## Maths CBSE 9th

We have covered each chapter of Maths CBSE class 9th under the heads such as Concept Map, Revision Notes, NCERT Textbook (Clean PDF), NCERT Solution, Case Study Questions, Assertion reason, HOTS (Higher Order Thinking Skills), Multiple Choice Questions, Very Short Answer Type Questions, True False, Fill in the Blanks, Short Answer Type Questions, Long Answer Type Questions, Practicals/Activities (PDF) and Chapter wise Solved Test Papers (PDF).

Along with the above materials we have also uploaded various Sample Papers which are strictly according to the pattern given by CBSE.

All the study material is completely FREE.

## Syllabus

The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society. The present revised syllabus has been designed in accordance with National Curriculum Framework 2005 and as per guidelines given in the Focus Group on Teaching of Mathematics which is to meet the emerging needs of all categories of students. For motivating the teacher to relate the topics to real life problems and other subject areas, greater emphasis has been laid on applications of various concepts. The curriculum at Secondary stage primarily aims at enhancing the capacity of students to employ Mathematics in solving day-to-day life problems and studying the subject as a separate discipline. It is expected that students should acquire the ability to solve problems using algebraic methods and apply the knowledge of simple trigonometry to solve problems of height and distances. Carrying out experiments with numbers and forms of geometry, framing hypothesis and verifying these with further observations form inherent part of Mathematics learning at this stage. The proposed curriculum includes the study of number system, algebra, geometry, trigonometry, mensuration, statistics, graphs and coordinate geometry, etc. The teaching of Mathematics should be imparted through activities which may involve the use of concrete materials, models, patterns, charts, pictures, posters, games, puzzles and experiments.

**Objectives: **The broad objectives of teaching of Mathematics at secondary stage are to help the learners to:

- consolidate the Mathematical knowledge and skills acquired at the upper primary stage;
- acquire knowledge and understanding, particularly by way of motivation and visualization, of basic concepts, terms, principles and symbols and underlying processes and skills;
- develop mastery of basic algebraic skills;
- develop drawing skills;
- feel the flow of reason while proving a result or solving a problem;
- apply the knowledge and skills acquired to solve problems and wherever possible, by more than one method;
- to develop ability to think, analyze and articulate logically;
- to develop awareness of the need for national integration, protection of environment, observance of small family norms, removal of social barriers, elimination of gender biases;
- to develop necessary skills to work with modern technological devices and mathematical software’s.
- to develop interest in mathematics as a problem-solving tool in various fields for its beautiful structures and patterns, etc.
- to develop reverence and respect towards great Mathematicians for their contributions to the field of Mathematics;
- to develop interest in the subject by participating in related competitions;
- to acquaint students with different aspects of Mathematics used in daily life;
- to develop an interest in students to study Mathematics as a discipline.

**UNIT I: NUMBER SYSTEMS**

**1. Real Numbers**

- Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.
- Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as `\sqrt{2}, \sqrt{3}`, and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.
- Definition of `n^{th}` root of a real number.
- Rationalization (with precise meaning) of real numbers of the type `\frac{1}{a+b\sqrt{x}}` and `\frac{1}{\sqrt{x}+\sqrt{y}}` (and their combinations) where `x` and `y` are natural number and `a` and `b` are integers.
- Recall of laws of exponents with integral 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**

- Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of `ax^{2}+bx+c=0, a≠0` where `a`, `b` and `c` are real numbers, and of cubic polynomials using the Factor Theorem.
- Recall of algebraic expressions and identities. Verification of identities:
- `(x+y+z)^{2}`=`x^{2}`+`y^{2}`+`z^{2}`+`2xy`+`2yz`+`2zx`;
- `(x\pm y)^{3}`=`x^{3}\pm y^{3}\pm 3xy (x\pm y)`;
- `x^{3}+y^{3}`= `(x+y)(x^{2}- xy+y^{2})`;
- `x^{3}-y^{3}`= `(x-y)(x^{2}+xy+y^{2})`; and their use in factorization of polynomials.

**2. Linear Equations 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 linear 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 a 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**

**1. 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:**Given two distinct points, there exists one and only one line through them.**Theorem: (Prove)**Two distinct lines cannot have more than one point in common.

**2. Lines and Angles**

**(Motivate)**If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse.**(Prove)**If two lines intersect, vertically opposite angles are equal.**(Motivate)**Lines which are parallel to a given line are parallel.

