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Math Solutions for class 10

  • Mathematics NCERT Grade 10, Chapter 1, REAL NUMBERS: This chapter starts with an introduction about real numbers. A partial discussion was done in class 9th about real numbers. Further going deep into this topic, two very important properties of a real number are discussed in the first two sections namely:
    1. Euclid's division algorithm
    2. The fundamental theorem of arithmetic

    Euclid's division algorithm consists of a brief description of Euclid's division lemma. The next exercise is based on the topic of revisiting irrational numbers. Various theorems are given to make the student understand that how a specific number can be proved irrational. Several examples are given for the same. Exercise 1.3 is a short exercise consisting of 3 questions. Moving further to the next exercise which is based on topic-revisiting rational numbers and their decimal expansions. 3 theorems are required to solve questions of exercise 1.4. Thus, students need to understand these theorems thoroughly.
    At the end of the chapter, for recapitulation, a summary of the chapter is provided.         

  •  Mathematics NCERT Grade 10, Chapter 2, Polynomials chapter starts by citing about degrees of polynomial and differentiation of polynomials based on its degree. The next section describes the geometrical meaning of the zeroes of a polynomial. Various graphs are shown to explain how plotting of equations is done. Parabolas are introduced in this section.
    After that, solved examples are given to show how the number of zeroes can be determined by using the graphical method.
    Exercise 2.1 is based on the same concept. In the next section, the relationship between zeroes and coefficients of a polynomial is explained.
    Exercise 2.2 consists of questions in which students need to find the sum and product of zeroes of the given polynomial.
    Moving further, students will learn about division algorithms for polynomials.
    Exercise 2.3 is based on the same.
    Additionally, an optional exercise is given but that exercise is just for practice and not given from the examination point of view.
    For a perfect finish, a few important points are given to give a summary of the chapter.

  • Mathematics NCERT Grade 10, Chapter 3- Pair of linear equations in two variables: This chapter starts with an introduction followed by a short description on the topic Pair of linear equations in two variables. The very first exercise contains 3 problems which are based on the concept on how a given situation can be represented algebraically and graphically.
    Next section is about the topic- Graphical Method of Solution of a Pair of Linear Equations. Here the concept of the inconsistent and consistent pair of linear equations is explained. Here the students have to check whether the given lines intersect at a point, are parallel or coincident.
    After that, Algebraic Methods of Solving a Pair of Linear Equations is explained. This includes

    • Substitution method
    • Elimination method
    • Cross Multiplication
    Next exercise is based on the topic- Equations Reducible to a Pair of Linear Equations in Two Variables. Concept of upstream and downstream is cited in this section. In the end, an optional exercise and key points of the chapter are given.

  • Mathematics NCERT Grade 10, Chapter 4 Quadratic Equations. We all have dealt with so many real-life situations where we come across quadratic equations. A short example is given to explain how we can deal with real-life problems with quadratic equations
                                                   ax+ bx + c = 0 ; a 0
    In the first exercise, students will learn how to check whether the given equation is quadratic or not and representing the same.
    Next part of the chapter deals with finding the solution of the quadratic equation by the following methods:

    • Factorisation
    • Completing the Squares
    • Quadratic formula
    Later students will study the nature of roots using the determinant. How to determine the nature of roots is made more clear by giving certain examples. In exercise, 4.4 questions are given in which the nature of roots has to be determined and also finding the real roots if they exist. In the end, all the important points of the chapters are covered in the summary.

  • Mathematics NCERT Grade 10, Chapter 5 Arithmetic Progressions: This chapter starts with the explanation of the arithmetic progression and how it is related to our daily routine. Once it is understood, the next step is to study the various terms related to arithmetic progression.

    a, a + d, a + 2d, a + 3d, ...
    This is the general form of an AP.
    Exercise 5.1 tests the students on the basic concept of arithmetic progression. Moving ahead to the next section, students will find a detailed description on the nth term of an AP. Various solved examples are given to make the problems of this chapter more comprehensible. Exercise 5.2 is given with multiple problems so that students can practice more about the same. The next section explains how to find the Sum of First n Terms of an AP.  In exercise 5.3 different types of word problems are given which lays emphasis on the same concept. A few diagrammatic questions are also given which makes the exercise interesting. Exercise 5.4 is an optional exercise and is not from the examination point of view.  Later, a summary of the chapter is provided followed by a note to the reader.    

