# NCERT Solutions for Class 10 Maths

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NCERT - Mathematics

NCERT Solutions for Class 10 Maths made available by Goprep help students build their fundamentals in the subject. Prepared by our expert faculty, NCERT Class 10 Maths Solutions assist students in CBSE Class 10 Maths board exam preparations.

So, students who are looking for Class 10 NCERT Maths Solutions can access our chapter-wise solutions to solve each question in the textbook with ease. We have prepared Class 10 Maths NCERT Book Solutions in a detailed and methodical way so as to help you get to the logic behind every solution.

Moreover, these solutions are sure to help students have an adequate practice of complex Maths topics to approach board exams confidently. Further, you can use the links given below to browse through our chapter-wise questions and solutions for Class 10 Maths NCERT book.

## NCERT Class 10 Maths Solutions - All Chapters

If you have a Maths exam coming up then there is no better way to revise all the concepts in a step-wise manner than reviewing our NCERT Solutions provided here. Get over your exam stress by learning key formulas of each chapter of Class 10 Maths NCERT book. Along with Class 10 Maths formulas, you can also go through the brief explanation of all the chapters.

All-in-all, you have landed on the right page where you can resolve your doubts, access CBSE NCERT Solutions, learn formulas, and view the summary of all the chapters. Guess what? You can also find the CBSE Class 10 Maths exam pattern and chapter-wise marks weightage. We leave you no choice but to wholly rely on our platform to prepare for your Maths exam.

## NCERT Class 10 Maths Solutions (Chapter-wise description)

Goprep's subject experts have listed down the list of important topics and formulas for all the chapters of Class 10 Maths NCERT Book below. It is highly recommended that you study the chapter-wise topics and formulas as these are expected to carry the maximum weightage in CBSE Class 10 Maths board exam.

### NCERT Class 10 Maths Chapter 1- Real Numbers

Real numbers comprise of natural numbers, whole numbers, rational & irrational numbers, and integers. In class 10 Maths NCERT book, you will also learn about advanced topics, including properties of positive integers- Euclid's division algorithm and the fundamental theorem of arithmetic. Besides, you will also get to know several theorems of rational and irrational numbers.

Topics

• Euclid’s Division Lemma
• The Fundamental Theorem of Arithmetic
• Irrational Numbers
• Rational Numbers and Their Decimal Expansions

Important Formulas of Real Numbers

1. Euclid’s Division Lemma: For two positive integers a and b, there exist whole numbers q and r respectively, which forms the equation shown below

a= bq + r, where 0 ≤ r < b

2. Euclid’s Division Algorithm: To find the HCF of two positive integers a and b where a > b, you have to consider the following steps.

Step 1: Use division lemma to find the values of q and r (a= bq + r, where 0 ≤ r < b).

Step 2: HCF of a and b = b, if r = 0. In case, r ≠ 0 then apply Euclid’s lemma to b and r.

Step 3: Follow the process until the remainder is zero. Take divisor as HCF (a, b). Also, HCF (a, b) = HCF (b,r)

### NCERT Class 10 Maths Chapter 2- Polynomials

Polynomials are expressions that contain variables and coefficients. A polynomial can be linear, quadratic, cubic, etc. based on their highest degree. This year, you will study the zeroes of a polynomial, the relationship between zeros and coefficients of a polynomial, and division algorithm for polynomials.

Note:In total, there are 7 units in NCERT Class 10 Maths textbook. Out of these, algebra carries the highest weightage as seen in CBSE class 10 maths previous year question papers.

Topics

• Geometrical Meaning of the Zeros of a Polynomial
• Relationship between Zeros and Coefficients of a Polynomial
• Division Algorithm for Polynomials

Important Formulas of Polynomials

1. If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then

α + β = -b/a, αβ = c/a

2. If α, β and γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d, then

• α + β + γ = -b/a
• αβ+ β γ + γ α = c/a
• αβ γ = -d/a

Division Algorithm: For a given polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x), such that

p(x) = g(x) q(x) + r(x), where r(x) = 0

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

A linear equation comprises of two variables and appears as a straight line on a graph. In this chapter of NCERT class 10 maths solutions, you have to represent different situations algebraically and graphically using a pair of equations. Also, you will get to learn about how to reduce equations to a pair of linear equations.

