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Class 9 Maths

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Class 9 Maths

  • Class 9 Chapter 1 – Number Systems
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  • Class 9_Maths Chapter 1 – Number systems contd.
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  • Class 9 Maths – Number System- Irrational Numbers
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  • Class 9 Maths Number System – Real Numbers
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  • Class 9 Maths- Changing a Decimal number into a Fractional Number
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  • Class 9 Maths – Operations on Real Numbers – Question Papers
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  • Class 9 Maths Non-Terminating Repeating Number
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  • Class 9 Maths – Representing a number on the Number line
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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 Chapter 2 Polynomials
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  • Class 9 Maths – Zeros of a Polynomial
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  • Class 9 Maths Remainder Theorem
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  • Class 9 Maths Chapter 2 Polynomials demo
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  • Class 9 Maths Chapter 2 – Polynomial contd
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  • Class 9 Math Chapter 2 – Polynomials 1
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  • Class 9_Maths Co-ordinate Geometry
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  • CLASS 9 Maths- Algebraic Expression, Factor Theorem
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  • CLASS 9 Maths – Factor Theorem
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  • Class 9 Maths Polynomials – Past Paper Solutions
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  • Class 9 Maths- Past Papers part 2
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  • Class 9 Maths Algebra Continued
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  • Class 9 Maths – Rationlisation
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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 – Polynomials Revision Class
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  • Class 9 Maths- Polynomia Revision and Summary
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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 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_Euclid Geometry
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  • Class 9 Maths – Chapter 5- Euclid’s Geometry Question Answer Revision
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  • Class 9 Maths – Chapter 5- Euclid’s Geometry Question Answer Revision 2
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  • Class 9 Maths – Euclid Geometry- Exam Revision contd 3
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  • Class 9_Maths_Euclid’s Geometry 4
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  • Class 9_Maths – Euclid Geometry Revision 5
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  • Class 9_Maths – Euclid Geometry Revision 6
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  • Class 9 Maths- Introduction to Lines & Angles 1
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  • Class 9 Maths- Introduction to Lines & Angles 2
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  • Class 9 Maths: Chapter 6 – Lines and Angles 3
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  • Class 9 Maths: Chapter 6 – Lines and Angles 4
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  • Class 9 Maths: Chapter 6 – Lines and Angles 5
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  • Class 9 Maths Lines and Angles 6
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  • Class 9 Maths Ch 6 Lines and Angles 7
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  • Class 9 Maths Ch 6 Lines and Angles 8
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  • Class 10 Maths: Trigonometric ratios for 90 Deg Angles
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  • Class 9 Maths: Chapter 7 – Triangles Part 1
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  • Class 9_Maths: Chapter 7 – Triangles- Solved Equations
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  • Class 9 Maths: Chapter 7- Triangles – Solved Equations contd 2
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  • Class 9 Maths: Chapter 7- Triangles Continued 3
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  • Class 9 Maths: Chapter 7 – Triangles Exercises- Problem-solving 4
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  • Class 9_Maths: Chapter 7 – Triangles Exercises continued 5
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  • Class 9_Maths: Chapter 7- Triangles 6 – Final Topics
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  • Class 9_Maths: Chapter 7 Triangles – Exercises and Problem-Solving 7
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  • Class 9_Maths: Triangles by Hirok 1
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  • Class 9_Maths_Traingles Hirok 2
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  • Class 9_Maths_Triangles Hirok 3
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  • Class 9_Maths_Triangles Hirok 4
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  • Class 9_Maths_Triangles Hirok 5
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  • Class 9_Maths Triangles 6
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  • Class 9_Maths Triangles 7
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  • Class 9_Maths_Quadrilaterals 1
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  • Class 9_Maths_Quadrilaterals 2
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  • Class 9_Maths_Quadrilaterals 3
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  • Class 9_Maths_Quadrilaterals 4
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  • Class 9_Maths_Quadrilaterals 5
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  • Class 9_Maths_Quadrilaterals 6
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  • Class 9_Maths_Quadrilaterals 8
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  • Class 9_Chapter 10_Circles 1
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  • Class 9_Chapter 10_Circles 2
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  • Class 9 Maths- Chapter 10 – Circles. Exam revision
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  • Class 9 Maths Ch 10 – Circles 3
