 # NCERT SOLUTIONS FOR CLASS 10 MATHS

## Introduction About NCERT Solutions for Class 10 Maths

NCERT Solutions for Class 10 Maths, available online and can be easily downloaded. It is the best study material for class 10 students that is available online. Students can find solutions to their textbook exercises and references for detailed comprehension. Class 10 is an essential time for every students’ school time and achieving better marks is important. NCERT Solutions for Class 10 Maths ensures a good result and helps students to have a great career ahead.

This PDF designed by an expert team of Fliplearn helps students with detailed explanations of every concept. Solutions to all textbook exercises with sample papers and references are a few offerings that makes NCERT Solutions the best for Class 10 students.

## Why students prefer NCERT Solutions for Class 10 Maths

Students trust in NCERT Solutions, hence they prefer it over any other study material available online. It provides students with references and sample papers that not only solve the problems but also offer a good explanation of the concepts. It makes it easy for students to retain the concepts.

The access to vast study material, sample papers, previous year’s question papers, references and books, helps students to gain good grades in exams and achieve their respective dreams in future.

## Topics Covered in NCERT Solutions for Class 10 Maths with brief details

NCERT Solutions of class 10 math contains all chapter solutions in pdf. Solutions can be downloaded chapter wise. NCERT Solutions for class 10 cover all the chapters including Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Triangles, Coordinate Geometry, Introduction to Trigonometry, Some Applications of Trigonometry, Circles, Constructions, Areas Related to Circles, Surface Areas and Volumes, Statistics, Probability.

Students can easily use the PDF of the chapter-wise solutions and gain conceptual knowledge to solve the problems according to the NCERT Maths textbook for Class 10. This helps students enhance their confidence, which is required to master concepts and perform well in exams.

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

1. Euclid’s Division Lemma

• Euclid’s Division Lemma
• Euclid’s Division Algorithm

2. The Fundamental Theorem of Arithmetic

• Prime Factorisation
• Fundamental Theorem of Arithmetic
• Highest Common Factor (HCF)
• Least Common Multiple (LCM)
• HCF X LCM = Product of Numbers

3. Revisiting Irrational Numbers

• Irrational number
• Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
• The sum of the difference between a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational and an irrational number is irrational.

4. Revisiting Rational Numbers and Their Decimal Expansions

• Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form of p/q, where p and q are coprime, and the prime factorization of q is of the form 2n5m, where n and m are non-negative integers.
• Any rational number of the form a/b, where b is a power of 10, will have a terminating decimal expansion.
• Let x = p/q be a rational number such that the prime factorization of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion that terminates.
• Let x = p/q be a rational number such that the prime factorization of q is not of the form 2n5m, where n and m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).
• Decimal expansion of every rational number is either terminating or non-terminating repeating.

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

1. Geometrical Meaning of the zeroes of a polynomial

• Geometrical meaning of the zeroes of a polynomial
• Geometrically zero of linear polynomial
• Shape of the graph of quadratic polynomial
• Geometrically zeroes of quadratic polynomial
• Geometrically zeroes of a cubic polynomial

2. Relationship between zeroes and coefficients of polynomial

• Relationship between zeroes and coefficients of linear polynomial
• Relationship between zeroes and coefficients of quadratic polynomial
• Relationship between zeroes and coefficients of a cubic polynomial

3. Division algorithm for polynomials

• Division of a polynomial by a polynomial
• Division algorithm for polynomials

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

1. Pair of Linear Equations in Two Variables

• The general form of a pair of linear equations
• Three possibilities with two lines in a plane
-Intersecting
-Parallel
-Coincident
• Algebraic and geometrical representation of a pair of linear equations in two variables

2. Graphical Method of Solution of a pair of linear equations

• The behavior of lines representing a pair of linear equations in two variables
• Existence of solutions

3. Algebraic methods of solving a pair of linear equations

• Substitution method
• Elimination method
• Cross multiplication method

4. Equations reducible to a pair of linear equations in two variables
If a pair of equations in two variables is not linear, then reduce it to linear form by making some suitable substitutions.