Topics

• Pair of Linear Equations in Two Variables
• Graphical Method of Solution of a Pair of Linear Equations
• Algebraic Methods of Solving a Pair of Linear Equations
• Equations Reducible to a Pair of Linear Equations in Two Variables

Important Formulas of Pair of Linear Equations in Two Variables

For a pair of linear equations p (x) = a1x + b1y + c1= 0 and q (x) = a2x + b2y + c2= 0, the following situations may arise:

i). a1/a2≠ b1/b2(In this case, the pair of linear equations is said to be consistent)

ii). a1/a2= b1/b2≠ c1/c2(In this case, the pair of linear equations is said to be inconsistent)

iii). a1/a2= b1/b2= c1/c2(In this case, the pair of linear equations is said to be consistent and dependent)

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

A quadratic equation is represented by an equation having the highest degree as 2. In NCERT class 11, you will revisit a few concepts of quadratic equations from previous classes. You will also learn to solve a quadratic equation by various methods including, factorisation, completing the square, and using the quadratic formula.

Topics

• Solution of a Quadratic Equation by Factorisation
• Solution of a Quadratic Equation by Completing the Square
• Nature of Roots

Quadratic formula: To find the roots of a quadratic equation ax2 + bx + c = 0, you have to apply the values of a, b and c in the following formula.

(-b ± √b2 - 4ac)/ 2a, where b2 - 4ac, c ≥ 0

### NCERT Class 10 Maths Chapter 5- Arithmetic Progression

It will be the first time you will study Arithmetic Progression (AP). AP is a list of numbers where you can obtain successive terms by adding a fixed number to the preceding terms. In this chapter, there will be questions asking you to find the nth term of an AP, sum of first (n) terms of an AP, and the last term of the finite AP using relevant formulas.

Topics

• Arithmetic Progressions
• nth Term of an AP
• Sum of First n Terms of an AP

Important Formulas of Arithmetic Progression

1. An AP can expressed in the form of a, a + d, a + 2d, a + 3d, . . .

2. The nth term of an AP with first term a and common difference d can be represented as

An = a + (n – 1) d

3. To find the sum of the first n terms of an AP, use the following equation

Sn = ½. n [2a + (n - 1)d]

4. To find the sum of all terms where the value of the last term of the finite AP is given, use the following equation

Sn = ½. n [a + l]

### NCERT Class 10 Maths Chapter 6- Triangle

As you are already familiar with Triangles and its Properties, you will learn to judge whether two triangles are similar or not, using Basic Proportionality Theorem. Further, you will be able to figure out the similarity of two triangles using different criteria, some of which you have already learned in the previous NCERT maths books.

Topics

• Similar Figures
• Similarity of Triangles
• Criteria for Similarity of Triangles
• Areas of Similar Triangles
• Pythagoras Theorem

### NCERT Class 10 Maths Chapter 7- Coordinate Geometry

Coordinate Geometry involves locating the position of a point on a plane using a pair of coordinate axes. This year, you will learn to find the distance between the two points using Distance Formula. To find the coordinates of the point dividing a line segment, you will have to apply Section Formula. Also, the Area of Triangle will come to use for finding the coordinates of points forming a triangle.

Topics

• Distance Formula
• Section Formula
• Area of a Triangle

Important Formulas of Coordinate Geometry

1. To find the distance between two points P (x1, y1) and Q (x2, y2), you can apply the following formula.

√(x2- x1)2 + (y2- y1)2

2. The distance of a point P (x,y) from the origin can be found by-

√x2 + y2

3. The coordinates of the point P (x,y) which divides the line segment with points A (x1, y1) and B (x2, y2) internally in the ratio m1: m2are

[(m1x2+ m2x1)/ (m1+ m2), (m1y2+ m2y1)/ (m1+ m2)]

4. The midpoint of the line segment having end-points P (x1, y1) and Q (x2, y2) is given by

[(x1+ x2)/ 2, (y1+ y2)/ 2]

5. To find the area of triangle with points (x1, y1), (x2, y2) and (x3 , y3), you can use the following expression

½. [x1(y2- y3) + x2(y3- y1) + x3(y1- y2)

### NCERT Class 10 Maths Chapter 8- Introduction to Trigonometry

The concept of Trigonometry is applied to study the relationships between the sides and angles of a triangle. For the first time, you will learn to find the Trigonometric Ratios of the Angle and prove two equations equal using Trigonometric Identities.

Topics

• Trigonometric Ratios
• Trigonometric Ratios of Some Specific Angles
• Trigonometric Ratios of Complementary Angles
• Trigonometric Identities

Important Formulas of Introduction to Trigonometry

1. In the right-angled triangle ABC, where ∠B = 90°

• sin A = Perpendicular the (side opposite to angle A)/ hypotenuse
• cos A = base (side adjacent to angle A)/ hypotenuse
• tan A = Perpendicular (the side opposite angle A)/ base (side adjacent to angle A)

2. Some more relations

• cosec A = 1/ sin A
• sec A = 1/cos A
• tan A= sin A/cos A

3. 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

4.sin2 A + cos2 A = 1,

sec2 A – tan2 A = 1 for 0° ≤ A < 90°,

cosec2 A = 1 + cot2 A for 0° < A ≤ 90º

### NCERT Class 10 Maths Chapter 9- Some Applications of Trigonometry

After covering the previous chapter of NCERT class 10 maths, you will now study how trigonometry is used in our surroundings. While finding heights and distances in questions, you will have to examine a few terms before solving them. These include the Line of Sight, the Angle of Elevation, the Angle of Depression. The height or length of an object or distance between two objects can be found out using trigonometric ratios.

