Find here CBSE Exemplar Solutions for CBSE Class 9 Mathematics prepared for benefits of students to score more. These Exemplar solutions for class IX Maths are value based questions prepared by our team of teachers and subject experts.

## Chapter-1 Number Systems

- Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating / non-terminating recurring decimals, on the number line through successive magnification. Rational numbers as recurring/terminating decimals.
- Examples of non-recurring / non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √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, every point on the number line represents a unique real number.
- Existence of √x for a given positive real number x (visual proof to be emphasized).
- Definition of nth root of a real number.
- Rationalization (with precise meaning) of real numbers 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.
- 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.)

## Chapter-2 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 ax² + bx + c, 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 = x² + y² + z² + 2xy + 2yz + 2zx

(x ± y)³ = x³ ± y³ ± 3xy (x ± y)

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.

## Chapter-3 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. Prove 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. Graph of linear equations in two variables. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.

## Chapter-4 Coordinate Geometry

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.

## Chapter-5 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. Equivalent versions of the fifth postulate. 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.

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