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### Class 9 RD Sharma Book :

Class 9 R.D Sharma Book पूरी तरह से NCERT based  किताब है जिसका अध्यन करके आप अपनी कक्षा में 100 % मार्क्स को भी प्राप्त कर सकते है |

Class 9 RD Sharma Maths में आपको हर Chapter के लास्ट में बहुत से Question Answers भी मिलते है जो की आपकी तयारी को और भी ज्यादा मजबूत बनाने की क्षमता रखते है |

Class 9 RD Sharma Maths Book की pdf download करने से पहले आपका यह जानना जरुरी है की इसके अन्दर आपको कौन कौन से Chapters मिलने वाले है | जिसके लिए हमने नीचे Class 9 RD Sharma Maths book के INDEX को दिया हुआ है |

1. Number System

2. Exponents of Real Numbers

3. Rationalisation

4. Algebraic Identities

5. Factorisation of Algebraic Expressions

6. Factorisation of Polynomials

7. Introduction to Euclid’s Geometry

8. Lines and Angles

9. Triangle and Its Angles

10. Congruent Triangles

11. Co-Ordinate Geometry

12. Herons Formula

13. Linear Equations in Two Variables

15. Areas of Parallelograms and Triangles

16. Circles

17. Constructions

18. Surface Area and Volume of a Cuboid and Cube

19. Surface Area and Volume of a Right Circular Cylinder

20. Surface Area and Volume of a Right Circular Cone

21. Surface Area and Volume of Sphere

22. Tabular Representation of Statistical Data

23. Graphical Representation of Statistical Data

24. Measures of Central Tendency

25. Probability

Chapter 1 Number System

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The number the numeral represents is called its value. THE NATURAL NUMBERS: The natural (or counting) numbers are from 1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Natural numbers are infinite numbers. Chapter 1 of Class 9 Maths mainly deals with problems based on rational and irrational numbers, natural numbers, whole numbers, representation of real numbers, rationalisation and laws of exponents and many more.

Chapter 2 Exponents of Real Numbers

Exponents are superscript numerals that let you know how many times you should multiply a number by itself. Some real-world applications include understanding scientific scales like the pH scale or the Richter scale, using scientific notation to write very large or tiny numbers and taking measurements. The three laws of exponent are firstly, multiplication of identical bases and addition of exponents. Secondly, dividing the similar bases and subtracting the exponent. Thirdly, multiplication of exponent when two or more exponents and just one base is present. If a is a positive real number and n is a positive integer, then the principal nth root of a is the unique positive real number x such that xn = a. The principal nth root of a is denoted by a (1/n). Chapter 2 helps students to understand concepts like Integral Exponents of a Real Number, Laws of Integral Exponents and Rational Exponents of a Real Number. We have prepared the Solution to facilitate easy learning and help students understand the concepts of exponents of Real Numbers, and free RD Sharma solutions are provided here.

Chapter 3 Rationalisation

A fraction whose denominator is a surd can be simplified by making the denominator rational. This process is called rationalizing the denominator. If the denominator has just one term that is a surd, the denominator can be rationalized by multiplying the numerator and denominator by that surd. “Rationalizing the denominator” is when we move a root (like a square root or cube root) from the bottom of a fraction to the top. Chapter 3 is about different algebraic identities and rationalization of the denominator. Rationalization is a process by which radicals in the denominator of a fraction are eliminated. In this chapter, students will learn to simplify algebraic expressions using identities.

Chapter 4 Algebraic Identities

The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and solving different polynomials. The algebraic identities are verified using the substitution method. In this method, substitute the values for the variables and perform the arithmetic operation. The three algebraic identities in Maths are:

Identity 1: (a+b)^2 = a^2 + b^2 + 2ab

Identity 2: (a-b)^2 = a^2 + b^2 – 2ab

Identity 3: a^2 – b^2 = (a+b) (a-b)

The chapter consists of exercise wise problems that are solved based on concepts like Identities, Identity for the square of a trinomial, Identity for the cube of a binomial, Sum and Difference of Cubes and One More Identity.

Chapter 5 Factorisation of Algebraic Expressions

The process of finding components (or factors) which when multiplied give the original expression is called factorisation. A number or quantity that when multiplied with another number produces a given number or expression. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. That means you can create 12 using some of these ingredients. Similarly, algebraic terms also have factors. These are the ingredients that make that term. The process of factorisation can be defined as the disintegration of a term into smaller factors. Whereas, the algebraic expressions are built up of variables, integer constants, and basic arithmetic operations of algebra. The chapter explains basic concepts like types of factorisation, factorisation as a sum or difference of two cubes, factorisation using the formulae for the cube of a binomial and factorisation of algebraic expressions.

