NCERT Solutions For Class 9 Maths

NCERT Solutions For Class 9 Maths Solutions In Hindi & English

NCERT Solutions for class 9 Maths all chapters in PDF form are given below for session 2021-22. A separate PDF of all exercises are given to download. Download Apps for class 9 Maths in Hindi and 9 Maths in English Free. NCERT Books 2021-22 as well as Exemplar problems books both are equally important for the CBSE exams. Chapter wise assignments Test Papers, Previous year Question Papers issued by CBSE and other schools, Chapter wise tests, Syllabus for the academic year 2021-22 and other online study material.

NCERT Solutions For Class 9 Maths

Class:9
Subject:Maths – गणित
Contents:NCERT Solutions

NCERT Solutions for class 9 Maths Free PDF Download

CHAPTER 1: NUMBER SYSTEMS

Representation of terminating/non-terminating recurring decimals on the number line (and successive magnification method). Presentation of square roots of 2, 3 and other non-rational numbers. Rationalization of real numbers, laws of integral powers and rational exponents with positive real bases in Number Systems.

CHAPTER 2: POLYNOMIALS

Examples and definition of a polynomial, coefficient, degrees, zeroes and terms of a polynomial. Constant, linear, quadratic and cubic polynomials, monomials, binomials, trinomials. Factors and multiples, Remainder and factor theorems, factorization of a polynomials using factor theorem.

CHAPTER 3: COORDINATE GEOMETRY

In this chapter Coordinate Geometry we will study the concept of cartesian plane, coordinates of a point in this xy – plane, name, terms, notations and other terms associated with the coordinate plane. Abscissa and ordinate of a points. Plotting a point in xy – plane and naming it.

CHAPTER 4: LINEAR EQUATIONS IN TWO VARIABLES

This chapter provides the introduction to the equation in two variables of the type ax + by + c = 0. Proving a linear equation has infinite number of solutions. Plotting a linear equation on graph and justification of any point on line. Problems based on Linear Equations in Two Variables in daily life.

CHAPTER 5: INTRODUCTION TO EUCLID’S GEOMETRY

Euclid’s geometry and history of Indian geometry. Introduction to Euclid’s Geometry provides a way of defining the common geometrical shapes and terms. Euclid’s five postulates and redefining (equivalent versions) the fifth postulates.  Relationship between axiom, postulates and theorems.

CHAPTER 6: LINES AND ANGLES

There are two theorems in Lines and Angles chapter which may be asked for proof. First is “If two lines intersect, vertically opposite angles are equal” and second is “The sum of the angles of a triangle is 180.”. Rest theorems are given for motivations and questions will be asked on the basis of all these theorems (Motivate/Prove).

CHAPTER 7: TRIANGLES

In this chapter Triangles, two theorems (“Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence)” and “The angles opposite to equal sides of a triangle are equal.”) are given for proof and the rest will be asked in the form of applications/problems.

CHAPTER 8: QUADRILATERALS

The chapter Quadrilaterals contains only one theorem( “The diagonal divides a parallelogram into two congruent triangles”) for proof. Other will be asked in the form of application and conceptual questions. Questions are on the basis of properties of quadrilaterals and combinations of it with triangles.

CHAPTER 9: AREAS OF PARALLELOGRAMS AND TRIANGLES

The combinations of areas of parallelograms and triangles are given to prove in most of the question. Proof and questions based on “Parallelograms on the same base and between the same parallels have the same area” will be asked. Example of median may be used as theorem in most of the questions.

CHAPTER 10: CIRCLES

“Equal chords of a circle subtend equal angles at the center” and “The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle” are given for proof in Circles. The other theorems are important for solving questions based on triangle, quadrilateral and circles.

CHAPTER 11: CONSTRUCTIONS

There are two categories of constructions. One is Construction of bisectors of line segments and angles of measure 60, 90, 45 etc. The other is construction of a triangle given its base, sum/difference of the other two sides and one base angle & given perimeter and base angles.

CHAPTER 12: HERON’S FORMULA

Use of Heron’s  formula to find the area of triangles, quadrilaterals (dividing it into two triangles) and other polygons. Knowledge of formulae of plane figures will also help in doing questions.