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  • Class 9 Maths Ch 10 – Circles 4
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  • Class 9 Maths Ch 10 Circles 5
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  • Class 9 Maths Chapter – 13 Surface Areas and Volumes 1
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  • Class 9 Maths Chapter – 13 Surface Areas and Volumes 2
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  • Class 9 Maths Chapter – 13 Surface Areas and Volumes 3
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  • Class 9 Maths Chapter – 13 Surface Areas and Volumes 4
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  • Class 9 Maths Chapter – 13 Surface Areas and Volumes 5
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  • Class 9 Maths Ch 13 – Surface Area & Volume 6
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  • Class 9 Maths Ch 13 – Surface Area & Volume 7
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  • Class 9 Maths Ch 13 – Surface Area & Volume 8
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  • Class 9 Maths Ch 14 Statistics 1
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  • Class 9 Maths Ch 14 Statistics 2
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  • Class 9 Maths Ch 14 Statistics 3
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  • Class 9 Maths Ch 14 Statistics 4
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  • Class 9 Mathematics Chapter 15- Probability Lecture 1
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  • Class 9 Mathematics Chapter 15- Probability Lecture 2
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  • Class 9 Maths Chapter 4 – Linear equation in two variables Lecture 1
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  • Class 9 Maths Linear equation in two variables Vreegu 2
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  • Class 9 Maths Chapter 4 – Linear equation in two variables Lecture 3
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  • Class 9 Maths Chapter 12 Heron’s Formula Lecture 1
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  • Class 9 Maths Chapter 12 Heron’s Formula Lecture 2
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  • Class 9 Maths Chapter 2 – Polynomials Lecture 1
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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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  • Class 9 Maths Ch 6 – Lines and Angles 1
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  • Class 9 Maths Ch 6 – Lines and Angles 2
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  • Class 9 Maths Ch 6 – Lines & Angles 3
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  • Class 9 Maths Ch 6 – Lines & Angles 4
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  • Class 9 Maths Ch 6 – Lines & Angles 5
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 2
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 3
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 4
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 5
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 6
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 7
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 8
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 9
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 10
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 11
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  • Class 9 Maths Ch 13 Surface Are & Volumes Revision 12
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  • Class 9 Maths Ch 7 Triangles 1
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  • Class 9 Maths Ch 7 Triangles 2
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  • Class 9 Maths Ch 7 Triangles 3
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  • Class 9 Maths Ch 7 Triangles 4
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  • Class 9 Maths Ch 7 Triangles 5
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  • Class 9 Maths Ch 7 Triangles 6
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  • Class 9 Maths Ch 9 – Area of Parallelograms and Triangles 1
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  • Class 9 Maths Ch 9 – Area of Parallelograms and Triangles 2
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  • Class 9 Maths Ch 9 – Area of Parallelograms and Triangles 3
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  • Class 9 Maths Ch 9 – Area of Parallelograms and Triangles 4
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MCQs on Surface Area and Volume
Here are some multiple-choice questions (MCQs) with answers based on NCERT Class 9 Mathematics chapter "Surface Area and Volume." **1.** What is the total surface area of a cube with a side length of 4 cm? a. 16 cm² b. 32 cm² c. 48 cm² d. 64 cm² **Answer:** b. 32 cm² **2.** The volume of a cuboid is 180 cubic centimeters, and its length is 9 cm. What is its width if the height is 5 cm? a. 3 cm b. 4 cm c. 6 cm d. 8 cm **Answer:** c. 6 cm **3.** What is the lateral surface area of a cylinder with a radius of 4 cm and a height of 10 cm? a. 80 cm² b. 120 cm² c. 160 cm² d. 240 cm² **Answer:** c. 160 cm² **4.** The radius of a hemisphere is 7 cm. What is its curved surface area (in square centimeters)? a. 154 cm² b. 308 cm² c. 462 cm² d. 616 cm² **Answer:** a. 154 cm² **5.** A cone with a radius of 6 cm and a height of 8 cm is cut into two parts by a plane parallel to its base. What is the volume of the smaller cone? a. 96 cm³ b. 64 cm³ c. 32 cm³ d. 48 cm³ **Answer:** b. 64 cm³ **6.** If the volume of a cylinder is 3850 cubic centimeters and its radius is 7 cm, what is its height (in centimeters)? a. 10 cm b. 11 cm c. 12 cm d. 14 cm **Answer:** d. 14 cm **7.** A rectangular tank measures 12 cm by 8 cm by 5 cm. What is the volume of water it can hold (in cubic centimeters)? a. 480 cm³ b. 240 cm³ c. 960 cm³ d. 384 cm³ **Answer:** a. 480 cm³ **8.** The lateral surface area of a cone is 264 cm², and its slant height is 15 cm. What is the radius of the cone? a. 5 cm b. 8 cm c. 10 cm d. 12 cm **Answer:** b. 8 cm **9.** A cylinder and a cone have the same base area and the same height. Which one has a greater volume? a. Cylinder b. Cone c. Both have the same volume d. Cannot be determined with the given information **Answer:** a. Cylinder **10.** What is the total surface area of a sphere with a radius of 6 cm (in square centimeters)? a. 36π cm² b. 72π cm² c. 144π cm² d. 216π cm² **Answer:** c. 144π cm² These MCQs cover various concepts from the "Surface Area and Volume" chapter of NCERT Class 9 Mathematics and can be used for practice and self-assessment.