Topics

• Heights and Distances
• Line of Sight
• Angle of Elevation
• Angle of Depression

### NCERT Class 10 Maths Chapter 10- Circles

A Circle is a collection of all points in a plane which are placed at a constant radius from the centre. In this chapter, you will study the meaning of a Tangent to a circle and related theorems.

Topics

• Tangent to a Circle
• Number of Tangents from a Point on a Circle

### NCERT Class 10 Maths Chapter 11- Constructions

Basic Construction includes the knowledge of bisecting an angle, drawing a perpendicular bisector of a line segment, Construction of triangle etc. In this chapter, you will learn to divide a line segment in a given ratio; construct a triangle similar to a given triangle; construct a pair of tangents from an external point to a circle.

Topics

• Division of a Line Segment
• Construction of Tangents to a Circle

### NCERT Class 10 Maths Chapter 12- Areas Related to Circles

In chapter 12 “Areas related to circles” of NCERT class 10 maths book, you will first revisit the basic formulas related to a Circle, i.e. Circumference and Area of a Circle. Next, you will learn to find the Length of an Arc of a Sector of a Circle, Area of a Sector of a Circle, and Area of Segment of a Circle.

Topics

• Perimeter and Area of a Circle
• Areas of Sector and Segment of a Circle
• Areas of Combinations of Plane Figures

Important Formulas of Areas Related to Circles

1. Circumference of a circle = 2 π r

2. Area of a circle = π r2

3. Length of an arc of a sector of a circle with radius r and angle θ is given by =θ/360. 2 π r

4. Area of a sector of a circle with radius r and angle θ is given by = θ/360. π r2

### NCERT Class 10 Maths Chapter 13- Surface Areas and Volumes

With previous knowledge of finding surface areas and volumes of solids, you will learn to find the surface area and volume of objects formed by the combination of any two objects. This chapter also introduces you with formulae that helps you find the frustum of a cone.

Topics

• Surface Area of a Combination of Solids
• Volume of a Combination of Solids
• Conversion of Solid from One Shape to Another
• Frustum of a Cone

Important Formulas of Surface Areas and Volumes

1. Volume of a frustum of a cone = ⅓. Πh (r12 + r2 + r1 r2)

2. Curved surface area of a frustum of a cone = πl(r1 + r2), where l = √h2 + (r1 - r2)2

3. Total surface area of frustum of a cone = πl(r1 + r2 ) + π(r12 + r22)

### NCERT Class 10 Maths Chapter 14- Statistics

In this chapter, you will learn to find Mean for Grouped Data through three different formulae- Direct method, Assumed Mean method, and Step Deviation method. Also, the chapter introduces you to formulae to find the median and mode of grouped data. Obtaining Cumulative Frequency and representing it graphically another new edition to your previous knowledge.

Topics

• Mean of Grouped Data
• Mode of Grouped Data
• Median of Grouped Data
• Graphical Representation of Cumulative Frequency Distribution

Important Formulas of Statistics

1. The mean for grouped data can be obtained using the following methods:

• Direct method (x-bar) = ∑fixi/ ∑fi
• Assumed mean method (x-bar) = a + (∑fidi/ ∑fi)
• Step deviation method (x-bar) = a + (∑fiui/ ∑fi) x h

2.The mode for grouped data can be obtained using the formula:

Mode = [l + h{(f1 - f0)/(2f1- f0- f2)}]

3. The median for grouped data can be obtained by using the formula:

Median = [l + h {(n/2 - cf)/f}]

### NCERT Class 10 Maths Chapter 15- Probability

Having discussed experimental Probabilities in the earlier NCERT textbooks, you will study the difference between Experimental Probability and Theoretical Probability. Both these topics form the core of this chapter. Further readings will involve the Probability of a sure event, Probability of an impossible event, Probability of an event, Elementary event, and Complementary events.

Topics

• Probability
• Sure event
• Impossible event
• Complementary events

Important Formulas of Probability

Theoretical Probability: The theoretical probability of an event E, can be defined as

P (E) = Number of outcomes favorable to E/ Number of all possible outcomes of the experiment.