Chapter 6 Factorisation of Polynomials

In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Chapter 7 Introduction to Euclid’s Geometry

Euclid, a Greek Mathematician, employed some of his axioms and theorems to study planes and other solid figures. The word “geometry” comes from the Greek words “geo”, which means the “earth”, and “metron”, which means “to measure”. Euclidean geometry is a mathematical system attributed to Euclid, a teacher of mathematics in Alexandria in Egypt. Euclid gave us an exceptional idea regarding the basic concepts of geometry, in his book called “Elements”. Euclid’s system is considered an extremely deductive, comprehensive and logical approach towards geometry. Though Euclid’s geometry is also about shapes, lines and angles, the students need to have an in-depth understanding of the topics to be able to understand how the shapes, lines and angles interact with each other. The Chapter explains students the Axioms and Theorems, Incidence Properties, Parallel and Intersecting Lines, Line Segment, Length Axioms and Plane.

Chapter 8 Lines and Angles

The pairs of lines are nothing but the two lines, which may be intersecting or parallel or perpendicular. The region between two infinitely long lines pointing a specific direction (ray) from a common point (or vertex) is termed as an angle. The complete notes on lines and angles are given, which covers the various concepts such as parallel lines, transversal, angles, intersecting lines, interior angles are explained with the examples.

Chapter 9 Triangle and Its Angles

A triangle has three angles. The sum of the measures of the angles is always 180° in a triangle. A triangle that has one obtuse angle is called an obtuse triangle. When a triangle has three congruent sides, we call the triangle an equilateral triangle. A triangle is a 2D geometrical figure consisting of three edges and three vertices—the three angles made by the three sides of a triangle. Triangles are categorized based on their angles and sides.
Triangles categorized based on their angles: Acute Triangle, Right Angle and Obtuse Triangle.
Triangles categorized based on their sides: Scalene triangle, Isosceles triangle and Equilateral triangle.

Chapter 10 Congruent Triangles

When two triangles are congruent, they will have the same three sides and the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there. ASA stands for “angle, side, angle” and means that we have two triangles where we know two angles and the included side are equal. If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. This chapter has important concepts like congruence of line segments, angles, triangles, congruence criteria and inequality relations.

Chapter 11 Co-Ordinate Geometry

Coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates. Coordinate Geometry (or the analytic geometry) describes the link between geometry and algebra through graphs involving curves and lines. It provides geometric aspects in Algebra and enables them to solve geometric problems. It is a part of geometry where the position of points on the plane is described using an ordered pair of numbers. Here, the concepts of coordinate geometry (also known as Cartesian geometry). Coordinate geometry, it is possible to find the distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc.

Chapter 12 Heron’s Formula

In geometry, Heron’s formula (sometimes called Hero’s formula), named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first. For a quadrilateral, when one of its diagonal value and the sides are given. Then the area can be calculated by splitting the given quadrilateral into two triangles and use the Heron’s formula. The Chapter explains essential topics like introduction, Heron’s formula and its applications.

Chapter 13 Linear Equations in Two Variables

If a, b, and r are real numbers (and if a and b are not both equal to 0), then ax+by = r is called a linear equation in two variables. (The “two variables” are the x and the y.) The numbers a and b are called the coefficients of the equation ax+by = r. An equation of the form ax + by + c = 0; where a, b, c are real numbers, a, b are not equal to zero, and x, y are variable, is called a linear equation in two variables. For example, 2x + 3y = 0 and 2 – x + y = 0. This chapter consists of topics like Linear Equations in two variables, solution of a linear equation, Graph of a linear equation in two variables and equations of lines parallel to the X-axis and Y-axis.

In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices. A quadrilateral is a trapezoid or a trapezium if 2 of its sides parallel to each other. A quadrilateral is a parallelogram if two pairs of sides parallel to each other. Squares and Rectangles are special types of parallelograms. With four interior angles present, the sum of the interior angles of a quadrilateral is 360 degrees
,/br> Quadrilaterals are classified based on their intersecting nature. If they are not intersecting, they are called simple quadrilaterals. Otherwise, if they happen to self-intersect, it is a complex quadrilateral. Simple quadrilaterals are further classified into concave and convex quadrilaterals based on the position of diagonals and their interior angles. The Chapter has problems solved using the essential topics like Quadrilateral and some terms associated to it, Angle sum property, Types of quadrilaterals, Properties of a parallelogram, Properties of a rectangle, rhombus and a square and useful facts about the triangle.

Chapter 15 Areas of Parallelograms and Triangles

Parallelogram, in Geometry, is a non-self-intersecting quadrilateral with two pairs of parallel sides. A trapezoid is a quadrilateral with only one pair of parallel sides. In a parallelogram, the diagonals bisect each other. Whereas, a triangle is a polygon with three vertices and three edges. A triangle can be classified into the isosceles triangle, equilateral triangle, and scalene triangle. The diagonal of a parallelogram are of equal lengths, and each diagonal of a parallelogram separates it into two congruent triangles. The Chapter is based on essential topics like figures on the same base and between the exact parallels, polygonal regions and area axioms.

Chapter 16 Circles

Circles are geometric figures whose points all lie the same distance from a given point, the circle’s centre. In brief, a circle is a closed plane curve consisting of all points at a given distance from a fixed point within it, which is called the centre of the circle. The centre of a circle always lies in the interior of the circle. The distance between any points of the circumference of the circle and the centre of the circle is called the radius. Also, the longest chord of a circle is its diameter. Chapter 16 mainly deals with basic definitions, arcs of a circle, chord and segment of a circle, congruence of circles and arcs, some results on equal chords, arcs and angles subtended by them and cyclic quadrilateral.