Exam notes for 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:** - Geometry is the branch of mathematics that deals with shapes, sizes, properties of figures, and space. - In this chapter, we'll explore concepts related to the surface area and volume of various three-dimensional figures. **13.1 Surface Areas** - The surface area of a three-dimensional object is the total area of its surfaces, including the curved and flat surfaces. - Surface area is measured in square units (e.g., square centimeters, square meters). **13.1.1 Surface Area of Cubes and Cuboids:** - A cube has all its sides equal, and each face is a square. - The surface area of a cube is 6 times the area of one of its faces (A = 6l², where 'l' is the length of a side). - A cuboid has 6 rectangular faces. - The total surface area of a cuboid is the sum of the areas of its six faces (A = 2lw + 2lh + 2wh, where 'l' is length, 'w' is width, and 'h' is height). **13.1.2 Surface Area of Right Circular Cylinder:** - A right circular cylinder has two circular bases and one curved surface. - The lateral or curved surface area of a cylinder is 2πrh, where 'r' is the radius and 'h' is the height. - The total surface area includes the two circular bases, so it is 2πrh + 2πr². **13.1.3 Surface Area of Right Circular Cone:** - A right circular cone has one circular base and one curved surface. - The lateral or curved surface area of a cone is ½πrl, where 'r' is the radius and 'l' is the slant height. - The total surface area includes the base, so it is πr(r + l). **13.1.4 Surface Area of Sphere:** - A sphere has a curved surface. - The surface area of a sphere is 4πr², where 'r' is the radius. **13.2 Volume** - The volume of a three-dimensional object represents the space it occupies. - Volume is measured in cubic units (e.g., cubic centimeters, cubic meters). **13.2.1 Volume of Cubes and Cuboids:** - The volume of a cube is given by V = l³, where 'l' is the length of a side. - The volume of a cuboid is V = lwh, where 'l' is length, 'w' is width, and 'h' is height. **13.2.2 Volume of Right Circular Cylinder:** - The volume of a cylinder is V = πr²h, where 'r' is the radius and 'h' is the height. **13.2.3 Volume of Right Circular Cone:** - The volume of a cone is V = (1/3)πr²h, where 'r' is the radius and 'h' is the height. **13.2.4 Volume of Sphere:** - The volume of a sphere is V = (4/3)πr³, where 'r' is the radius. **Applications:** - These concepts of surface area and volume are applied in various real-life situations such as construction, packaging, manufacturing, and design. **Important Formulas:** - Summarize the key formulas for surface area and volume of different three-dimensional figures. - Practice solving problems to gain a better understanding of these concepts. **Conclusion:** - Understanding surface area and volume is crucial for solving real-world problems and working with three-dimensional objects. - Practice and application of these concepts will help in mastering geometry and its practical applications. These notes should provide you with a comprehensive overview of the key concepts covered in NCERT Class 9 Maths Chapter on Surface Area and Volume. Remember to practice solving problems to reinforce your understanding of these topics.

