CLASS 9 | MATHS | SEBA - Aspire Academy

Course Content

Class 9 Maths

Chapter 1 – Number systems (part1)

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Chapter 1 – Number systems .(Part 2)

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Chapter 1 – Number systems .(Part 3)

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Chapter 1 – Number systems .(Part 4)

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Chapter 1 – Number systems .(Part 5)

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Chapter 1 – Number systems .(Part 6)

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Chapter 1 – Number systems .(Part 7)

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Chapter 1 – Number systems .(Part 8)

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Class 9 Maths Chapter 2 Polynomials demo

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Chapter 2 – Polynomials (Part 1)

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Chapter 2 : Polynomials (Part 2)

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Chapter 2 : Polynomials (Part 3)

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Chapter 2 : Polynomials (Part 4)

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Chapter 2 : Polynomials (part 5)

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Chapter 2 : Polynomials ( Part 6)

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Class 9 Maths Chapter 2 (Extra class) – Polynomials Lecture 1 by bhrigu

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Class 9 Maths Chapter 2 – Polynomials Lecture 2

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Class 9 Maths Chapter 2 – Polynomials Lecture 3

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Class 9 Maths Chapter 2 – Polynomials Lecture 4

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Class 9 Maths Chapter 2 – Polynomials Lecture 5

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Class 9 Maths Ch 2 Polynomials 6

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Class 9 Maths Ch 2 Polynomials 7

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Class 9 Maths Ch 2 Polynomials 8

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Chapter 3 :Co-ordinate Geometry – Basori

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Chapter 4 :Linear Equations In Two Variable 1

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Chapter 4 :Linear Equations In two Variable 2

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Chapter 4 :Linear Equations In Two Variable 3

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Chapter 5 – Lines and Angles 1

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Chapter 5 – Lines and Angles 2

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Chapter 5 – Lines and Angles 3

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Chapter 5 – Lines and Angles 4

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Chapter 5 – Lines and Angles 5

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Chapter 5 – Lines and Angles 6

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Chapter 5 – Lines and Angles 7

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Chapter 5 – Lines and Angles 8

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Chapter 6 : Triangles 1

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Chapter 6 : Triangles 2

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Chapter 6 : Triangles 3

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Chapter 6 : Triangles 4

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Chapter 6 : Triangles 5

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Chapter 6 : Triangles 6

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Chapter 6 : Triangles 7

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Chapter 7 :Quadrilaterals 1

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Chapter 7 :Quadrilaterals 2

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Chapter 7 :Quadrilaterals 3

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Chapter 7 :Quadrilaterals 4

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Chapter 7 :Quadrilaterals 5

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Chapter 7 :Quadrilaterals 6

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Chapter 7 :Quadrilaterals 8

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Chapter 8 – Area of Parallelograms and Triangles 1 ||SEBA||

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Chapter 8 – Area of Parallelograms and Triangles 2 || SEBA||

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Chapter 8 – Area of Parallelograms and Triangles 3 || SEBA ||

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Chapter 8 – Area of Parallelograms and Triangles 4 ||SEBA||

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Chapter 9:Circles 1

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Chapter 9:Circles 2

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Chapter 9:Circles 3

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Chapter 9:Circles 4

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Chapter 9:Circles 5

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Chapter – 10 : Surface Areas and Volumes 1

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Chapter – 10 : Surface Areas and Volumes 2

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Chapter – 10 : Surface Areas and Volumes 3

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Chapter – 10 : Surface Areas and Volumes 4

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Chapter – 10 : Surface Areas and Volumes 5

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Chapter – 10 : Surface Areas and Volumes 6

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Chapter – 10 : Surface Areas and Volumes 7

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Chapter – 10 : Surface Areas and Volumes 8

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Chapter 11 : Statistics 1

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Chapter 11 : Statistics 2

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Chapter 11 : Statistics 3

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Chapter 11 : Statistics 4

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Chapter 12 – Probability (Lecture 1) || SEBA||

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Chapter 12 – Probability (Lecture 2) || SEBA||

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Class 9 Maths Revision Classes

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Class 9 Maths – Revision Classes 2

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Class 9 Maths – Revision Classes 3

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Class 9 Maths – Revision Classes 4

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Class 9 Maths – Number System – Exam Revision

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Class 9 Maths Numbers Exam Revision continued

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Class 9 Maths Number System – Exam Revision of Exponential Real Number

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Class 9 Maths Exam Revision contd – Exponential Real Numbers

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Class 9 Maths- Circles. Exam revision

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Class 9 Maths Polynomials – Past Paper Solutions part 1

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Class 9 Maths- Polynomials – Past Papers Part 2

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Class 9 Maths- Polynomia Revision and Summary

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Class 9 Maths – Polynomials Revision Class

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Class 9 Maths Polynomials – Questions & Answers Part 1

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Draft LessonClass 9 Maths – Polynomials Revision – Questions & Answers

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Class 9 Maths – Co-ordinate Geometry Exam revision

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Class 9 Maths- Co-ordinate Geometry Questions and Past Papers

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Class 9 Maths – Linear Equations – Revision Questions & Answers

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Class 10 Maths: Trigonometric ratios for 90 Deg Angles