  • Mathematics NCERT Grade 10, Chapter 6- Triangles: We all are familiar with triangles and are also aware that it has many properties. Factors like congruency and similarity are known to everyone. But going deep into the same, a short description of similar figures is given. 
    Certain factors are stated that helps in recognizing whether the figures are similar or not. An activity is given to make this concept more clear.
    The next section is on the topic- Similarity of Triangles. Basic proportionality theorem and its converse are important theorems to be studied in this chapter. The next section deals with different criteria that are used to prove the similarity of triangles:

    • AAA similarity criterion
    • AA  similarity criterion
    • SSS similarity criterion
    • SAS similarity criterion
    Several theorems and activities are also given in this section. Exercise 6.4 is based on the topic- Areas of similar triangles. The next section deals with Pythagoras theorem and has an exercise based on the same. In the end, there is an optional exercise that is not from the examination point of view and the summarization of key points is done.

  • Mathematics NCERT Grade 10, Chapter 7: Coordinate geometry - To start with, a short description of different coordinates are given with reference to what students have learned in class IX.
    This chapter lays emphasis on how to find the distance between the two points whose coordinates are given, and to find the area of the triangles formed by three given points. Last, but not least students will also learn how to find the coordinates of the points which divide a line segment joining two given points in a given ratio. 
    In the first section, readers will learn about the distance formula. It is explained in detail with the help of a diagram. Solved examples are given to make the concept more understandable to students.
    Moving further, section formula is given and a special case of finding the mid-point of the line segment is also explained which is important from the examination point of view.
    In the last section, area of a triangle is discussed followed by solved example and exercise.
    An optional exercise is given at the end of the chapter. After that, the chapter is concluded in 5 points for a quick revision.

  • Mathematics NCERT Grade 10, Chapter 8: Introduction to Trigonometry: To begin, some real-life examples are given in order to explain the need for trigonometry. In this chapter, students will study the trigonometric ratios of the angles. The discussion of the trigonometric ratios will be restricted to acute angles only.
    In section 8.2 various trigonometric ratios are explained i.e ratios of the sides of a right triangle with respect to its acute angles. Solved examples are given to explain how to find trigonometric ratios and how to prove certain given relations.
    In the next section, trigonometric ratios of some specific angles 0°, 30°, 45°, 60° and 90° are taught. Table 8.1 contains all the required trigonometric ratios value. Students must learn these values as they are the most important part of this chapter and are essential to solve the problems.
    In section 8.4 trigonometric ratios of complementary angles are taught. Exercise 8.3 is based on the same. 
    The next part of the chapter is about trigonometric identities. Three important identities are highlighted in this section. In the end key points of chapter are discussed.

  • Mathematics NCERT Grade 10, Chapter 9: Some applications of trigonometry - In this chapter students will study about some ways in which trigonometry is used in life around us. Trigonometry is the most ancient subjects studied by scholars all over the world.
    In section 9.2 the topic Height and Distances is discussed. Various terms are explained in this part of the chapter.  7 solved examples are included in this chapter to make the same concept comprehensible to students.
    This chapter contains only one exercise. This is a vast exercise containing 16 questions. In the first few questions, the height of various objects is unknown and students have to calculate the same. In some questions, the width of the object is to be calculated. Other questions demand calculation of the distance between two objects
    A short summarization of the chapter is done in the end. 

  • Mathematics NCERT Grade 10, Chapter 10- Circles: Starting by stating the definition of the circle which was studied in class IX, the chapter moves on to explain the secant and tangent of the circle. The first section discusses the topic- Tangent to a Circle. This section is explained with the help of two activities followed by the theorem that the tangent at any point of a circle is perpendicular to the radius through the point of contact. A few remarks are mentioned at the end of the section which should not be ignored.
    Exercise 10.1 is a brief exercise containing 4 questions.
    The next section is about the number of tangents from a point on a circle. Three different cases are discussed in this section and later the theorem that related to the lengths of tangents from an external point to a circle is explained. The last exercise of the chapter is 10.2 in which the first three questions are multiple-choice questions with one correct option. After that few questions are based on proving the given statement.
    Finally, the conclusion of the chapter is given in 3 points. 

