## Maharashtra State Board Solutions for Class 9 Maths

Welcome to one of the most important years of your academic life. The year-end would bring you in competition with your peers from across the country for the very first time in your life. Class 9th becomes the foundation stone on which the palace of your academics is built, and this is the year when students are under a lot of pressure from the upcoming board examinations.

Keeping all other subjects aside for a while, mathematics takes an important role because this is the time where the subjects transform drastically to include the core topics that would be taught in the higher classes and be used in real life. Thus, the understanding of these topics becomes all the more important because concepts leftout here would lead to problems in the entire units in further classes and professional courses. To help the students understand the concepts that are essential to the class 9th mathematics, we at Aakash Institute provide all the solutions with a detailed explanation in Maharashtra State Board Solutions for Class 9 Maths.

Maharashtra State Board Solutions for Class 9 Maths contains all the answers in a stepwise manner. The language used in Maharashtra State Board Solutions for Class 9 Maths is easy and understandable, which ensures the students become self-sufficient when equipped with the solutions. The Aakash Maharashtra State Board Solutions for Class 9 Maths is prepared by the industry experts and experienced faculty members which is the testimony of its quality.

### Maharashtra State Board Solutions for Class 9 Maths- Chapter-Wise

- Exercise 1.1
- Exercise 1.2
- Exercise 1.3
- Exercise 1.4

### CHAPTER 2 POLYNOMIALS

we will learn zeroes of polynomial, relationship between zeroes and coefficients of quadratic polynomial and Euclid’s division lemma.

- Exercise 2.1
- Exercise 2.2
- Exercise 2.3
- Exercise 2.4
- NCERT Book Chapter 2

### CHAPTER 3 LINEAR EQUATIONS IN TWO VARIABLES

Focus on linear equations solutions through four methods as graphical method, elimination method, substitution method and cross multiplication method following the word problems.

- Exercise 3.1
- Exercise 3.2
- Exercise 3.3
- Exercise 3.4
- Exercise 3.5
- Exercise 3.6
- Exercise 3.7
- NCERT Book Chapter 3

### CHAPTER 4 QUADRATIC EQUATIONS

gives an idea of solving quadratic equations (if real roots exist) by factorization, completing the square method and quadratic formula.

- Exercise 4.1
- Exercise 4.2
- Exercise 4.3
- Exercise 4.4
- NCERT Book Chapter 4

### CHAPTER 5 ARITHMETIC PROGRESSIONS

A P includes derivation of nth term, sum of first n terms and their application in solving daily life problems.

- Exercise 5.1
- Download Exercise 5.1 in English
- Exercise 5.2
- Download Exercise 5.2 in English
- Exercise 5.3
- Download Exercise 5.3 in English
- Exercise 5.4
- NCERT Book Chapter 5

### CHAPTER 6 TRIANGLES

Triangles the definitions and examples of similar triangles, proof of theorem (Basic Proportionality Theorem-BPT/Thales, Pythagoras, Similar triangle) and questions based on these theorem.

- Exercise 6.1
- Download Exercise 6.1 in English
- Exercise 6.2
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- Exercise 6.3
- Download Exercise 6.3 in English
- Exercise 6.4
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- Exercise 6.5
- Download Exercise 6.5 in English
- Exercise 6.6 (Optional)
- NCERT Book Chapter 6

### CHAPTER 7 COORDINATE GEOMETRY

Coordinate Geometry gives a brief introduction of Distance formula and questions based on equality of sides of triangle, types of quadrilateral, section formula and area of triangle.

- Exercise 7.1
- Download Exercise 7.1 in English
- Exercise 7.2
- Download Exercise 7.2 in English
- Exercise 7.3
- Download Exercise 7.3 in English
- Exercise 7.4
- Download Exercise 7.4 in English
- NCERT Book Chapter 7

### CHAPTER 8 INTRODUCTION TO TRIGONOMETRY

Introduction to Trigonometry describes about identities of trigo to find the missing term of triangle. Important identities proof through right triangle, proofs based on basic identities.

- Exercise 8.1
- Download Exercise 8.1 in English
- Exercise 8.2
- Download Exercise 8.2 in English
- Exercise 8.3
- Download Exercise 8.3 in English
- Exercise 8.4
- Download Exercise 8.4 in English
- NCERT Book Chapter 8

### CHAPTER 9 SOME APPLICATIONS OF TRIGONOMETRY

Applications on trigonometry and Trigonometry are quite new and important topic for class 9, which describe the practical application of trigo in daily life.

- Exercise 9.1(Q1 – Q 8)
- Exercise 9.1 (Q9 – Q16)
- NCERT Book Chapter 9

### CHAPTER 10 CIRCLES

Circles consists of tangent of a circle at point of contact. There are two theorems which will be asked for proof. Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.

- Exercise 10.1
- Exercise 10.2
- NCERT Book Chapter 10

### CHAPTER 11 CONSTRUCTIONS

Constructions gives an idea of dividing a line segment into given ratio, tangents to a circle from a point outside it and construction of a triangle similar to a given triangle. Steps of constructions must be written in order to understand easily.