Exam notes for NCERT Class 9 Maths Chapter on Irrational Numbers
Exam notes for NCERT Class 9 Maths Chapter on Irrational Numbers: **Chapter 1: Irrational Numbers** **Introduction:** - In this chapter, we will explore irrational numbers, which are numbers that cannot be expressed as fractions or terminating or repeating decimals. - Irrational numbers include square roots of non-perfect squares, cube roots of non-perfect cubes, and so on. **Understanding Irrational Numbers:** 1. **Rational Numbers vs. Irrational Numbers:** - Rational numbers can be expressed as fractions, while irrational numbers cannot. - Rational numbers can be written as either terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. 2. **Examples of Irrational Numbers:** - √2, √3, √5, π (Pi), and e (Euler's number) are common examples of irrational numbers. 3. **Non-Perfect Square Roots:** - √2, √3, √5, and so on, are non-perfect square roots and are irrational. - They cannot be expressed as fractions, and their decimal representations go on forever without repeating. 4. **Proving Irrationality:** - We can prove the irrationality of a number using the method of contradiction. - For example, to prove √2 is irrational, we assume it is rational, express it as a fraction, and derive a contradiction. **Operations with Irrational Numbers:** 1. **Addition and Subtraction:** - To add or subtract irrational numbers, simply combine like terms. - For example, √2 + √3 - √2 = √3. 2. **Multiplication and Division:** - To multiply or divide irrational numbers, use the distributive property and simplify as much as possible. - For example, √2 * √3 = √6, and √6 / √2 = √3. **Rationalizing Denominators:** - Rationalizing is the process of converting a fraction with an irrational denominator into a fraction with a rational denominator. - To rationalize a denominator with a square root (√x), multiply both the numerator and denominator by (√x) to eliminate the square root in the denominator. **Decimal Representation of Irrational Numbers:** - Irrational numbers, when expressed in decimal form, are non-terminating and non-repeating. - You can use calculators or long division to approximate their decimal values. **Real Numbers:** - Real numbers include both rational and irrational numbers. - The set of real numbers is represented by ℝ. **Summary:** - Irrational numbers are numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. - Common examples include square roots of non-perfect squares like √2, √3, and mathematical constants like π and e. - Operations with irrational numbers involve combining like terms, multiplying, and dividing. - Rationalizing the denominator is a technique to eliminate square roots from fractions. - Real numbers encompass both rational and irrational numbers. **Practice Problems:** 1. Prove that √2 is irrational using the method of contradiction. 2. Simplify: (√5 + √7) * (√5 - √7). 3. Convert 5√3 into a rational number by rationalizing the denominator. These notes should help you understand the concepts of irrational numbers covered in NCERT Class 9 Maths Chapter 1. Make sure to practice problems to reinforce your understanding.

Exam notes for the NCERT Class 9 Maths Chapter on Converting Decimal Numbers to Fractional Numbers
Exam notes for NCERT Class 9 Maths Chapter on Converting Decimal Numbers to Fractional Numbers: **Chapter 1: Converting Decimal Numbers to Fractional Numbers** **Introduction:** - In this chapter, we will learn how to convert decimal numbers to fractional numbers. - Decimal numbers have a finite or infinite number of decimal places, and converting them to fractions can be useful in various mathematical calculations. **Understanding Decimal Numbers:** 1. **Decimal Representation:** - Decimal numbers are represented in the form of a whole number part, a decimal point, and a decimal fraction part. - For example, 3.25 consists of the whole number 3, a decimal point, and the decimal fraction 25. 2. **Finite and Infinite Decimals:** - Decimal numbers can be finite or infinite. - Finite decimals have a fixed number of decimal places (e.g., 0.75). - Infinite decimals go on forever without repeating (e.g., 0.333...). **Converting Finite Decimals to Fractions:** 1. **Steps to Convert Finite Decimals:** - To convert a finite decimal to a fraction: 1. Count the number of decimal places. 2. Write the decimal number as the numerator. 3. The denominator is the power of 10 corresponding to the number of decimal places. - For example, to convert 0.75 to a fraction: - Count 2 decimal places. - Write 75 as the numerator. - The denominator is 10^2 = 100. - So, 0.75 = 75/100, which can be simplified to 3/4. **Converting Infinite Decimals to Fractions:** 1. **Recurring Decimal Patterns:** - Infinite decimals often have recurring patterns. - To convert a recurring decimal to a fraction: 1. Assign variables to the repeating part and non-repeating part. 2. Subtract the entire number from its repeating part. 3. Set up an equation and solve for the fraction. - For example, to convert 0.333... to a fraction: - Let x = 0.333... - Subtract x from 10x: 10x - x = 3.333... - 0.333... - Solve for x: 9x = 3 - x = 3/9, which simplifies to 1/3. **Decimal to Fraction Examples:** 1. Convert 0.6 to a fraction. - Decimal places = 1 - 0.6 = 6/10 - Simplify to 3/5. 2. Convert 0.125 to a fraction. - Decimal places = 3 - 0.125 = 125/1000 - Simplify to 1/8. 3. Convert 0.666... to a fraction. - Let x = 0.666... - 10x - x = 6.666... - 0.666... - 9x = 6 - x = 6/9, which simplifies to 2/3. **Summary:** - Decimal numbers can be converted to fractions. - For finite decimals, count the decimal places and use powers of 10. - For recurring decimals, set up equations to solve for the fraction. - Converting decimals to fractions is essential for various mathematical calculations and comparisons. **Practice Problems:** 1. Convert 0.25 to a fraction. 2. Convert 0.888... to a fraction. 3. Convert 0.05 to a fraction. These notes should help you understand how to convert decimal numbers to fractional numbers, which is a valuable skill in mathematics. Practice with different decimal numbers to strengthen your understanding.