CHAPTER 13: SURFACE AREAS AND VOLUMES

Students are familiar with surface areas and volumes as they have already studies mensuration in earlier classes. This chapter also contains problems based on surface areas and volumes of cube, cuboids, cylinders, cones, spheres and hemispheres. Conversion of one figure into the other comparing volumes is also given as an application of mensuration.

CHAPTER 14: STATISTICS

Introduction to statistics includes the presentation of data collected in raw form. Presentation of  data in tabular form by grouping them in proper intervals, drawing a bar graph, histogram or polygon. Finding the measure of central tendency mean, mode and median of raw data.

CHAPTER 15: PROBABILITY

Probability based on observation or frequency approach. Questions based on real life or day to day incidents. Toss of a coin, throwing a dice, based on deck of cards, etc. questions based on simple events.

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NCERT Solutions For Class 9 Maths FAQ

What Are The Theorems That May Be Asked For Proving In Exams For Class 9 Chapter 6: Lines And Angles?

The following theorems may be asked to prove in exams, rest are for application and motivation:
1. If two lines intersect, vertically opposite angles are equal.
2. The sum of the angles of a triangle is 180.

Which Theorems May Be Asked For Proving Form Class 9 Chapter 7: Triangles?

The following theorems may be asked to prove in exams, rest are for application and motivation:
1. Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).
2. The angles opposite to equal sides of a triangle are equal.

Is There Any Theorem In Chapter 10: Quadrilateral To Prove?

Yes, there is only one important theorem (The diagonal divides a parallelogram into two congruent triangles), that may be asked to prove.

How Many Theorems Are There In Chapter 9: Areas Of Parallelograms And Triangles To Prove?

There is only one theorem (Parallelograms on the same base and between the same parallels have equal area.) asked for proving. This theorem is quite important one for all examinations.

Is There Only One Theorem In Chapter 10: Circle, Which Is Asked To Prove In Examinations?

No, there are two theorems (1. Equal chords of a circle subtend equal angles at the center. 2. The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.) for the examination point of view.

Class 9 Maths Important Points Related To All Chapters.

Chapter 1: Number Systems

Class 9 Maths Chapter 1 Consists of the concepts of Natural number, Whole numbers, Integers, etc. and plotting on number lines. Find some rational or Irrational Numbers between two numbers are generally asked in CBSE Exams. For example, converting an irrational number into p/q is asked in most of CBSE Final Term exams 2019-20. Plotting a square root number on number line is also important as examination point of view.

Chapter 2: Polynomials

In 9th Maths Chapter 2 important Questions are based on remainder theorem and factor theorem. For polynomial in one variable, terms, factors and zeroes are to be evaluated. We also have to identify monomials, binomials or trinomials in the given polynomial expression. The concepts of Degree of constant polynomial and others should must be known to all students for periodic test or final exams. Classification of linear, quadratic or cubic polynomial are frequently asked in terminal exams. Factorisation by middle term splitting or using identities are generally asked in all exams.

Chapter 3: Coordinate Geometry

Class 9 Maths Chapter 3 Coordinate Geometry help us to find the location of any object by using two independent information (x-axis and y-axis). This chapter explains the plotting a point in Cartesian Plane as well as to find the coordinate to a point, using two dimensional coordinate system. Concepts about Abscissa, Ordinate, Origin, x-axis, y-axis, coordinate plane, quadrants, etc., are describe properly in the NCERT Solutions provided by Tiwari Academy.

Chapter 4: Linear Equations In Two Variables

Linear equations in two variables of Class 9 Maths provides the techniques to find the multiple solutions of an equations, so that a line on graph can be drawn. Introduction of standard form of linear equations help the students to understand the terms related to a linear equation, like coefficient of x or y or constant term. The application of linear equations in day to day life are also included in the contents provided by Tiwari Academy NCERT Solutions.

Chapter 5: Introduction To Euclid’s Geometry

This chapter explains that what is geometry, its origin and use, its initial fundamental provided by Euclid. Postulates and Axioms are main point to be studied in the chapter 5 Introduction to Euclid’s Geometry. The other version of Euclid’s Fifth postulate is frequently asked in unit test and terminal exams of class 9. Different definitions of geometrical terms like Point, Line, Straight Line, Surface, Curves, etc. are also important part of chapter 5 of class 9 Mathematics.