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

• The standard form of a quadratic equation

2. The solution of a Quadratic Equation by Factorisation

• Roots by factorization method

• If b2 – 4ac greater than equal to zero then roots of the quadratic equation are given by • If b2 – 4ac < 0 then the equation will have no real roots

4. Nature of Roots

• Discriminant
• Nature of roots
– Distinct real roots if b2 – 4ac > 0
– Two equal roots if b2 – 4ac = 0
– No real roots if b2 – 4ac < 0

### NCERT Solutions Class 10 Maths Chapter 5 – ARITHMETIC PROGRESSIONS

1. Arithmetic Progressions

• Meaning of AP
• The general form of AP
• Finite AP
• Infinite AP
• To identify a sequence is an AP

2. The Nth term of an AP

• Nth term of an AP = a + (n – 1)d

3. Sum of first n terms of an AP

• or • To find the nth term, given Sn:

an = Sn  –  Sn-1

• Sum of first n positive integers ### NCERT Solutions Class 10 Maths Chapter 6 – TRIANGLES

1. Similar Figures

• Two geometric figures which have the same shape but different sizes are called similar figures.
• All congruent figures are similar but all similar figures may or may not be congruent.
• Two polygons of the same number of sides are similar, if
– their corresponding angles are equal and
– their corresponding sides are in the same ratio.

2. Similarity of Triangles

• Two triangles are similar, if
– their corresponding angles are equal and
– their corresponding sides are in the same ratio
• Basic proportionality theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
• The converse of Basic Proportionality Theorem: If a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.

3. Criteria for the similarity of triangles

• A.An (Angle – Angle – Angle) Similarity Criterion
• An (Angle – Angle) Similarity Criterion
• S.S (Side-Side-Side) Similarity Criterion

4. Areas of similar triangles

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

5. Pythagoras Theorem

• 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.
• Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides.
• The converse of Pythagoras Theorem: 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 to the first side is a right angle.

### NCERT Solutions Class 10 Maths Chapter 7 – COORDINATE GEOMETRY

1. Distance Formula

• The distance between the points P(x1, y1) and Q(x2, y2) is 2. Section Formula

• The coordinates of the point P(x, y) which divides the line segment joining the points

An (x1, y1) and B(x2, y2) internally in the ratio m1: m2 is • If the ratio in which P divides AB is k: 1 then the coordinates of the point P will be • The coordinates of the mid-point P of the points A(x1, y1) and B(x2, y2) is 3. Area of a triangle

• The area of the triangle formed by the points (x1, y1), (x2, y2), and (x3, y3) is the numerical value of the expression • If the area of the triangle is 0 square units, then its vertices will be collinear.

### NCERT Solutions Class 10 Maths Chapter 8 – INTRODUCTION TO TRIGONOMETRY

1. Trigonometric Ratios

• Trigonometry
• Trigonometric Ratios
• Invariance of Trigonometric Ratio

2. Trigonometric Ratios of Some Specific Angles

• Trigonometric ratios of 0o, 30o, 45o, 60o and 90o

3. Trigonometric Ratios of Complementary Angles 4. Trigonometric Identities ### NCERT Solutions Class 10 Maths Chapter 9 – Some Applications of Trigonometry

1. Heights and Distances

• Line of sight
• Angle of elevation
• Angle of depression
• With the help of trigonometric ratios, the following can be easily found
(i) the height or length of an object
(ii) the distance between the distant objects

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

1. Tangent to a circle

• Meaning of tangent to the circle
• Tangent to a circle is a special case of the secant
• Point of contact
• The tangent at any point of a circle is perpendicular to the radius through the point of contact.

2. Number of tangents from a point on a circle

• Length of tangent
• The length of tangents drawn from an external point to a circle are equal
• The centre lies on the bisector of the angle between the two tangents.