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Exam notes for the NCERT Class 9 Maths Chapter on Number systems
Exam notes for NCERT Class 9 Maths Chapter on Number Systems: Chapter 1: Number Systems Introduction: - The concept of numbers is fundamental in mathematics. - Number systems help us represent and work with different types of numbers. Types of Numbers: 1. Natural Numbers (N): - Natural numbers are positive integers starting from 1. - They are used for counting and ordering. - Example: 1, 2, 3, 4, ... 2. Whole Numbers (W): - Whole numbers include all natural numbers along with zero. - Example: 0, 1, 2, 3, 4, ... 3. Integers (Z): - Integers include positive and negative whole numbers along with zero. - Example: ... -3, -2, -1, 0, 1, 2, 3, ... 4. Rational Numbers (Q): - Rational numbers are numbers that can be expressed as fractions (p/q), where p and q are integers, and q is not equal to zero. - Example: 1/2, -3/4, 7, -2, 0.5, ... 5. Irrational Numbers: - Irrational numbers are numbers that cannot be expressed as fractions. - They have non-repeating, non-terminating decimal expansions. - Example: √2, π (pi), e, ... 6. Real Numbers (R): - Real numbers include all rational and irrational numbers. - They form the complete number system. - Example: All numbers on the number line. Number Line: - The number line is a graphical representation of real numbers. - Numbers increase as you move to the right and decrease as you move to the left. Operations on Real Numbers: 1. Addition and Subtraction: Addition and subtraction of real numbers follow the rules of arithmetic. 2. Multiplication: Multiplying two real numbers with the same sign results in a positive number, while multiplying with different signs results in a negative number. 3. Division: Division of real numbers is defined, except for division by zero. Properties of Real Numbers: 1. Closure Property: The sum and product of any two real numbers are also real numbers. 2. Commutative Property: Addition and multiplication are commutative for real numbers (a + b = b + a, ab = ba). 3. Associative Property: Addition and multiplication are associative for real numbers ((a + b) + c = a + (b + c), (ab)c = a(bc)). 4. Distributive Property: Multiplication distributes over addition (a(b + c) = ab + ac). 5. Identity Property: The additive identity is 0, and the multiplicative identity is 1. Rational Number and Decimal Expansion: - Rational numbers can have terminating (finite) or recurring (infinite) decimal expansions. - For example, 1/4 = 0.25 (terminating), 1/3 = 0.333... (recurring). Irrational Numbers and Decimal Expansion: - Irrational numbers have non-terminating, non-recurring decimal expansions. - For example, √2 = 1.41421356... (non-recurring). Summary: - Number systems include natural, whole, integers, rationals, irrationals, and real numbers. - Real numbers encompass all other types of numbers. - Real numbers can be operated upon using standard arithmetic operations. - Properties of real numbers include closure, commutative, associative, distributive, and identity properties. Practice Problems: 1. Classify the following numbers as natural, whole, integers, rational, or irrational: -5, 1/2, √3, 0, 7.5, -10. 2. Perform the following operations: (a) 5 + (-3), (b) 2/3 - 1/6, (c) √5 × 2, (d) 7/8 ÷ 2/5. 3. Determine the decimal expansions of the following rational numbers: 2/7, 5/6, 1/9, 4/25. 4. Show that √3 is an irrational number. Understanding number systems is crucial in mathematics, as it provides the foundation for various mathematical concepts and operations. Practice working with different types of numbers and their properties to strengthen your understanding.

Exam notes for the NCERT Class 9 Maths Chapter on Exponential Real Numbers
Exam notes for NCERT Class 9 Maths Chapter on Exponents and Real Numbers: Chapter 1: Number Systems (Exponents and Real Numbers) Exponents: - An exponent represents how many times a number (base) is multiplied by itself. - Example: In 2^3, 2 is the base, and 3 is the exponent. It means 2 * 2 * 2 = 8. Laws of Exponents: 1. Product Law: When multiplying numbers with the same base and different exponents, add the exponents. - Example: a^m * a^n = a^(m + n). 2. Quotient Law: When dividing numbers with the same base and different exponents, subtract the exponents. - Example: a^m / a^n = a^(m - n). 3. Power Law: When raising an exponent to another exponent, multiply the exponents. - Example: (a^m)^n = a^(m * n). 4. Zero Exponent Law: Any non-zero number raised to the power of 0 is 1. - Example: a^0 = 1 (for a ≠ 0). 5. Negative Exponent Law: A number raised to a negative exponent is the reciprocal of the same number raised to the positive exponent. - Example: a^(-n) = 1 / (a^n). Real Numbers: - Real numbers include all rational and irrational numbers. - Rational numbers can be expressed as fractions (e.g., 3/4, -2/5). - Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions (e.g., √2, π). Prime Factorization: - Prime factorization breaks down a number into its prime factors. - Example: Prime factorization of 36 = 2^2 * 3^2 (expressing 36 as a product of prime numbers). HCF (Highest Common Factor) and LCM (Least Common Multiple): - HCF is the largest common factor of two or more numbers. - LCM is the smallest common multiple of two or more numbers. - Example: For numbers 12 and 18, HCF = 6, and LCM = 36. Rationalization: - Rationalizing the denominator involves removing radicals from the denominator. - For example, rationalize the denominator of 1/√3: Multiply by √3/√3 to get (√3) / 3. Irrational Numbers: - Irrational numbers have non-repeating, non-terminating decimal expansions. - They cannot be expressed as fractions. - Examples: √2, √5, π (pi). Summary: - Exponents represent repeated multiplication. - Laws of exponents help in simplifying expressions with exponents. - Real numbers include both rational and irrational numbers. - Prime factorization breaks down a number into prime factors. - HCF is the largest common factor, and LCM is the smallest common multiple. - Rationalization is used to remove radicals from the denominator. Practice Problems: 1. Simplify expressions: 2^4 * 2^2, 3^5 / 3^2, (4^3)^2. 2. Find the HCF and LCM of 16 and 24. 3. Express √8 as a simplified radical. 4. Determine if the following numbers are rational or irrational: √9, √11, 5/7, π. 5. Write the prime factorization of 72.