Chapter 17 Constructions

“Construction” in Geometry means to draw shapes, angles or lines accurately. These constructions use an only compass, straightedge (i.e. ruler) and a pencil. It is the “pure” form of geometric construction: no numbers involved! These constructions are considered as a pure form of geometric constructions since no other measuring devices like protractors are used. Euclid used this method of construction for various basic constructions including bisecting a line, drawing different angles, constructing shapes like triangles, etc. This chapter helps students understand basic constructions, construction of standard angles and construction of triangles.

Chapter 18 Surface Area and Volume of a Cuboid and Cube

Surface area and volume are calculated for any three-dimensional geometrical shape. The surface area of any given object is the area or region occupied by the surface of the object. At the same time, the volume is the amount of space available in an object. In geometry, there are different shapes and sizes, such as a sphere, cube, cuboid, cone, cylinder, etc.

A cuboid is a three-dimensional shape, so to find the volume of a cuboid, we need to know three measurements i.e. Length, Width and Height. The volume of the cuboid is equal to the product of the area of one surface and height. Volume of the cuboid = (length × breadth × height) cubic units. Volume of the cuboid = (l × b × h) cubic units. Example: If the length, breadth and height of a cuboid are 5 cm, 3 cm and 4 cm, then find its lateral surface area.

Chapter 19 Surface Area and Volume of a Right Circular Cylinder

The surface area of a closed right circular cylinder is the sum of the area of the curved surface and the area of the two bases. The curved surface that joins the two circular bases is said to be the lateral surface of the right circular cylinder. Chapter 19 is about right circular cylinder which is a 3D geometrical structure having two circular bases and two parallel faces. In this chapter, students will learn how to find the surface area and volume of a cylinder.

Chapter 20 Surface Area and Volume of a Right Circular Cone

The surface area of any right circular cone is the sum of the area of the base and lateral surface area of a cone. The surface area is measured in terms of square units.
Total surface area: The total area occupied by the surface, including the curved part and the base(s).
Curved surface area: The area occupied by the surface excluding the base(s) is known as curved surface area.
Volume: The space occupied by an object, which is measured in cubic units.

Chapter 21 Surface Area and Volume of Sphere

A sphere does have a surface and volume, and all we need to know to calculate the volume and surface area of a sphere is the measure of its radius! Now, just like with a circle, we can divide a sphere in half. Half a sphere is called a hemisphere, and it has precisely one-half the volume of the entire sphere. A sphere is a perfectly round geometrical 3-dimensional object. It can be characterized as the set of all points located distance r (radius) away from a given point (centre). It is mainly a three-dimensional object which has volume and surface area. Sphere, the section of a sphere by a plane, surface area and volume are the main concepts discussed in brief under this chapter.

Chapter 22 Tabular Representation of Statistical Data

Tabulation, i.e. Tabular Presentation of data is a method of presentation of data. It is a systematic and logical arrangement of data in the form of Rows and Columns to the characteristics of data. It is an orderly arrangement which is compact and self-explanatory. After the collection of data, the person has to find ways to arrange them in tabular form to observe their features and study them. Such an arrangement is called a presentation of data. The tabular representation of statistical data is necessary to make it easier for the observer to understand. In this chapter, students understand the steps to be followed to represent the data in tabular form. The chapter has topics like Statistics, statistical data, presentation of data, frequency distribution, construction of frequency distribution table and cumulative frequency distribution.

Chapter 23 Graphical Representation of Statistical Data

A chart is a graphical representation of data, in which “the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart”. A data chart is a type of diagram or graph that organizes and represents a set of numerical or qualitative data. The graphical representation is a better way to analyze any numerical data. A graph is a type of chart, which the statistical data are represented (In the form of lines or curves). Graphs help us to understand the relationship between variables and measure the position or values of one variable when absolute value changes the other variable. The Chapter has topics like graphical representation of data, bar graph, histogram and method of constructing a frequency polygon.

Chapter 24 Measures of Central Tendency

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. Associated with statistics, measures of central tendency is the typical or central value of a probability distribution. The measure of Central Tendency attempts to signify a group of data by a single value; it is also called measures of central location. We are all familiar with at least one of the measures of central tendency, wherein Mean is the most commonly used one. The mean of data by the simplest of words means the average of the given data. Other measures of central tendency are median and mode. The Chapter explains essential and basic concepts like measures of central tendency, the arithmetic mean of grouped data and median.

Chapter 25 Probability

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur, how likely they are to happen, using probability. A probability is a real number between 0 and 1 that describes how likely it is that an event will occur. A probability of 0 means that an event will never occur. A probability of 1 means that an event will always occur. A probability of 0,5 means that an event will occur half the time or one time out of every 2. The concepts which are explained here are some of the terms related to probability and operations. “Probability is a measure of the possibility that an event will occur”. It is qualified as a number between zero and one. A simple example of probability is tossing of the coin. A coin consists of two sides, head and a tail, which means there are only two outcomes. The probability of tails equals the probability of heads. The probability of tails or heads is 1/2 since there are no other outcomes.

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