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

1. Division of a line segment

• To divide a line segment in a given ratio
• To construct a triangle similar to a given triangle as per the given scale factor

2. Construction of tangents to a circle
To construct the tangents to a circle from a point outside it.

### NCERT Solutions Class 10 Maths Chapter 12 – Areas related to circles

1. Perimeter and Area of a Circle – A Review

• Circumference of circle
• The relation between circumference and diameter of a circle
• Area of a circle

2. Areas of sector and segment of a circle

• Sector
• Area of the minor sector and major sector
• Length of an arc of a sector
• Segment of circle
• Area of minor and major segment

3. Areas of combinations of plane figures

• Composite shapes
• Area of composite shapes

### NCERT Solutions Class 10 Maths Chapter 13 – SURFACE AREAS AND VOLUMES

1. The surface area of a combination of solids

• Surface area
• Methods to find the surface area of a combination of solids

2. The volume of a combination of solids

• Volume
• The volume of the combination of solids

3. Conversion of solids from one shape to another

• When a solid is converted from one shape to another, its volume remains the same.
• When the liquid of one container of any shape is poured into another container of a different size or shape, its value remains unaltered.

4. Frustum of a cone

• Frustum
• Surface area and volume of a frustum

### NCERT Solutions Class 10 Maths Chapter 14 – STATISTICS

1. Mean of grouped data

• Arithmetic Mean – Grouped Frequency Distribution
(i) Direct Method (ii) Assumed mean method (iii) Step deviation method 2. Mode of grouped data

• Mode-Grouped Frequency Distribution 3. Median of grouped data

• Median
• Cumulative Frequency
• Median-ungrouped frequency distribution
• Median – Grouped Frequency Distribution 4. Graphical representation of cumulative frequency distribution

• Less than type ogive
• More than type ogive

### NCERT Solutions Class 10 Maths Chapter 15 – PROBABILITY

1. Probability – A Theoretical Approach

• Probability
• Experiment
• Trial
• Outcome
• Sample Space
• Event
• Simple Event
• Compound Event
• Equally likely outcomes
• Empirical Probability
• Formula to find the empirical probability
• Theoretical probability
• Favorable outcomes
• Formula to find the theoretical probability
• Elementary event
• Sure event
• Impossible event

Students can easily find solutions to their problems and make maths-learning an enjoyable experience.

## Benefit of NCERT Solutions for Class 10 Maths

• Students can benefit from the NCERT Solutions for Class 10 by getting a great scope of studying and referring material online.
• Simplified topics with easy to grasp and retain graphs and illustrations help students to remember things for a long.
• These solutions are prepared by an expert team of Fliplearn that focuses on accuracy.
• The study material helps students to bridge the knowledge gap with the exercises and solutions.
• Providing simple and easy to understand solutions.
• Detailed study material is available for class 10 students

Fliplearn simplifies your problems and makes learning easy and enjoyable. Download the Fliplearn app to find more help and a personalized experience.

### Is NCERT Solutions the best for class 10 exam preparations?

Choose NCERT Solutions for Class 10 to prepare for exams and score good marks. Find a number of reference exercises, sample papers, and textbook solutions. It is very helpful for the students as it makes it very simple to solve problems and understand concepts.

### What are the advantages of NCERT Class 10 Solutions?

There are numerous advantages of studying from NCERT Class 10 Solutions. From a vast online study material to sample papers, references, and self-analytical tests, there is a lot to help students.

### What is the best source to prepare for Class 10 exams?

The best Class 10 exam preparation source is NCERT Solutions. This is the most authentic online learning material that also provides ways to analyze your own performance. Apart from the regular textbook problems and their solutions, NCERT Solutions also offer references that cover an understanding of concepts and formulas.

### NCERT Solutions for other Classes

NCERT Solutions is also available for other classes. Students preparing for class 6th, 7th, 8th, 9th, 11th, and 12th will find it the best way to score good marks. NCERT Solutions for class 6th, 7th, 8th, 9th, 11th, and 12th provides study material and references that help students trust it the most.