Exam notes for the NCERT Class 9 Maths Chapter on Polynomials
Exam notes for NCERT Class 9 Maths Chapter on "Polynomials": Chapter 2: Polynomials Introduction: - A polynomial is an algebraic expression consisting of variables, coefficients, and exponents. - It can have one or more terms, and each term can be a constant, a variable, or a combination of both. Degree of a Polynomial: - The degree of a polynomial is the highest power of the variable in any term. - Example: In the polynomial 3x^2 - 5x + 2, the degree is 2. Types of Polynomials: 1. Monomial: A polynomial with only one term. - Example: 4x, 3y^2, 7 2. Binomial: A polynomial with two unlike terms. - Example: 2x + 3, 5y - 7 3. Trinomial: A polynomial with three unlike terms. - Example: 4x^2 - 3x + 1, 2a^3 + 5a - 7 Addition and Subtraction of Polynomials: - To add or subtract polynomials, combine like terms (terms with the same variables and exponents). Multiplication of Polynomials: - Multiplying a polynomial by a monomial involves distributing the monomial to each term in the polynomial. - Example: (3x + 2) * 4 = 12x + 8 - Multiplying two polynomials involves applying the distributive property for each term in one polynomial with each term in the other polynomial. - Example: (2x + 3)(4x - 5) = 8x^2 - 10x + 12x - 15 = 8x^2 + 2x - 15 Special Products: 1. Square of a Binomial: (a + b)^2 = a^2 + 2ab + b^2 2. Cube of a Binomial: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 Division of Polynomials: - Polynomial division involves long division or synthetic division. - Long division is similar to numerical long division but applied to polynomials. Remainder Theorem: - If a polynomial 'f(x)' is divided by (x - a), the remainder is equal to 'f(a)'. Factor Theorem: - If (x - a) is a factor of a polynomial 'f(x)', then 'a' is a zero of 'f(x)'. Synthetic Division: - A shortcut method to divide a polynomial by a linear divisor of the form (x - a). - Useful for finding zeros and factors. Quadratic Polynomials: - Quadratic polynomials are of the form ax^2 + bx + c. - They can have zero, one, or two real solutions based on the discriminant (D = b^2 - 4ac). - D > 0: Two distinct real solutions. - D = 0: One real solution (repeated). - D < 0: No real solutions (complex roots). Summary: - Polynomials are algebraic expressions with variables, coefficients, and exponents. - The degree of a polynomial is the highest power of the variable. - Polynomials can be added, subtracted, multiplied, and divided. - Special products like squares and cubes of binomials have specific formulas. - The Remainder Theorem and Factor Theorem help in polynomial factorization. - Quadratic polynomials can have zero, one, or two real solutions based on the discriminant. Practice Problems: 1. Add the polynomials: 3x^2 - 2x + 5 and 2x^2 + 4x - 1. 2. Multiply the polynomials: (x + 3)(x - 2). 3. Divide the polynomial 4x^3 - 9x^2 + 5x - 7 by (x - 2) using synthetic division. 4. Find the zeros of the quadratic polynomial: 2x^2 - 7x + 6 and determine their nature. 5. Factorize the polynomial: x^3 - 8y^3.

Exam notes for the NCERT Class 9 Maths Chapter on Zeros of a Polynomial
Exam notes for NCERT Class 9 Maths Chapter on "Zeros of a Polynomial": Chapter 2: Polynomials - Zeros of a Polynomial Introduction: - A polynomial is an algebraic expression with one or more terms. - A polynomial equation is an equation in which a polynomial is set equal to zero. Zeros of a Polynomial: - A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. Factor Theorem: - If 'a' is a zero of a polynomial 'f(x)', then (x - a) is a factor of 'f(x)'. - Conversely, if (x - a) is a factor of 'f(x)', then 'a' is a zero of 'f(x)'. Remainder Theorem: - If 'f(x)' is divided by (x - a), the remainder is equal to 'f(a)'. Finding Zeros of a Polynomial: 1. Synthetic Division: A method for dividing polynomials to find zeros. - Example: Find the zeros of p(x) = 2x^3 - 5x^2 - 3x + 6. 2. Long Division: Another method for dividing polynomials to find zeros. - Example: Find the zeros of q(x) = 3x^4 - 2x^3 - 5x^2 + 7x - 2. Fundamental Theorem of Algebra: - Every polynomial of degree 'n' has exactly 'n' zeros, counting multiplicities (some zeros may repeat). Quadratic Polynomials: - Quadratic polynomials (degree 2) have at most two zeros. - The discriminant (D) determines the nature of the zeros. - D > 0: Two distinct real zeros. - D = 0: One real zero with multiplicity 2. - D < 0: Two complex (non-real) zeros. Cubic Polynomials: - Cubic polynomials (degree 3) have at most three zeros. - Rational zeros are found using the Rational Root Theorem. Summary: - Zeros of a polynomial are values that make the polynomial equal to zero. - The Factor Theorem and Remainder Theorem are used to find zeros. - Synthetic division and long division help in finding zeros. - The Fundamental Theorem of Algebra states the number of zeros. - Quadratic polynomials can have two real or complex zeros. - Cubic polynomials can have up to three zeros. Practice Problems: 1. Find the zeros of the polynomial f(x) = 2x^3 - 3x^2 - 2x + 1. 2. Use the Factor Theorem to factorize p(x) = x^3 - 4x^2 - 5x + 6 and find its zeros. 3. Determine the nature of zeros for the quadratic polynomial q(x) = x^2 - 6x + 9. 4. Find the rational zeros of the polynomial r(x) = 2x^3 - 5x^2 - 3x + 6 using the Rational Root Theorem.