**3. 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.

**4. 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.

**5. Circles**

**(Prove)**Equal chords of a circle subtend equal angles at the center and**(motivate)**its converse.**(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.**(Motivate)**Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.**(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.**(Motivate)**Angles in the same segment of a circle are equal.**(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.**(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. Areas**

- Area of a triangle using Heron’s formula (without proof)

**2. Surface Areas and Volumes**

- Surface areas and volumes of spheres (including hemispheres) and right circular cones.

**UNIT VI: STATISTICS & PROBABILITY**

**Statistics**

- Bar graphs, histograms (with varying base lengths), and frequency polygons.

## Marks Distribution

Units | Unit Name | Marks |
---|---|---|

I | NUMBER SYSTEMS | 10 |

II | ALGEBRA | 20 |

III | COORDINATE GEOMETRY | 04 |

IV | GEOMETRY | 27 |

V | MENSURATION | 13 |

VI | STATISTICS & PROBABILITY | 06 |

Total | 80 |

### Internal Assessment

INTERNAL ASSESSMENT | 20 MARKS |
---|---|

Pen Paper Test and Multiple Assessment (5+5) | 10 Marks |

Portfolio | 05 Marks |

Lab Practical (Lab activities to be done from the prescribed books) | 05 Marks |

## Design of PAPER

**QUESTION PAPER DESIGN****CLASS – IX** **(2022-23)**

Time: 3 Hrs.

Max. Marks: 80

Typology of Questions | Total Marks | % Weightage (approx.) |
---|---|---|

Remembering: Exhibit memory of previously learned material by recalling facts, terms, basic concepts, and answers.Understanding: Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas | 43 | 54 |

Applying: Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way. | 19 | 24 |

Analysing: Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizationsEvaluating: Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria.Creating: Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions | 18 | 22 |

Total | 80 | 100 |

## Prescribed Books

**PRESCRIBED BOOKS:**

**Mathematics**– Textbook for class IX – NCERT Publication**Mathematics exemplar**problems for class IX, NCERT publication.**Laboratory Manual**– Mathematics, secondary stage – NCERT Publication**Guidelines for Mathematics Laboratory**in Schools, class IX – CBSE Publication

**Latest NCERT Mathematics Textbook of class 9 is available in PDF. Click on the chapter to download:**

**Chapter 1:** Number systems **(PDF)****Chapter 2:** Polynomials **(PDF)****Chapter 3:** Coordinate Geometry **(PDF)****Chapter 4:** Linear Equations in Two Variables **(PDF)****Chapter 5:** Introduction to Euclid Geometry **(PDF)****Chapter 6:** Lines and Angles **(PDF)****Chapter 7:** Triangles **(PDF)****Chapter 8:** Quadrilaterals **(PDF)****Chapter 9:** Areas of Parallelograms and Triangles **(PDF)****Chapter 10:** Circles **(PDF)****Chapter 11:** Constructions **(PDF)****Chapter 12:** Heron’s Formula **(PDF)****Chapter 13:** Surface Areas and Volumes **(PDF)****Chapter 14:** Statistics **(PDF)****Chapter 15:** Probability **(PDF)**

**Latest NCERT Mathematics Exemplar Problems of class 9 is available in PDF. Click on the chapter to download:**

**Chapter 1:** Number Systems **(PDF)****Chapter 2:** Polynomials **(PDF)****Chapter 3:** Coordinate Geometry **(PDF)****Chapter 4:** Linear Equations in Two Variables **(PDF)****Chapter 5:** Introduction to Euclid’s Geometry **(PDF)****Chapter 6:** Lines and Angles **(PDF)****Chapter 7:** Triangles **(PDF)****Chapter 8:** Quadrilaterals **(PDF)****Chapter 9:** Areas of Parallelograms and Triangles **(PDF)****Chapter 10:** Circles **(PDF)****Chapter 11:** Constructions **(PDF)****Chapter 12:** Heron’s Formula **(PDF)****Chapter 13:** Surface Areas and Volumes **(PDF)****Chapter 14:** Statistics and Probability **(PDF)**

## NCERT Solution

**Chapter 1:** Number systems**Chapter 2:** Polynomials**Chapter 3:** Coordinate Geometry**Chapter 4:** Linear Equations in Two Variables**Chapter 5:** Introduction to Euclid Geometry**Chapter 6:** Lines and Angles**Chapter 7:** Triangles**Chapter 8:** Quadrilaterals**Chapter 9:** Areas of Parallelograms and Triangles**Chapter 10:** Circles**Chapter 11:** Constructions**Chapter 12:** Heron’s Formula**Chapter 13:** Surface Areas and Volumes**Chapter 14:** Statistics**Chapter 15:** Probability