## CBSE Class 10 Maths Unit-wise Weightage

Check the marking scheme for Class 10 Maths unit-wise here.

 Unit Name Marks 1 Number Systems 06 2 Algebra 20 3 Coordinate Geometry 06 4 Geometry 15 5 Trigonometry 12 6 Mensuration 10 7 Statistics & Probability 11 Total 80

## CBSE Class 10 Maths Exam Pattern

CBSE Board had introduced changes in the question paper format and the type of questions. From now onward, 25% of the questions will be objective type. The total number of questions has been revised from 30 to 40. However, the weightage of the question paper will remain the same, i.e. 80 marks.

We have tabulated the section-wise paper pattern of CBSE Class 10 Maths below.

 Section Question Type Weightage of Marks per question A Objective Type (1-20) 1 B Very Short Answer Type (21-26) 2 C Short Answer Type (27-34) 3 D Long Answer Type (35-40) 4

## Best Maths Reference Books for Class 10 | Maths Guides

As per the previous year trend, most questions are asked from CBSE NCERT Class 10 Maths Book itself. 20-30% questions of the paper are given from reference books. Based on this observation, we have prepared the list of best Maths guides for CBSE Class 10.

 Class 10 Maths Reference Books Link RD Sharma RD Sharma Class 10 Solutions RS Aggarwal RS Aggarwal Class 10 Solutions KC Sinha KC Sinha Class 10 Solutions NCERT Exemplar NCERT Exemplar Class 10 Maths Solutions

## Benefits of NCERT Class 10 Maths Solutions

With more than a dozen of online study resources offering free NCERT Solutions, what makes Goprep the best platform? Get over this confusion by referring to the pointers mentioned below.

• Use of common methods: Our experts have hands-on experience in teaching students from different schools. This has enabled them to apply a common method in each question that can be understood by all.
• Organized and clear explanation: Step-by-step solutions with lucid explanation gives you no headache in understanding the logic.
• Get doubts resolved by uploading a screenshot: It is quite natural to come across a doubt when you solve Maths problems. No worries as you can upload the screenshot of any question in which you are facing doubt. To do that, click on the ‘camera’ icon on the homepage of Goprep website.
• Chapter-wise summary: As you can see, we have provided a brief explanation for every chapter of NCERT Class 10 Maths book. This will allow you to take a quick walkthrough of the chapter.
• Revise important Maths formulas: It becomes a tedious task to look for different formulas when practicing CBSE previous year question papers. With formulas of all chapters at one place, we save you a lot of time and energy.

• Is Class 10 Maths NCERT book enough for CBSE 10th Board Exam preparation?

Depending on the time and goal you have in your mind, you can choose to study only from NCERT or pick reference books too.

• If your aim is to score 70-80 marks and have less time, you should practice all in-text questions, examples, and back-exercise questions of every chapter. Do not let go of even a single question or example unsolved in NCERT Class 10 Maths textbook.
• To score above 80 marks, you need to solve extra questions from one of the Class 10 Maths reference books mentioned above.
• How to score 100/100 marks in CBSE Class 10 Maths?

For scoring a good CGPA or percentage in Class 10, you should aim at securing 100/100 marks in CBSE Class 10 Maths. To accomplish this, you may follow our preparation strategy.

• Create a time table and devote at least 1.5 hours to study Mathematics.
• Prepare a rough notebook for practicing questions and a small pocket notebook to jot down important formulas and graphs.
• In the beginning, study from NCERT textbook only and solve all in-text, examples, and back-exercise questions.
• Refer to our Class 10 Maths NCERT Solutions if you come across a doubt in any textbook question.
• If you have time left, pick any one reference book among RD Sharma, RS Aggarwal, KC Sinha or NCERT Exemplar to practice extra questions.
• Do not forget to solve CBSE Class 10 Maths previous year question papers as these will help you know the exam pattern.
• How to prepare for CBSE Class 10 Maths Board exam?

Preparation strategy for all subjects remains the same in any Class.

• You should stick to NCERT books for the entire preparation.
• Choose a reference book if you think you require extra practice.
• Practice previous year papers to understand the exam pattern and type of questions asked.
• Spend at least 1.5 hours practicing questions and working on your weak areas simultaneously.
• Revise important formulas and NCERT Solutions for Class 10 Maths provided above.
• Stick to the study plan throughout the preparation to get the desired result.
• What is the list of chapters covered in NCERT Maths Class 10 book?

Class 10 Maths NCERT book consists of 15 chapters in total. The list of chapters along with the summary is provided above.

• Which is the best study resource for studying Maths in Class 10?

NCERT book is the best study resource for clearing your basic concepts. To support your preparation and revise key concepts, you can access NCERT Solutions for Class 10 Maths at Goprep.