Exam notes for the NCERT Class 9 Maths Chapter on Remainder Theorem
Exam notes for NCERT Class 9 Maths Chapter on the Remainder Theorem: Chapter 2: Polynomials Topic: The Remainder Theorem Introduction: - The Remainder Theorem is a fundamental concept in polynomial algebra. - It helps in finding the remainder when a polynomial is divided by a linear divisor of the form (x - a). - This theorem is useful for finding the value of a polynomial at a specific point. Statement of the Remainder Theorem: - If a polynomial 'f(x)' is divided by (x - a), then the remainder is equal to 'f(a)'. Explanation: - Let 'f(x)' be a polynomial, and we want to divide it by (x - a). - According to the Remainder Theorem, if we divide 'f(x)' by (x - a), the remainder 'R' will be equal to 'f(a)'. Example 1: - Consider the polynomial f(x) = 2x^3 - 5x^2 + 3x - 7. - If we want to find the remainder when f(x) is divided by (x - 3), we substitute 'x = 3' into f(x). - f(3) = 2(3)^3 - 5(3)^2 + 3(3) - 7 - R = 54 - 45 + 9 - 7 - R = 11 So, when f(x) is divided by (x - 3), the remainder is 11. Example 2: - Let's take another polynomial, g(x) = x^2 - 4x + 4. - To find the remainder when g(x) is divided by (x - 2), we substitute 'x = 2' into g(x). - g(2) = (2)^2 - 4(2) + 4 - R = 4 - 8 + 4 - R = 0 In this case, when g(x) is divided by (x - 2), the remainder is 0, which means (x - 2) is a factor of g(x). Applications of the Remainder Theorem: 1. Finding Remainders: The theorem helps find the remainder when a polynomial is divided by a specific linear divisor. 2. Factorization: If the remainder is zero, it implies that (x - a) is a factor of the polynomial. This can aid in polynomial factorization. 3. Evaluation: The Remainder Theorem allows us to evaluate polynomials at specific values of 'x'. Summary: - The Remainder Theorem states that when a polynomial 'f(x)' is divided by (x - a), the remainder is equal to 'f(a)'. - It is a useful tool for finding remainders, factoring polynomials, and evaluating them at specific points. - If the remainder is zero, (x - a) is a factor of the polynomial. Practice Problems: 1. Find the remainder when the polynomial p(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 2 is divided by (x - 1). 2. Determine whether (x - 3) is a factor of the polynomial q(x) = x^3 - 5x^2 + 6x - 18. If so, find the remainder. 3. Evaluate the polynomial r(x) = 4x^2 - 2x + 7 at x = 2 using the Remainder Theorem. 4. If a polynomial s(x) has a remainder of 3 when divided by (x - 4), what is the value of s(4)?

Exam notes for the NCERT Class 9 Maths Chapter on Co-ordinate Geometry
Exam notes for NCERT Class 9 Maths Chapter on Coordinate Geometry: Chapter 3: Coordinate Geometry Introduction: - Coordinate Geometry, also known as Analytical Geometry, is a branch of mathematics that deals with the study of geometry using algebraic concepts. - It involves representing geometric figures and objects on a coordinate plane using coordinates (x, y). - This chapter introduces the Cartesian Coordinate System, plotting points, and understanding the distance formula. The Cartesian Coordinate System: - The Cartesian Coordinate System is a grid system used to locate points in a plane. - It consists of two perpendicular lines called the x-axis and y-axis, which intersect at the origin (0, 0). - The x-axis represents horizontal movement, and the y-axis represents vertical movement. - Any point on the plane is represented as (x, y), where 'x' is the distance from the y-axis (horizontal) and 'y' is the distance from the x-axis (vertical). Plotting Points: - To plot a point, start from the origin and move along the x-axis first (right for positive, left for negative) to reach the 'x' coordinate. - Then, move along the y-axis (up for positive, down for negative) to reach the 'y' coordinate. - Plot the point where the two lines intersect. Distance Formula: - The distance between two points (x1, y1) and (x2, y2) on a coordinate plane can be found using the distance formula: - Distance (d) = √[(x2 - x1)^2 + (y2 - y1)^2] Slope of a Line: - The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by: - Slope (m) = (y2 - y1) / (x2 - x1) Forms of Linear Equations: - The equation of a straight line in the form 'y = mx + c' represents a linear equation, where 'm' is the slope, and 'c' is the y-intercept (the point where the line intersects the y-axis). Summary: - Coordinate Geometry uses the Cartesian Coordinate System to locate points in a plane. - The distance between two points can be found using the distance formula. - The slope of a line helps determine its inclination. - Linear equations can be represented in the form 'y = mx + c.' Practice Problems: 1. Plot the points A(3, 4), B(-2, 1), and C(0, 0) on the coordinate plane. 2. Find the distance between points P(2, 3) and Q(5, 7). 3. Calculate the slope of a line passing through points X(1, 2) and Y(3, 5). 4. Write the equation of a line with a slope of -2 passing through the point (4, 6). 5. Determine the point where the line with the equation y = 3x - 2 intersects the x-axis. 6. If a line has a slope of 1/2 and passes through the point (2, -3), write its equation. 7. Find the coordinates of the midpoint of the line segment joining points A(1, 2) and B(5, 8).