- Exercise 11.1
- Exercise 11.2
- NCERT Book Chapter 11

### CHAPTER 12 AREAS RELATED TO CIRCLES

Areas related to circles suggest the methods to find area of sectors, segments, area and perimeter of plane figure (triangles, simple quadrilateral and circles). Questions based on sectors or segments are restricted to central angles 60, 90, 120 etc., for class x.

- Exercise 12.1
- Exercise 12.2
- Exercise 12.3
- NCERT Book Chapter 12

### CHAPTER 13 SURFACE AREAS AND VOLUMES

Surface areas and volumes: In this chapter we will calculate the surface area and volumes of combinations of any two/three figures (cubes, cuboids, spheres, hemispheres, cones, cylinders and frustum of cone). Converting one type of shape into another type using the concepts of volume.

- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
- Exercise 13.4
- Exercise 13.5
- NCERT Book Chapter 13

### CHAPTER 14 STATISTICS

Statistics has the general methods of finding mean, mode, median, cumulative frequency graph and relations between three values of central tendency (3 Median = 2 mean + Mode) .

- Exercise 14.1
- Exercise 14.2
- Exercise 14.3
- Exercise 14.4
- NCERT Book Chapter 14

### CHAPTER 15 PROBABILITY

- Probability has classical definition of probability and problems on single events only.
- Exercise 15.1
- Exercise 15.2
- NCERT Book Chapter 15

## Chapter 1 Real Numbers

The chapter starts with Euclid’s division theorem, which states that “for given set of integers a and b, there exist precise integers q and r, which fulfil the equation “a = bq + r”, where ‘r’ lies between 0 and b. HCF and LCM are based on divisibility. Other included topics are Fundamental Arithmetic Theorem, properties of positive integers, conversion of a fraction to decimal and decimal to a fraction. The chapter has four exercises. 1st exercise discusses Euclid’s division lemma, 2nd exercise is based on HCF & LCF, the 3rd exercise is talking about proving the rational or irrational number and the last mean 4th exercise is based on fractions to decimal and decimal to fractions.

### Chapter 2 Polynomials

The chapter starts with the definition of the degree of a polynomial, linear polynomial, quadratic polynomial and cubic polynomial. In this chapter, there are four exercises, including one optional exercise as well. The chapter teaches about the use of a graph to calculate the roots of the polynomial or the value at which the polynomial would return zero value. The chapter also delves into the relationship between zeroes and the coefficient of a polynomial. It helps students locate the zeroes of a quadratic polynomial and cubic polynomial. The chapter also teaches the remainder theorem to divide a polynomial by another polynomial.

### Chapter 3 Pair of Linear Equations in one Variable

This chapter explains the idea of pairs of linear equations. This chapter begins with linear equations in two variables, using the graphical method of their solution, algebraic conditions for linear equations. The chapter further elaborates on the solutions of a pair of linear equations in two variables algebraically by substitution, elimination and cross multiplication method. Students can find different methods of solving these equations easily. This chapter contains seven practice sets, including an algebraic method, elimination method, cross-multiplication methods, and substitution method. Maharashtra State Board Solutions for Class 9 Maths help the students get a comprehensive understanding of such topics.

### Chapter 4 Quadratic Equations

In this chapter, the student will be able to know about the writing of quadratic equations in a standard form. The chapter proceeds to explain the method of solving the quadratic equations through the factorisation method and Shreedharacharya’s formula. This chapter also talks about the basics of a quadratic equation. It additionally integrated everyday sports primarily based totally on the situational troubles of quadratic equations. This chapter includes four exercises, where you have to solve whether the equation is quadratic or not and conversion of the word problem in quadratic equation, find the roots of the quadratic equation factorisation, and the chapter also tells about the nature of roots whether they are real and distinct, real and equal or non-real roots. The chapter also defines the relationships between the roots and the coefficient in the quadratic equation.

### Chapter 5 Arithmetic Progressions

This chapter introduces students to a new topic which is an arithmetic progression. This chapter deals with the concept of sequence and series. The chapter elaborates on arithmetic progression, which is a series in which two consecutive terms have a constant difference. The chapter increases the analytical and problem-fixing talents of students. This chapter constitutes four exercises related to the general form of arithmetic progression, finding the 1st term and the difference of an AP. Questions are also based on finding whether the series is arithmetic progression or not, and it also includes finding out the nth term of arithmetic progression by using the formula provided in the chapter. This chapter will help you get the concepts and help you build the capability to solve problems related to sequences and series. The chapter forms the base of various advanced concepts to come in the higher classes.

