts which are important for the exams. Students must solve the NCERT Textbook problems by choosing the correct study material which satisfies their needs. Practising the problems on a regular basis by referring to the NCERT Solutions for Class 10 Maths improves the problem-solving abilities of students.

Q6

### Is BYJU’S **NCERT Solutions for Class 10 Maths** important?

The NCERT Solutions for Class 10 Maths at BYJU’S are created in such a way that the students can grasp the concepts and clear their doubts instantly. The solutions contain comprehensive explanations to make the subject more interesting for the students. The PDF of solutions can be accessed by the students and can be used to improve their conceptual knowledge.

Q7

### Is the **NCERT Solutions for Class 10 Maths** the best reference guide for CBSE students?

Among the numerous study materials available in the market, students should select the right one based on their requirements. The NCERT Solutions for Class 10 Maths from BYJU’S is considered to be the best reference guide for the CBSE students. The solutions are available in both chapter-wise and exercise-wise formats, which can be downloaded and used by the students. The links of the PDF can be accessed by the students without any time constraints.