Chapter 6: Lines And Angles

Class 9 Maths Chapter 6 – Lines and Angles tells us about the concepts of line segments, rays, angles, vertex and its arms, collinear and non-collinear points. It involves the applications of acute, right, obtuse, straight angles also including adjacent angle, reflex angle, complementary & Supplementary angles and vertically opposite angles. Theorem on Vertically Opposite angles is also important as per terminal exams. Exercise 6.1 is based on the application of vertically opposite angles.

Chapter 7: Triangles

Class 9 Maths Chapter 7 Triangles is based on mainly the criteria of congruency of triangles. There are two theorems which may be asked for proof but axioms based on theorems are also important for problem solving. Exercise 7.1 is totally based on SAS and ASA criteria of Congruence Triangles. Exercise 7.2 and 7.3 includes the questions based on RHS and SSS including sum applications of other theorems also. Exercise 7.4 is based on inequalities of sides and angles.

Chapter 8: Quadrilaterals

Chapter 8 Quadrilaterals of Class 9 Maths is based on angle sum property of quadrilateral, types of quadrilateral (Rectangles, Trapezium, Square, Rhombus, Parallelograms, etc.) and its properties. Applications based on parallelogram including theorems and axioms are given to prove. Application of Mid-Point Theorem and its converse is one of the important segment of Chapter 8. Mid-Point Theorem is generally asked to prove in terminal exams as well as unit tests of most of the CBSE School Examinations.

Chapter 9: Areas Of Parallelograms And Triangles

In 9 Maths Chapter 9 the area of parallelograms and triangles is to be determined using theorems and corollaries. The correlation between a triangle and a parallelogram, on the same base and between same parallel line, is also explained with multiple practice questions. The result of examples also can be used as a corollary to prove some questions. Median divides triangle into two triangles in equal area. This result helps a lot in solving the questions of Exercise 9.3 onwards.

Chapter 10: Circles

In Class 9 Maths Chapter 10 Circles and its components – like segment, arc, semicircle, chords, diameter, minor and major sectors are to be discussed. The theorem, Perpendicular from the centre bisect the chord and Equal chords are equidistant from the centre, are widely used for solving questions in Exercise 10.2, 10.3 and 10.4. These are also useful in other exercises of Chapter 10 of class 9 Maths. Exercise 10.5 contains most of the important questions as per examination point of view.

Chapter 11: Constructions

Constructions chapter of Class 9 Maths is generally based on making angles and triangles only using ruler and compass. Exercise 11.1 illustrate the questions related to making various angles like 15, 30, 60, 120, 150, 45, 90, etc., with the help of ruler and compass only i.e. without using protector. Exercise 11.2 provide practice of constructing a triangle when sum of two side or difference of two side and a base angle is given or perimeter of triangle and two based angle is given.

Chapter 12: Heron’s Formula

Class 9 Maths Chapter 12 Heron’s Formula help us to find out the area of a triangle or quadrilateral (splitting it into two triangles). Heron’s Formula is derived for area of all type of triangle when its three sides are known. Area of other polygons also can be evaluated by divided it into suitable triangles. In this chapter, various application based Heron’s Formula are given which are related to day to day life.

Chapter 13: Surface Areas And Volumes

Class 9 Maths Chapter 13 explains the concepts of mensuration. Exercise 13.1, 13.2, 13.3 and 13.4 describes the surface areas of solid figures like Cube, Cuboid, Cone, Cylinder, Sphere and Hemisphere, etc., whereas the Exercise 13.5, 13.6, 13.7 and 13.8 illustrates the concepts of finding volumes of these solid figures. Exercise 13.9 is optional exercise contains very good set of questions for practice. To do this chapter properly, students must derive all the formulae and then learn them.

Chapter 14: Statistics

Exercise 14.1 of class 9 Maths Chapter 14 is based on Primary Data and Secondary Data concepts. Exercise 14.2 emphasize the data in organised way to representation it graphically by using Histogram, Bar Charts or Polygons. Exercise 14.3 is based on questions with miscellaneous type of variation in data. Exercise 14.4 includes the concepts of central tendency of statistical data like Mean, Mode and Median of ungrouped data.

