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The set of real numbers is denoted by R. Thus every real number is either a rational number or an irrational number. In either case, it has a non�terminating decimal representation.
In the case of rational numbers, the decimal representation is repeating including repeating zeroes and if the decimal representation is non�repeating, it is an irrational number.
From the above discussion we may conclude that: To every real number there corresponds a unique point on the number line and conversely, to every point on the number line there corresponds a real number. We have studied the following points: 1. The divisor at this stage will be HCF a, b. The Fundamental Theorem of Arithmetic: Every composite number can be expressed factorized as a product of primes and this factorization is unique, apart from the order in which the prime factors occur.
In the next section, the relationship between zeroes and coefficients of a polynomial is explained. Moving further, students will learn about division algorithms for polynomials. 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.
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. Concept of upstream and downstream is cited in this section. In the end, an optional exercise and key points of the chapter are given. 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.
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. In the end, all the important points of the chapters are covered in the summary.
Once it is understood, the next step is to study the various terms related to arithmetic progression. This is the general form of an AP. Exercise 5. Moving ahead to the next section, students will find a detailed description on the n th term of an AP. Various solved examples are given to make the problems of this chapter more comprehensible.
In exercise 5. A few diagrammatic questions are also given which makes the exercise interesting. Later, a summary of the chapter is provided followed by a note to the reader. 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. 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. 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. 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. Solved examples are given to explain how to find trigonometric ratios and how to prove certain given relations.
Table 8. Students must learn these values as they are the most important part of this chapter and are essential to solve the problems. Exercise 8. 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. Various terms are explained in this part of the chapter. 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. 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 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 After that few questions are based on proving the given statement.
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