  • Mathematics NCERT Grade 10, Chapter 11: Constructions- The chapter starts with a short introduction.
    Following constructions are to be studied in this chapter:

    • To divide a line segment in the given ratio.
    • To construct a triangle similar to the given triangle as per the given scale factor.
    • To construct the pair of tangents from an external point to a circle.
    The division of a line segment is explained first. Given steps help in understanding the construction in a better and effective manner. The steps of construction are followed by justification which give an insight into the logic used behind the construction.
    Exercise 11.1 contains 7 questions all are related to construction of similar triangles and division of line segment. This is followed by the construction of tangents to a circle. In this section student will not only draw tangents to circle but will also measure their lengths, construct tangents inclined to each other at some angle etc. Exercise 11.2 is based on the same. In the end, a summary of the chapter is provided.     

  • Mathematics NCERT Grade 10, Chapter 12: Areas related to circles- We all are familiar with the methods of finding perimeters and areas of simple plane figures such as rectangles, squares, parallelograms, triangles, and circles.
    The first section is about the perimeter and area of a circle. The concept of (pie) π is discussed. One solved example is discussed in this section. First exercise 12.1 is a short exercise containing 5 questions.
    Second section deals with areas of sector and segment of a circle. Two examples are discussed in this section to make the concept clear to students. The following exercise 12.2 contains 14 questions based on the same.
    The next section is about the topic- Areas of combination of plane figures. This particular section contains 3 solved examples and then the last exercise of the chapter 12.3 is given.
    Later the summary of the chapter is given. 

  • Mathematics NCERT Grade 10, Chapter 13: Surface areas and volumes- Students have already learnt about solids like cuboid, cone, cylinder, and sphere. They have also learned how to find their surface areas and volumes. In this chapter, students will see how to find surface areas and volumes of combination of solids. The first two exercises of this chapter are based on the same.
    The next topic is about the Conversion of solid from one shape to another. 4 solved examples are given to make the concept more clear.
    The next exercise 13.3 contains different questions based on the same concept. Section 13.5 is on the topic- Frustum of a cone. In this chapter a student will study how to find the slant height, curved surface area, total surface area and volume of the frustum of a cone. This chapter contains activity, solved examples and exercise questions to explain this topic in a better way.
    Exercise 13.5 is an optional exercise that is not given from the examination point of view. The conclusion of the chapter is given at the end of the chapter.

  • Mathematics NCERT Grade 10, Chapter 14: Statistics- Concept of classification of given data into ungrouped as well as grouped frequency distributions was explained to students in class IX. Bar graphs, histograms, and frequency polygons were also introduced earlier. In this chapter, the study of three measures namely mean, median and mode from ungrouped data to that of grouped data will be extended. 
    Concept of cumulative frequency, the frequency distribution (cumulative) and how to draw cumulative frequency curves (ogives) will also be explained in this chapter.
    The very first section covers the topic Mean of Grouped Data. Upper limit and the lower limit are explained. There are three methods of finding mean:

    1. Direct method
    2. Assumed mean method
    3. Step deviation method

    The next section deals with the concept of Mode of grouped data. 3 solved examples and one activity will help the students to understand the topic. After that students will learn about Median of Grouped Data and can practice the same in the following exercise. The last topic of the chapter is Graphical Representation of Cumulative Frequency Distribution followed by an exercise. Later key points of the chapter are explained in brief. 

  • Mathematics NCERT Grade 10, Chapter 15: Probability- In the starting of chapter empirical or experimental and theoretical or classical probabilities are discussed. The first section is about Probability- A Theoretical Approach. One will come across the term equally likely outcomes in this section.
    In this chapter, it is assumed that all the experiments have equally likely outcomes. Different solved examples are given to explain the concept of probability.  Certain remarks are given, which should not be ignored.
    After that, complementary event is another topic which requires one's attention. Impossible and sure or certain events are also discussed in this chapter. There are a total of 13 solved examples in this section out of which example number 10 and 11 are not from the examination point of view.
    This chapter contains only one exercise, 15.1 which is a vast exercise comprising of 25 questions.
    An optional exercise is also given which however is not from the examination point of view. And in the end, students will find a summary of the chapter in 7 points. 

RS Aggarwal 2020, 2021 Solutions for class 10