CBSE Class 9th Mathematics NCERT Solution PDF

Description

Boost your CBSE exam preparation with Easy Study Notes’ Class 9th Mathematics NCERT solution, crafted to help students understand, memorize, and revise key concepts faster. These handwritten and chapter-wise organized notes simplify every topic from Algebra,Geometry and Trigonometry — making learning easy and effective for every Class 9th student.

Each note is carefully prepared as per the latest CBSE syllabus (2025–26), focusing on concept clarity, formulas, diagrams, and NCERT-based questions to ensure maximum marks in exams.

Key Features & Benefits

• Complete CBSE Class 9th Mathematics syllabus covered
• Handwritten & easy-to-understand format for visual learning
• Includes Algebra, Geometry, and Trigonometry chapters
• Concepts explained pointwise for quick revision
• NCERT questions, formulas, and diagrams included
• Exam-focused content prepared by experienced teachers
• Instant PDF download – study anytime, anywhere
• Created by Easy Study Notes, trusted by thousands of students

Subjects Covered

• Number Systems : Real Numbers,Review of Representation of Natural,Numbers,Integers
• Algebra: Polynomials,Linear Equations, 
• Geometry: Coordinate Geometry,Euclid’s Geometry,Lines and         Angles,Triangles,Quadrilaterals and Circles
• Measuration: Areas,Surface Area and Volumes
• Statistics: Bar Graphs,Histogram and Frequency Polygon.

Each chapter includes key terms, solved examples, and visual illustrations to make complex topics simple and engaging.

Chapter-Wise Coverage (As Per Latest CBSE Syllabus)

UNIT-I-NUMBER SYSTEM

Real Numbers:

Review of Representation of Natural Numbers

Integers

Rational Number on the Number Line

 Rational Number as a recurring/non-terminating decimals

Operations on Real Numbers

Example of non-recuring/non-terminating

decimals

Existance of Non-rational

Numbers(irrational numbers) such as √2

and √3 and their representation on number

line and conversely,viz every point on the

number line represents a unique real

number

Defination of nth root of real number

Rationalization (with precise meaning) of

real number of the type 1/a+B√x and 1/√x+√y and

their combinations) where x and y are

natural number and a and b are integers

Recalls of laws of exponents with integeral

powers.Rational exponents with positive

real bases (to be done by particular cases

,allowing learner to arrive at the general

laws)

UNIT-II ALGEBRA

1. POLYNOMIALS

Defination of a polynomial in one variable

with one example and counter examples

Coefficients of a polynomial

Terms of a polynomial and zero

polynomial

Degree of Polynomial

Constant,Linear,Quadratic and cubic

polynomial

Monomial,Binomials,Trinomials

Factors and Multiples

Zeros of a polynomial

Motivate and State The Remainder

Theoram with examples

Statement and proof of the Factor Theoram

Factorization of ax2+bx+c,a≠0 where a,b

and c are real number

Cubic Polynomial using the factor theorem

Recall of algebraic expression and

identities.Verification of Identities:

(x+y+z)2=x2+y2+z2+2xy+2yz+2zx

(x+y)3=x3+y3+3xy(x+y)

(x-y)3=x3-y3-3xy(x-y)

x3+y3=(x+y)(x2+xy+y2)

x3-y3=(x-y)(x2-xy+y2)

x3+y3+z3-3xyz=(x+y+z)(x2+y2+z2-xy-yz-zx)

and their use in factorization of polynomials

2. LINEAR EQUATION IN TWO

VARIABLES

Recall of linear equations in one variable

Introduction to the equation in two variables

Focus on linear equations of the type

ax+by+c=0

Explain that a linar equation in two

variables has infinitely many solutions and

justify their being written as ordered pairs

of real numbers

Plotting them and showing that they lie on

line .

UNIT-III

COORDINATE GEOMETRY

Coordinate Geometry

The Cartesian Plane

Coordinates of a point

Names and terms associated with the

coordinate plane

Notations

UNIT-IV:

GEOMETRY

Introduction to Euclid’s Geometry

History – Geometry in India and Euclid’s

geometry. Euclid’s method of formalizing

observed phenomenon into rigorous

Mathematics with definitions,

common/obvious notions,

axioms/postulates and theorems.

The five postulates of Euclid.

Showing the relationship between axiom

and theorem, for example:

(Axiom) 1. Given two distinct points, there

exists one and only one line through them.

(Theorem) 2. (Prove) Two distinct lines

cannot have more than one point in

common.

Lines and Angles

(Motivate) If a ray stands on a line, then

the sum of the two adjacent angles so

formed is 180O and the converse.

(Prove) If two lines intersect, vertically

opposite angles are equal.

(Motivate) Lines which are parallel to a

given line are parallel.

TRIANGLES

(Motivate) Two triangles are congruent if

any two sides and the included angle of

one triangle is equal to any two sides and

the included angle of the other triangle

(SAS Congruence).

(Prove) 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).

(Motivate) Two triangles are congruent if

the three sides of one triangle are equal to

three sides of the other triangle (SSS

Congruence).

(Motivate) Two right triangles are

congruent if the hypotenuse and a side of

one triangle are equal (respectively) to the

hypotenuse and a side of the other triangle.

(RHS Congruence)

(Prove) The angles opposite to equal sides

of a triangle are equal.

(Motivate) The sides opposite to equal

angles of a triangle are equal.

QUADRILATERALS

(Prove) The diagonal divides a

parallelogram into two congruent triangles.

(Motivate) In a parallelogram opposite

sides are equal, and conversely.

(Motivate) In a parallelogram opposite

angles are equal, and conversely.

(Motivate) A quadrilateral is a

parallelogram if a pair of its opposite sides

is parallel and equal.

(Motivate) In a parallelogram, the

diagonals bisect each other and conversely.

(Motivate) In a triangle, the line segment

joining the mid points of any two sides is

parallel to the third side and in half of it

and (motivate) its converse.

CIRCLES

1.(Prove) Equal chords of a circle subtend

equal angles at the center and (motivate) its

converse.

2.(Motivate) The perpendicular from the

center of a circle to a chord bisects the

chord and conversely, the line drawn

through the center of a circle to bisect a

chord is perpendicular to the chord.

 

3. (Motivate) Equal chords of a circle (or of

congruent circles) are equidistant from the

center (or their respective centers) and

conversely.

4.(Prove) 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.

5.(Motivate) Angles in the same segment

of a circle are equal.

6.(Motivate) If a line segment joining two

points subtends equal angle at two other

points lying on the same side of the line

containing the segment, the four points lie

on a circle.

7.(Motivate) The sum of either of the pair

of the opposite angles of a cyclic

quadrilateral is 180° and its converse.

 

UNIT-V MENSURATION

1.AREA

Area of a triangle using Heron’s formula

(without proof)

2.SURFACE AREA AND VOLUMES

Surface areas and volumes of spheres

(including hemispheres) and right circular

cones.

UNIT-V STATISTICS

STATISTICS

Bar graphs,

Histograms (With Varying Base Lengths)

Frequency Polygons

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