Exam notes for the NCERT Class 9 Maths Chapter on Lines and Angles
Exam notes for NCERT Class 9 Maths Chapter on Lines and Angles: Chapter 6: Lines and Angles Introduction: - Geometry is a branch of mathematics that deals with the study of shapes, sizes, and properties of figures and spaces. - In this chapter, we will explore the concepts related to lines and angles. Basic Definitions: - Line: A line is a straight path that extends indefinitely in both directions. It has no endpoints. - Ray: A ray is a part of a line that starts at one endpoint (called the initial point) and goes on indefinitely in one direction. - Line Segment: A line segment is a part of a line that has two distinct endpoints. Types of Angles: - Acute Angle: An acute angle is an angle that measures less than 90 degrees. - Right Angle: A right angle is an angle that measures exactly 90 degrees. - Obtuse Angle: An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees. - Straight Angle: A straight angle is an angle that measures exactly 180 degrees. - Reflex Angle: A reflex angle is an angle that measures more than 180 degrees but less than 360 degrees. Pair of Lines: - Intersecting Lines: Lines that cross or meet at a point are called intersecting lines. - Parallel Lines: Lines that are equidistant and will never intersect, no matter how far they are extended, are called parallel lines. - Transversal: A line that intersects two or more parallel lines is called a transversal. - Corresponding Angles: Pairs of angles in matching positions when a transversal intersects two parallel lines. - Alternate Interior Angles: Pairs of angles on opposite sides of the transversal and inside the parallel lines. - Alternate Exterior Angles: Pairs of angles on opposite sides of the transversal and outside the parallel lines. - Consecutive Interior Angles: Pairs of angles on the same side of the transversal and inside the parallel lines. Sum of Angles: - Linear Pair: Two angles are said to form a linear pair if their sum is 180 degrees. - Vertically Opposite Angles: When two lines intersect, the pairs of opposite angles are called vertically opposite angles and are equal. Parallel Lines and Transversals: - The sum of the angles on the same side of a transversal is 180 degrees (supplementary). - The sum of the interior angles on the same side of a transversal is 180 degrees. - The alternate interior angles are equal. - The alternate exterior angles are equal. Applications of Lines and Angles: - Understanding lines and angles is fundamental to solving geometric problems and real-world applications. - Knowledge of angles is essential in fields like architecture, engineering, navigation, and design. Summary: - This chapter introduces the basic concepts of lines, angles, and their properties. - It covers various types of angles and their relationships, including parallel lines and transversals. - Understanding lines and angles is essential in geometry and has practical applications in various fields. Practice Problems: 1. If two angles are complementary, and one angle measures 40 degrees, find the measure of the other angle. 2. If the measure of one angle in a linear pair is 120 degrees, what is the measure of its adjacent angle? 3. Find the value of x in the figure where a transversal intersects two parallel lines. 4. If two angles are vertically opposite, and one angle measures 75 degrees, what is the measure of the other angle? 5. In the figure, if ∠1 = 3x and ∠2 = 4x, find the values of x, ∠1, and ∠2. 6. If ∠AOC = 150 degrees and ∠BOC = 30 degrees, find ∠AOB in the figure where OA and OB are rays. 7. Determine the types of angles formed when a transversal intersects parallel lines in different scenarios. 8. Prove that the sum of the angles of a triangle is 180 degrees. 9. If two lines are parallel, what can you conclude about the measures of corresponding angles? 10. Given ∠ABC = 70 degrees and ∠BCD = 110 degrees, determine the type of angles formed by lines AB and CD.

Exam notes for the NCERT Class 9 Maths Chapter on Triangles
Exam notes for NCERT Class 9 Maths Chapter on Triangles: Chapter 7: Triangles Introduction: - Geometry is the branch of mathematics that deals with the study of shapes, sizes, and properties of figures. - In this chapter, we will explore the concepts related to triangles, one of the fundamental shapes in geometry. Basic Definitions: - Triangle: A triangle is a closed figure with three sides, three angles, and three vertices. - Vertex: The point where two sides of a triangle meet is called a vertex (plural: vertices). - Side: Each of the three line segments that form a triangle is called a side. Classification of Triangles: - Based on Sides: - Scalene Triangle: A triangle in which all three sides have different lengths. - Isosceles Triangle: A triangle in which at least two sides have the same length. - Equilateral Triangle: A triangle in which all three sides have the same length. - Based on Angles: - Acute-angled Triangle: A triangle in which all three angles are acute (measure less than 90 degrees). - Right-angled Triangle: A triangle in which one angle is a right angle (measures 90 degrees). - Obtuse-angled Triangle: A triangle in which one angle is obtuse (measures more than 90 degrees but less than 180 degrees). Properties of Triangles: - Angle Sum Property: The sum of the three angles of a triangle is always 180 degrees. - Exterior Angle Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. - Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is represented as a^2 + b^2 = c^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. - Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. Congruence of Triangles: - Two triangles are congruent if their corresponding sides and angles are equal. - The criteria for the congruence of triangles include SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), RHS (Right-angle-Hypotenuse-Side), and SAA (Side-Angle-Angle). Applications of Triangles: - Understanding properties of triangles is crucial for solving problems related to construction, engineering, architecture, and navigation. - Triangles are fundamental to trigonometry, which is used in various fields, including physics and astronomy. Summary: - This chapter introduces the basic concepts of triangles, their classification based on sides and angles, and properties such as the angle sum property and Pythagoras theorem. - Understanding triangles is fundamental in geometry and has practical applications in real-world scenarios. Practice Problems: 1. Determine whether the following triangles are scalene, isosceles, or equilateral: (a) All sides have different lengths (b) Two sides have the same length (c) All sides have the same length. 2. Calculate the measure of an exterior angle of a triangle if the other two angles are 40 degrees and 60 degrees. 3. Use Pythagoras theorem to find the length of the hypotenuse in a right-angled triangle with side lengths 3 cm and 4 cm. 4. Prove that the sum of the angles of a triangle is 180 degrees. 5. Given two triangles with sides and angles, determine whether they are congruent using the appropriate criteria (SSS, SAS, ASA, RHS, SAA). 6. In a right-angled triangle, one acute angle measures 30 degrees. Determine the measures of the other two angles. 7. Prove the converse of Pythagoras theorem: If the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle. 8. Solve real-life problems involving the congruence of triangles and the application of Pythagoras theorem.

