NCERT Solutions for Class 10 Maths 2021 Board Exam

NCERT Class 10 Maths Chapters and Exercises

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, rational number and decimal expansion of rational numbers are explained with the help of theorem.

Topics Covered:

Euclid’s division lemma, Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5 Decimal representation of rational numbers in terms of terminating/non-terminating recurring decimals. 

Chapter 2 – Polynomials

In Polynomials, the chapter begins with the definition of 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 the questions on finding the number of zeroes through a graph. It requires the understanding of 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 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:

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithms for polynomials with real coefficients. 

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. The Exercise 3.2 explains the methods of solving the pair of the linear equation through Graphical Method. Exercise 3.3, 3.4, 3.5, 3.6 describes the Algebraic Method, Elimination Method, Cross-Multiplication Method, Substitution Method respectively. Exercise 3.7 is an optional exercise which contains all types of questions. Students must practice these exercises to master the method of solving the linear equations.

Topics Covered:

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 and by cross multiplication method. Simple situational problems. Simple problems on equations reducible to linear equations. 

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

1. Two distinct real roots, if b² – 4ac > 0
2. Two equal roots, if b² – 4ac = 0
3. No real roots, if b² – 4ac < 0

Topics Covered:

Standard form of a quadratic equation ax² + bx + c = 0, (a ≠ 0). Solutions of quadratic equations (only real roots) by factorization, and by using quadratic formula. Relationship between discriminant and nature of roots. Situational problems based on quadratic equations related to day to day activities to be incorporated.

Chapter 5 – Arithmetic Progressions

This chapter introduces students to a new topic that is  Arithmetic Progression i.e AP. This 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 first n terms of an AP. The last exercise includes higher-level questions based on AP to enhance students analytical and power solving skills.

Topics Covered:

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

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 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 converse of Pythagoras Theorem is described.

Topics Covered:

Definitions, examples, 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. 
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 
9. (Prove) In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, the angle opposite to the first side is a right angle.

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 Distance Formula, Section Formula and Area of a Triangle in this chapter.

Topics Covered:

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

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 0⁰ and 90⁰. 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:

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 300 , 450 and 600 . Relationships between the ratios. Proof and applications of the identity sin²A + cos²A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles. 

Chapter 9 – Some Applications of Trigonometry

This chapter is the continuation of the previous chapter as here the students will study the applications of trigonometry. It is used in geography, navigation, construction of maps, 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, angle of depression.

Topics Covered:

Simple problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30°, 45°, 60°. 

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 through with the concept of Tangent to a Circle and Number of Tangents from a Point on a Circle.

Topics Covered:

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.

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 more clearer to the students.

Topics Covered:

1. Division of a line segment in a given ratio (internally). 
2. Tangents to a circle from a point outside it. 
3. Construction of a triangle similar to a given triangle.

Chapter 12 – Areas Related to Circles

This chapter begins with the concepts of 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 finding the areas of some combinations of plane figures involving circles or their parts.

Topics Covered:

Motivate the area of a circle; 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 a segment of a circle, problems should be restricted to the central angle of 60°, 90° and 120° only. Plane figures involving triangles, simple quadrilaterals and circles should be taken.)

Chapter 13 – Surface Areas and Volumes

In Chapter 13, there are a total of 5 exercises. The first exercise consists of the 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. The Exercise 13.3 deals with the questions in which solid is converted from one shape to another. The 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:

1. Surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone. 
2. Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combinations of not more than two different solids are taken).

Chapter 14 – Statistics

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

Topic Covered:

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

Chapter 15 – Probability

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

Topic Covered:

Classical definition of probability. Simple problems on finding the probability of an event.

CBSE Class 10 Maths Exam Pattern 2020

The Central Board of Secondary Education conducts the CBSE Class 10 Maths Exam for a total of 100 marks where 80 marks are for theory paper and 20 marks for internal assessment. By going through the CBSE Maths exam pattern 2020 for class 10, students can understand how the marks are awarded to every answer based on the steps. The board releases the sample paper and marking scheme before the board exam each year so that students can get well versed with the exam pattern and question paper design. Students who are appearing for the exams must go through the Marking Scheme 2020. According to CBSE revised pattern 25% of questions from theory paper are objective questions. While the number of questions has increased, marks per question have been reduced, converting maximum questions to objective types or very short answer type. Here you can find the CBSE Class 10 Maths Exam pattern.

CBSE Class 10 Chapter-wise marks weightage

Every year the board releases the sample paper and chapter-wise marks weightage before the board exam so that students get well versed with the exam pattern and question paper design. The marking scheme for the exam 2020 is released by the CBSE on the official site. Here you can find the CBSE Class 10 Chapter-wise marks weightage according to the previous year question papers.

Question Paper Design- Standard Maths and Basic Maths

MATHEMATICS – Standard Question Paper Design Class X (2020-2021)

Time – 3 hours

Max. Marks – 80

MATHEMATICS – Basic Question Paper Design Class X (2020-2021)

Time – 3 hours

Max. Marks – 80