### Chapter 6 Triangles

This chapter deals with the definitions and examples of similar triangles. Through this chapter, a student will learn about the same shape but not essentially the same size. It starts with the concept of a similar and congruent figure. The chapter also explains the condition for the similarity of two triangles and theorems according to the similarity of triangular shapes. Further, the areas of similar triangles are explained with a theorem, and at last, Pythagoras theorem and converse of Pythagoras theorem are also covered in the chapter. In this chapter, there are six exercises based on the properties of triangles. Various theorems are also there, which form the basis of the exercises. This chapter also helps you with the concepts involved in higher studies.

### Chapter 7 Coordinate Geometry

This chapter starts with standards of coordinate geometry, graphs of linear equations. The chapter in the class 9th maths revolves around the Distance formula and Section formula. There are sections on the area of the triangle as well. From this chapter, students will know how to calculate the distance between the two points whose coordinates are given. The chapter also teaches them to discover the location and area of the triangle fashioned out of three particular given points that do not lie on the same line. Also, students will get to know how to find the coordinates of the point which divides a line segment joining two given points in a particular ratio. Maharashtra State Board Solutions for Class 9 Maths introduce the students to Distance Formula, Section Formula and Area of a Triangle in this chapter.

### Chapter 8 Introduction to Trigonometry

From this chapter, students will be able to study some ratios of a right triangle with respect to one of its acute angles, which is known as trigonometric ratios of the angles. The chapter also constitutes the trigonometric ratios for angles of 00 to 900. Students also get to know how to calculate trigonometric ratios for some particular angles and establish some particular points involving these ratios, known as trigonometric specification. In this chapter, four exercises consist of various questions on finding Trigonometric Ratios, Trigonometric Identities, and Trigonometric Ratios of Complementary Angles are primary subtopics of this chapter.

### Chapter 9 Some Applications of Trigonometry

In this chapter, students will be able to study the applications of trigonometry. It is mainly 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 locating the heights and distances of diverse objects without measuring them. Students get to learn about the line of sight, attitude of elevation, attitude of depression. Questions are mainly based on the realistic applications of trigonometry. The angle of elevation, Angle of Depression are considered tougher portions of this chapter. In Chapter 9, students should know about trigonometry to find the heights and distances of various objects without real-time measuring of these objects.

### Chapter 10 Circles

Students have a basic idea about circles and various terms related to a circle like a chord, segment, arc, etc. In this chapter, students will be able to study the different situations that arise when a circle and a line are given in a plane. They get accustomed to the idea of Tangent to a Circle and Number of Tangents from a Point on a Circle. The chapter deals with the circle theorems, which help solve various questions in geometry. The chapter contains various questions based on the tangent to the circle and the chords in a circle which make use of the circle theorems, and the solutions of these have been provided in an understandable way in the Maharashtra State Board Solutions for Class 9 Maths.

### Chapter 11 Constructions

This chapter starts with the division of a line segment in a given ratio. Then tangents to a circle from a point outside it and after that the construction of a triangle similar to a given triangle. From this chapter, students can learn about the construction of various geometrical figures. There are two exercises in this chapter from that, student will able to study how to divide a line segment, and also they will understand the construction of tangents to a circle. Methods and steps for construction are explained in understandable language, and also some examples are additionally incorporated to make it clearer to the students.

### Chapter 12 Areas Related to Circles

This chapter constitutes the concepts of perimeter and area of a circle. Applying this concept, the chapter further explains how to find the area of sector and segment of a circular region. Moreover, students will be able to get clarity on finding the areas of some combinations of plane figures involving circles. This chapter has Three exercises regarding questions of the area of a circle and real-life questions, and cost-related questions on areas related to the circular region.

### Chapter 13 Surface Areas and Volume

This chapter deals with finding areas and volumes of different solids such as cube, cuboid and cylinder, and so on. The chapter includes a total of five exercises, consisting of questions based on finding the surface area of an object formed by combining any two of the basic solids like the cuboid, cone, cylinder, sphere and hemisphere. The questions in the chapter are based on finding the volume of objects formed by combining various solid figures or conversion of one shape to another. The Maharashtra State Board Solutions for Class9 Maths provided by the Aakash institute provides the answers to the easy as well as the tough question of the topic.

### Chapter 14 Statistics

Here, students will learn the numerical representation of individual data to grouped data and find the Mean, Mode and Median. Also, this chapter contains the idea of cumulative frequency, cumulative frequency distribution and a way to draw cumulative frequency curves. This chapter forms the basis of data science and analytics. The chapter teaches the students the pictorial representation of the raw, unorganised data from which tangible results can be obtained. There are generally four exercises in the chapter wherein you need to resolve questions primarily based totally on locating mean, median and mode from individual records to that of group records. You must be able to recognise and resolve the questions associated with cumulative frequency, the cumulative frequency distribution, and a process to draw cumulative frequency curves after going through this chapter.

### Chapter 15 Probability

The final chapter deals with Probability. The chapter begins with the general approach of probability. It is the study of obtaining the chances of success or failure in a particular experiment. The chapter also includes the different aspects of experimental probability and theoretical probability. There are many examples given to explain it effectively. But before solving the exercise problems, students have to solve the examples of textbooks, and after that, it will be easy to understand and solve.