Exam notes for the NCERT Class 9 Maths Chapter on Quadrilaterals
Exam notes for NCERT Class 9 Maths Chapter on Quadrilaterals: Chapter 8: Quadrilaterals Introduction: - A quadrilateral is a polygon with four sides and four angles. - In this chapter, we will explore different types of quadrilaterals, their properties, and the conditions for the special types of quadrilaterals. Types of Quadrilaterals: 1. Parallelogram: - A quadrilateral in which opposite sides are parallel and equal in length. - Opposite angles are also equal. - Diagonals bisect each other. 2. Rectangle: - A parallelogram with all angles equal to 90 degrees (right angles). - Opposite sides are equal in length. 3. Square: - A rectangle with all sides equal in length. - All angles are right angles. 4. Rhombus: - A parallelogram with all sides equal in length. - Opposite angles are equal but not necessarily right angles. - Diagonals bisect each other at right angles. 5. Trapezium: - A quadrilateral with one pair of opposite sides parallel. 6. Kite: - A quadrilateral with two pairs of adjacent sides equal. - Diagonals intersect at right angles. Properties of Quadrilaterals: - The sum of interior angles: The sum of interior angles of any quadrilateral is 360 degrees. - Opposite sides in a parallelogram: Opposite sides are equal in length and parallel. - Diagonals of a parallelogram: Diagonals bisect each other. - Diagonals of a rectangle: Diagonals are equal in length. - Diagonals of a square: Diagonals are equal in length and bisect each other at 90 degrees. - Diagonals of a rhombus: Diagonals are equal in length and bisect each other at 90 degrees. - The property of a trapezium: One pair of opposite sides is parallel. - The property of a kite: Two pairs of adjacent sides are equal. - Converse of the property of a parallelogram: If opposite sides of a quadrilateral are equal and parallel, it is a parallelogram. Special Parallelograms: - A square is a special type of parallelogram. - A rhombus is a special type of parallelogram. - A rectangle is a special type of parallelogram. Conditions for Special Parallelograms: 1. A Quadrilateral is a Parallelogram If: - Both pairs of opposite sides are equal and parallel. - Diagonals bisect each other. 2. A Quadrilateral is a Rectangle If: - Both pairs of opposite sides are equal and parallel. - All angles are equal to 90 degrees. 3. A Quadrilateral is a Square If: - Both pairs of opposite sides are equal and parallel. - All angles are equal to 90 degrees. - Diagonals bisect each other at right angles. 4. A Quadrilateral is a Rhombus If: - All sides are equal. - Diagonals bisect each other at right angles. Summary: - This chapter introduces the concept of quadrilaterals, their properties, and the conditions for special types of quadrilaterals like parallelograms, rectangles, squares, and rhombuses. - Understanding these properties is essential for solving problems involving geometric figures. Practice Problems: 1. Determine whether the given quadrilateral is a parallelogram based on the properties of opposite sides and diagonals. 2. Find the value of an unknown angle in a quadrilateral using the property that the sum of interior angles is 360 degrees. 3. Prove that the opposite sides of a quadrilateral are equal and parallel based on the given conditions. 4. Given the lengths of the sides of a quadrilateral, determine whether it is a rectangle, square, or rhombus. 5. Calculate the measures of unknown angles in special parallelograms like squares and rectangles. 6. Solve real-life problems involving the properties of quadrilaterals, such as finding the dimensions of a rectangular garden or the length of a diagonal in a rhombus.

Exam notes for the NCERT Class 9 Maths Chapter on Circles
Exam notes for NCERT Class 9 Maths Chapter on Circles: Chapter 10: Circles Introduction: - A circle is a closed figure in which all points on the boundary are equidistant from a fixed point called the center. - The fixed distance from the center to any point on the circle is called the radius. - The distance across the circle passing through the center is called the diameter. The diameter is twice the radius. - The circumference of a circle is the distance around its boundary. It is equal to (2pi) times the radius, where (pi) is approximately 3.14. - The symbol (pi) (pi) represents the ratio of the circumference of a circle to its diameter, and it is an irrational number. Terms Associated with Circles: 1. Arc: A part of the circumference of a circle. 2. Sector: The region enclosed between two radii and the corresponding arc. 3. Segment: The region enclosed between an arc and a chord of a circle. 4. Chord: A line segment joining any two points on the circle. 5. Secant: A line that intersects the circle at two distinct points. 6. Tangent: A line that touches the circle at exactly one point, called the point of tangency. 7. Cyclic Quadrilateral: A quadrilateral whose all vertices lie on the circumference of a circle. Theorems and Concepts: 1. Equal Chords and Their Distances from the Center: - Chords equidistant from the center of a circle are equal in length. 2. Angles Subtended by an Arc: - An angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle. 3. Angle Subtended by a Chord: - The angle subtended by a chord at the center is double the angle subtended by it at any point on the circumference. 4. Tangent to a Circle: - A line drawn perpendicular to a radius at the point of contact is called the tangent to the circle. 5. Number of Tangents from a Point Outside the Circle: - A point outside a circle has two tangents drawn to it from the same point. 6. Cyclic Quadrilaterals: - The opposite angles of a cyclic quadrilateral add up to 180 degrees. Important Formulae: 1. Circumference of a circle: (C = 2pi r) 2. Area of a circle: (A = pi r^2) Summary: - This chapter introduces the concept of circles, their properties, and various terms associated with circles. - It explores the relationship between chords, angles subtended by arcs, and angles subtended by chords at the center and on the circumference. - Understanding these concepts is essential for solving problems involving circles, such as finding the length of chords, the area of sectors, and the angles formed by tangents. Practice Problems: 1. Find the radius and circumference of a circle if its diameter is given. 2. Calculate the area of a sector or segment of a circle. 3. Determine the length of a chord based on the radius and angle subtended at the center. 4. Find the angle subtended by a chord at the center or on the circumference. 5. Solve real-life problems involving circles, such as finding the length of a wire required to make a circular fence or the area of a circular garden.