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 Terminating Repeating Number
Exam notes for NCERT Class 9 Maths Chapter on Terminating and Repeating Decimals: **Chapter 1: Number Systems (Terminating and Repeating Decimals)** **Terminating Decimals:** - A terminating decimal is a decimal number that has a finite number of decimal places. - When you divide a whole number by a power of 10, the result is a terminating decimal. - For example, 0.75, 2.5, 0.8 are all terminating decimals. **Repeating Decimals:** - A repeating decimal is a decimal number that repeats the same sequence of digits infinitely. - When you divide a number by a prime number (other than 2 or 5), the result is a repeating decimal. - For example, 1/3 = 0.333... (repeating 3s), 4/7 = 0.571428... (repeating 571428). **Conversion from Fraction to Decimal:** - To convert a fraction to a decimal, divide the numerator by the denominator. - For terminating decimals, the division will eventually terminate. - For repeating decimals, you'll notice a recurring pattern. **Conversion from Decimal to Fraction:** - To convert a decimal to a fraction, express it in the form "p/q," where "p" is the given decimal, and "q" is a power of 10 that makes the decimal whole. - Simplify the fraction to its lowest terms if necessary. **Expressing Recurring Decimals as Fractions:** - To express a repeating decimal as a fraction, let "x" be the repeating decimal, and "n" be the length of the repeating part. - Multiply "x" by 10^n to create another number "y." - Subtract "x" from "y." - Solve for the fraction as (y - x) / (10^n - 1). **Example:** - Express 0.25 (a terminating decimal) as a fraction: 0.25 = 25/100 = 1/4. **Example:** - Express 0.333... (repeating decimal) as a fraction: Let x = 0.333... - Multiply x by 10 to get 10x = 3.333... - Subtract x from 10x: 10x - x = 3.333... - 0.333..., which simplifies to 9x = 3. - Solve for x: x = 3/9 = 1/3. **Rational and Irrational Numbers:** - Rational numbers can be expressed as fractions, either terminating or repeating decimals. - Irrational numbers have non-repeating, non-terminating decimal expansions. - Example of an irrational number: √2 = 1.41421356... (non-repeating). **Summary:** - Terminating decimals have a finite number of decimal places. - Repeating decimals repeat the same sequence of digits infinitely. - Conversion between fractions and decimals is a useful skill. - Recurring decimals can be expressed as fractions. **Practice Problems:** 1. Convert 0.6, 0.125, and 0.375 to fractions in their lowest terms. 2. Express 0.272727... as a fraction. 3. Convert the fraction 5/6 to a decimal. 4. Determine whether the following numbers are rational or irrational: √7, 0.666..., 1/8, π (pi). Understanding terminating and repeating decimals is important for working with real numbers and solving various mathematical problems. Practice converting between fractions and decimals to strengthen your skills.