Chapter 15: Probability

Class 9 Chapter 15 Probability is based on outcomes of events using favorable outcomes and total number of outcomes in any experiment. Questions based on coins, deck of cards, dice and day to day life experiences are given in the chapter. Some of the questions given in exercise 15.1 are based on statistical data also. Probability is one of the important chapters for further classes also.

NCERT Solutions For Class 9 Maths Solutions In Hindi & English

कथन सत्य हैं या असत्य? कारण के साथ अपने उत्तर दीजिए। प्रत्येक अपरिमेय संख्या एक वास्तविक संख्या होती है।

सत्य, क्योंकि सभी परिमेय और अपरिमेय संख्याओं का संग्रह ही वास्तविक संख्या होती है।

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Diagonals of parallelogram bisect each other.
Therefore, PO = OR and SO = OQ
In ΔPQS, PO is median. [∵ SO = OQ]
Hence, ar(PSO) = ar(PQO) … (1)
[∵ A median of a triangle divides it into two triangles of equal areas.]
Similarly, in ΔPQR, QO is median. [∵ PO = OR]
Hence, ar(PQO) = ar(QRO) … (2)
And in ΔQRS, RO is median. [∵ SO = OQ]
Hence, ar(QRO) = ar(RSO) … (3)
From the equations (1), (2) and (3), we get
ar(PSO) = ar(PQO) = ar(QRO) = ar(RSO)
Hence, in parallelogram PQRS, diagonals PR and QS divide it into four triangles in equal area.

A circle has only finite number of equal chords. True or False?

False.
Because, there are infinite number of equal chords in a circle.

सीधे गुणा किए बिना निम्न्लिखित मान ज्ञात कीजिए: 103×107

103 × 107
= (100 + 3)(100 + 7)
= (100)^2 + (3 + 7)100 + 3 × 7
[∵〖(x + a)(x + b) = x〗^2 + (a + b)x + ab]
= 10000 + 1000 + 21
= 11021

x-अक्ष और y-अक्ष रेखाओं से बने तल के प्रत्येक भाग के नाम बताइए।

चतुर्थांश

If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a

Given equation of line:
3y = ax + 7.
Putting x = 3 and y = 4,
we have, 3 × 4 = a × 3 + 7
⇒ 12 = 3a + 7
⇒ 12 – 7 = 3a
⇒ a = 5/3

क्या यूक्लिड की पाँचवीं अभिधारणा से समांतर रेखाओं के अस्तित्व का औचित्य निर्धारित होता है? स्पष्ट कीजिए।

हाँ, यूक्लिड की पाँचवीं अभिधारणा से समांतर रेखाओं के अस्तित्व का औचित्य निर्धारित होता है। क्योंकि यदि दो रेखाओं को कोई तीसरी रेखा इस प्रकार कटती है कि एक ही ओर बने आंतरिक कोणों का योग 180° से कम हो, तो दोनों रेखाएँ एक दूसरे को प्रतिच्छेद करती हैं। परन्तु यदि कोणों का योग 180° हो तो रेखाएँ परस्पर प्रतिच्छेद नहीं करती हैं अर्थात समांतर होती हैं।

What do you mean by collinear points or non-collinear points?

If three or more points lie on the same line, they are called collinear points; otherwise they are called non-collinear points.

In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.

Draw the bisectors of ∠A, ∠B and ∠C, which intersect each other at O. Point O is equidistant from the three sides of ΔABC i.e.
OP = OQ = OR.

ABCD एक चतुर्भुज है जिसमें P, Q, R और S क्रमशः भुजाओं AB, BC, CD और DA के मध्य-बिंदु हैं । AC उसका एक विकर्ण है। दर्शाइए कि SR || AC और SR=1/2 AC है।

ACD में,
S भुजा DA का मध्य-बिंदु हैं [∵ दिया है]
R भुजा DC का मध्य-बिंदु हैं [∵ दिया है]
अतः, SR || AC और SR = 1/2 AC
[∵ मध्य-बिंदु प्रमेय]

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