Exam notes for the NCERT Class 9 Maths Chapter on Surface Area and Volume
Exam notes for NCERT Class 9 Maths Chapter on Surface Area and Volume: Chapter 13: Surface Area and Volume Introduction: - This chapter deals with the concepts of surface area and volume of various 3D geometrical shapes. - These concepts are essential in real-life applications, such as calculating the amount of material needed for construction or finding the capacity of containers. Surface Area: - The surface area of a 3D object refers to the total area of all its surfaces. - It is measured in square units (e.g., square meters, square centimeters). - Surface area can be calculated differently for different 3D shapes: - For a cube, the surface area is (6a^2), where (a) is the length of a side. - For a cuboid, the surface area is (2(lw + lh + wh)), where (l) is length, (w) is width, and (h) is height. - For a cylinder, the surface area is (2pi r^2 + 2pi rh), where (r) is the radius, and (h) is the height. - For a cone, the surface area is (pi r^2 + pi rl), where (r) is the radius, and (l) is the slant height. - For a sphere, the surface area is (4pi r^2), where (r) is the radius. Volume: - The volume of a 3D object refers to the space enclosed by its surfaces. - It is measured in cubic units (e.g., cubic meters, cubic centimeters). - Volume can be calculated differently for different 3D shapes: - For a cube, the volume is (a^3), where (a) is the length of a side. - For a cuboid, the volume is (lwh), where (l) is length, (w) is width, and (h) is height. - For a cylinder, the volume is (pi r^2h), where (r) is the radius, and (h) is the height. - For a cone, the volume is (frac{1}{3}pi r^2h), where (r) is the radius, and (h) is the height. - For a sphere, the volume is (frac{4}{3}pi r^3), where (r) is the radius. Summary: - This chapter introduces the concepts of surface area and volume and provides formulas for calculating them for various 3D shapes. - It covers shapes like cubes, cuboids, cylinders, cones, and spheres. - Understanding these concepts is crucial for solving problems related to construction, packaging, and storage. Practice Problems: 1. Calculate the surface area and volume of a cube given its side length. 2. Find the surface area and volume of a cuboid with given dimensions. 3. Determine the surface area and volume of a cylinder or cone based on its radius and height. 4. Solve real-life problems involving the calculation of surface area and volume, such as painting the walls of a room or filling a cylindrical tank with water.

Exam notes for the NCERT Class 9 Maths Chapter on Statistics
Exam notes for NCERT Class 9 Maths Chapter on Statistics: Chapter 14: Statistics Introduction: - Statistics is a branch of mathematics that deals with the collection, presentation, analysis, and interpretation of data. - It plays a crucial role in various fields, including economics, science, business, and social sciences. Key Concepts: 1. Data: - Data is a collection of facts, figures, or information. - Data can be classified into two types: primary data (collected firsthand) and secondary data (obtained from other sources). - Data can be represented in various forms, including tables, graphs, and charts. 2. Presentation of Data: - Data can be presented using various methods: - Tabulation: Organizing data in the form of tables. - Frequency Distribution: Grouping data into classes or intervals and showing the frequency of each class. - Bar Graphs: Representing data using rectangular bars of different heights. - Histograms: A special type of bar graph where the data is continuous and the bars touch each other. - Pie Charts: Representing data in a circular chart where each sector's angle is proportional to the data it represents. 3. Measures of Central Tendency: - Central tendency measures help us find the center or average of a set of data. - Mean: The mean is the sum of all data values divided by the total number of values. - Median: The median is the middle value of a data set when arranged in ascending or descending order. - Mode: The mode is the data value that occurs most frequently in a dataset. 4. Measures of Dispersion: - Measures of dispersion tell us how data is spread out. - Range: The range is the difference between the highest and lowest data values. - Mean Absolute Deviation (MAD): MAD measures the average deviation of each data point from the mean. - Variance and Standard Deviation: Variance and standard deviation quantify the spread of data points around the mean. 5. Probability: - Probability is the likelihood of an event occurring. - Probability can be expressed as a fraction, decimal, or percentage. - Probability rules, such as the addition rule and multiplication rule, help calculate the probability of combined events. Summary: - Statistics is crucial for summarizing and analyzing data. - It includes data collection, presentation, measures of central tendency, measures of dispersion, and probability. - Understanding statistics is essential for making informed decisions in various fields. Practice Problems: 1. Create a frequency distribution table for a given dataset. 2. Calculate the mean, median, and mode of a dataset. 3. Calculate the range, MAD, variance, and standard deviation for a set of data. 4. Solve probability problems involving coin tosses, dice, and card decks. 5. Analyze real-life data sets using statistical techniques to draw conclusions or make predictions.