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 Algebraic Expression
Exam notes for NCERT Class 9 Maths Chapter on Algebraic Expressions: **Chapter 8: Algebraic Expressions** **Introduction:** - Algebraic Expressions are mathematical expressions that contain variables (letters) and constants (numbers) connected by mathematical operations. - In this chapter, you will learn about algebraic expressions, terms, factors, coefficients, and operations involving algebraic expressions. **Algebraic Expressions:** - An algebraic expression is a combination of variables, constants, and mathematical operations. - Examples of algebraic expressions: 3x + 2y, 4a - 5b, 2x^2 - 3xy + 7. **Terms in an Expression:** - A term is a part of an algebraic expression separated by the plus (+) or minus (-) signs. - In 3x + 2y, '3x' and '2y' are terms. **Factors and Coefficients:** - In the term '3x,' '3' is the coefficient, and 'x' is the variable. - In the term '2y,' '2' is the coefficient, and 'y' is the variable. **Operations on Algebraic Expressions:** 1. **Addition and Subtraction:** You can add or subtract like terms in algebraic expressions. - Example: (3x + 2y) + (2x - 4y) = 5x - 2y 2. **Multiplication:** You can multiply a term by a constant or another term. - Example: 2(3x + 2y) = 6x + 4y 3. **Division:** You can divide a term by a constant. - Example: (4x^2 - 6xy) / 2 = 2x^2 - 3xy **Monomials, Binomials, and Polynomials:** - A monomial is an algebraic expression with one term (e.g., 3x). - A binomial is an algebraic expression with two terms (e.g., 4a - 5b). - A polynomial is an algebraic expression with more than two terms (e.g., 2x^2 - 3xy + 7). **Like and Unlike Terms:** - Terms that have the same variables with the same powers are called like terms (e.g., 3x and 2x are like terms). - Terms that have different variables or the same variables with different powers are called unlike terms (e.g., 3x and 2y are unlike terms). **Operations on Polynomials:** - You can add, subtract, multiply, and divide polynomials using the same rules as for algebraic expressions. **Summary:** - Algebraic expressions consist of variables, constants, and mathematical operations. - Terms are parts of expressions separated by + or - signs. - Factors are parts of terms, and coefficients are the numerical part of terms. - You can perform operations like addition, subtraction, multiplication, and division on algebraic expressions. - Monomials have one term, binomials have two terms, and polynomials have more than two terms. **Practice Problems:** 1. Simplify the expression: 2x + 3y - (x - 2y). 2. Multiply the binomials: (3a + 4b)(2a - 5b). 3. Divide the polynomial 2x^3 - 6x^2 + 8x by 2x. 4. Identify the like terms in the expression: 5x^2 - 3xy + 2x^2 + 4xy - 7y. 5. Write a binomial that represents the difference between 'p' and 'q.' 6. Add the polynomials: (2x^2 - 3xy + 5) and (-x^2 + 4xy - 2). 7. Determine the coefficient of 'a' in the expression: 3ab^2c - 2a^2bc + 4abc. 8. Simplify the expression: 4x^2 - 2x(3x - 5). 9. Find the product of 3x^2y and (-2xy^2).

Exam notes for the NCERT Class 9 Maths Chapter on Factor Theorem
Exam notes for NCERT Class 9 Maths Chapter on the Factor Theorem: **Chapter 2: Factor Theorem** **Introduction:** - The Factor Theorem is an important concept in algebra that helps in finding factors of polynomial expressions. - It provides a criterion for determining whether a given expression is a factor of a polynomial. **Factor Theorem Statement:** - Let 'f(x)' be a polynomial function, and 'a' be a real number. 'x - a' is a factor of 'f(x)' if and only if 'f(a) = 0.' - In other words, if substituting 'x' with 'a' in the polynomial results in 'f(a) = 0,' then 'x - a' is a factor of 'f(x).' **Illustration of Factor Theorem:** - Consider a polynomial 'f(x)' and 'a' such that 'f(a) = 0.' This implies that 'x - a' is a factor of 'f(x).' **Using Factor Theorem:** 1. **Verify if 'x - a' is a factor:** Substitute 'x' with 'a' in the polynomial expression. If 'f(a) = 0,' then 'x - a' is a factor. - Example: For 'f(x) = x^2 - 5x + 6,' if 'f(2) = 0,' then 'x - 2' is a factor. 2. **Using Synthetic Division:** Divide the polynomial 'f(x)' by 'x - a' using synthetic division. If the remainder is zero, 'x - a' is a factor. - Example: Divide 'x^3 - 7x^2 + 14x - 8' by 'x - 2.' If the remainder is zero, 'x - 2' is a factor. **Factorizing Polynomials:** - Once you find a factor using the Factor Theorem, you can use it to factorize the given polynomial. - Example: If 'x - 3' is a factor of 'f(x),' you can write 'f(x)' as '(x - 3)g(x),' where 'g(x)' is another polynomial. **Application of Factor Theorem:** - The Factor Theorem is useful in solving polynomial equations, finding roots of polynomials, and simplifying complex expressions. **Summary:** - The Factor Theorem states that 'x - a' is a factor of 'f(x)' if and only if 'f(a) = 0.' - You can verify if 'x - a' is a factor by substituting 'x' with 'a' and checking if 'f(a) = 0.' - Once a factor is identified, it can be used to factorize the polynomial. - The Factor Theorem has practical applications in solving polynomial equations. **Practice Problems:** 1. Determine if 'x - 4' is a factor of the polynomial '2x^3 - 8x^2 + 3x - 12.' 2. Use the Factor Theorem to factorize 'x^3 - 8x^2 + 17x - 10' if 'x - 2' is a factor. 3. Find the values of 'k' for which 'x - k' is a factor of 'x^3 - 6x^2 + 11x - k.' 4. Check if 'x - 3' is a factor of '2x^4 - 7x^3 + 5x^2 - 11x + 6.' 5. Use the Factor Theorem to factorize 'x^3 + 4x^2 - 11x - 30' if 'x + 3' is a factor. 6. Determine if 'x - 1' is a factor of '3x^4 - 7x^3 + 5x^2 - 8x + 4.' 7. Find the values of 'a' for which 'x - a' is a factor of '2x^3 - 5x^2 + 4x - a.' 8. Factorize the polynomial 'x^3 - 6x^2 + 11x - 6' using the Factor Theorem.

Exam notes for the NCERT Class 9 Maths Chapter on Euclid’s Geometry
Exam notes for NCERT Class 9 Maths Chapter on Euclid's Geometry: **Chapter 5: Euclid's Geometry** **Introduction:** - Geometry is a branch of mathematics that deals with the study of shapes, sizes, properties, and dimensions of figures and spaces. - Euclid, a Greek mathematician, made significant contributions to the development of geometry. - Euclid's geometry is based on a set of axioms (self-evident truths) and deductive reasoning. **Euclid's Axiomatic Approach:** - Euclid's geometry is built on five postulates or axioms: 1. A straight line can be drawn from any two points. 2. A finite straight line can be extended indefinitely. 3. A circle can be drawn with any center and radius. 4. All right angles are equal to each other. 5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, then the two lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles. **Illustrations of Euclid's Geometry:** - Euclid's geometry provides a framework for proving various geometric theorems using deductive reasoning. - Examples include the construction of a perpendicular bisector, the construction of an angle bisector, and the construction of congruent angles. **Geometry Proofs:** - Euclid's geometry emphasizes the importance of proofs in mathematics. - Proofs are logical arguments that establish the truth of a geometric statement or theorem. - Geometric proofs involve stating the given information, defining what needs to be proved, and providing a step-by-step logical argument to reach the conclusion. **Geometry Constructions:** - Euclid's geometry also involves geometric constructions, which are methods for creating geometric figures using only a compass and straightedge. - Common constructions include drawing perpendiculars, bisecting angles, and constructing triangles with specified sides. **Geometry Theorems:** - Euclid's geometry contains numerous theorems that describe the relationships between geometric figures and properties. - Examples of theorems include the Pythagorean Theorem, the congruence of triangles, and the properties of circles. **Applications of Euclid's Geometry:** - Euclid's principles and theorems are foundational in various fields such as architecture, engineering, and computer graphics. - They are essential for solving real-world problems involving measurement and design. **Summary:** - Euclid's Geometry is based on five axioms and emphasizes the use of proofs to establish the truth of geometric statements. - It includes the construction of geometric figures and theorems that describe relationships between shapes and properties. - Euclid's work has had a profound influence on the development of mathematics and its practical applications. **Practice Problems:** 1. Prove that the sum of the angles of a triangle is 180 degrees. 2. Construct an equilateral triangle given one side. 3. Prove the Pythagorean Theorem. 4. Construct a perpendicular bisector of a line segment. 5. Prove that opposite sides of a parallelogram are equal and parallel. 6. Construct an angle bisector for a given angle. 7. Prove that the diagonals of a rectangle are equal in length and bisect each other. 8. Construct a triangle given its three sides (SSS congruence). 9. Prove that the angles opposite equal sides of a triangle are equal. 10. Construct a triangle given one angle, one side adjacent to the angle, and the perimeter.

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.

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