Exam notes for the NCERT Class 9 Maths Chapter on Probability
Exam notes for NCERT Class 9 Maths Chapter on Probability: Chapter 15: Probability Introduction: - Probability is a branch of mathematics that deals with uncertainty and randomness. - It helps us analyze and predict the likelihood of events occurring in various situations. Key Concepts: 1. Experiments and Outcomes: - An experiment is an action or process that results in an outcome. - Outcomes are possible results of an experiment. 2. Sample Space: - The sample space (S) is the set of all possible outcomes of an experiment. - Sample space can be finite, countably infinite, or uncountably infinite. 3. Event: - An event is a subset of the sample space. - It represents a specific outcome or a collection of outcomes. 4. Probability of an Event: - The probability of an event (P(A)) is a measure of the likelihood of that event occurring. - It is expressed as a fraction, decimal, or percentage and ranges from 0 (impossible) to 1 (certain). 5. Equally Likely Outcomes: - When all outcomes in a sample space are equally likely, the probability of an event A can be calculated as: - P(A) = (Number of favorable outcomes for A) / (Total number of outcomes in S) 6. Probability of an Impossible Event: - The probability of an impossible event is 0 (P(∅) = 0). 7. Probability of a Certain Event: - The probability of a certain event is 1 (P(S) = 1). 8. Complementary Events: - The complement of an event A (denoted as A') consists of all outcomes not in A. - P(A') = 1 - P(A) (probability of the event not happening). 9. Addition Rule of Probability: - For two mutually exclusive events A and B (events that cannot occur simultaneously): - P(A or B) = P(A) + P(B) 10. Multiplication Rule of Probability: - For two independent events A and B (events that do not influence each other): - P(A and B) = P(A) * P(B) 11. Conditional Probability: - Conditional probability is the probability of an event occurring given that another event has already occurred. - P(A | B) represents the probability of A given B. - P(A | B) = (P(A and B)) / (P(B)) 12. Probability Distribution: - A probability distribution is a table or graph that shows the probabilities of various outcomes of an experiment. - It is often used for random variables. Summary: - Probability is used to quantify uncertainty and randomness in various situations. - Sample space, events, and probabilities are fundamental concepts. - Equally likely outcomes and probability rules help calculate probabilities. - Conditional probability is used to find probabilities when certain conditions are met. - Probability distributions are used to represent random variables. Practice Problems: 1. Calculate the probability of drawing an ace from a standard deck of cards. 2. Find the probability of rolling a prime number on a fair six-sided die. 3. Calculate the probability of getting heads or tails when flipping a coin. 4. Use conditional probability to find the probability of drawing a red card from a deck after drawing a black card. 5. Create a probability distribution for the outcomes of rolling two dice and finding the sum.

Course Content

Class 9 Maths

Chapter 1 – Number systems (part1)

Chapter 1 – Number systems .(Part 2)

Chapter 1 – Number systems .(Part 3)

Chapter 1 – Number systems .(Part 4)

Chapter 1 – Number systems .(Part 5)

Chapter 1 – Number systems .(Part 6)

Chapter 1 – Number systems .(Part 7)

Chapter 1 – Number systems .(Part 8)

Class 9 Maths Chapter 2 Polynomials demo

Chapter 2 – Polynomials (Part 1)

Chapter 2 : Polynomials (Part 2)

Chapter 2 : Polynomials (Part 3)

Chapter 2 : Polynomials (Part 4)

Chapter 2 : Polynomials (part 5)

Chapter 2 : Polynomials ( Part 6)

Class 9 Maths Chapter 2 (Extra class) – Polynomials Lecture 1 by bhrigu

Class 9 Maths Chapter 2 – Polynomials Lecture 2

Class 9 Maths Chapter 2 – Polynomials Lecture 3

Class 9 Maths Chapter 2 – Polynomials Lecture 4

Class 9 Maths Chapter 2 – Polynomials Lecture 5

Class 9 Maths Ch 2 Polynomials 6

Class 9 Maths Ch 2 Polynomials 7

Class 9 Maths Ch 2 Polynomials 8

Chapter 3 :Co-ordinate Geometry – Basori

Chapter 4 :Linear Equations In Two Variable 1

Chapter 4 :Linear Equations In two Variable 2

Chapter 4 :Linear Equations In Two Variable 3

Chapter 5 – Lines and Angles 1

Chapter 5 – Lines and Angles 2

Chapter 5 – Lines and Angles 3

Chapter 5 – Lines and Angles 4

Chapter 5 – Lines and Angles 5

Chapter 5 – Lines and Angles 6

Chapter 5 – Lines and Angles 7

Chapter 5 – Lines and Angles 8

Chapter 6 : Triangles 1

Chapter 6 : Triangles 2

Chapter 6 : Triangles 3

Chapter 6 : Triangles 4

Chapter 6 : Triangles 5

Chapter 6 : Triangles 6

Chapter 6 : Triangles 7

Chapter 7 :Quadrilaterals 1

Chapter 7 :Quadrilaterals 2

Chapter 7 :Quadrilaterals 3

Chapter 7 :Quadrilaterals 4

Chapter 7 :Quadrilaterals 5

Chapter 7 :Quadrilaterals 6

Chapter 7 :Quadrilaterals 8

Chapter 8 – Area of Parallelograms and Triangles 1 ||SEBA||

Chapter 8 – Area of Parallelograms and Triangles 2 || SEBA||

Chapter 8 – Area of Parallelograms and Triangles 3 || SEBA ||

Chapter 8 – Area of Parallelograms and Triangles 4 ||SEBA||

Chapter 9:Circles 1

Chapter 9:Circles 2

Chapter 9:Circles 3

Chapter 9:Circles 4

Chapter 9:Circles 5

Chapter – 10 : Surface Areas and Volumes 1

Chapter – 10 : Surface Areas and Volumes 2

Chapter – 10 : Surface Areas and Volumes 3

Chapter – 10 : Surface Areas and Volumes 4

Chapter – 10 : Surface Areas and Volumes 5

Chapter – 10 : Surface Areas and Volumes 6

Chapter – 10 : Surface Areas and Volumes 7

Chapter – 10 : Surface Areas and Volumes 8

Chapter 11 : Statistics 1

Chapter 11 : Statistics 2

Chapter 11 : Statistics 3

Chapter 11 : Statistics 4

Chapter 12 – Probability (Lecture 1) || SEBA||

Chapter 12 – Probability (Lecture 2) || SEBA||

Class 9 Maths Revision Classes

Class 9 Maths – Revision Classes 2

Class 9 Maths – Revision Classes 3

Class 9 Maths – Revision Classes 4

Class 9 Maths – Number System – Exam Revision

Class 9 Maths Numbers Exam Revision continued