Topics
Number Systems
Algebra
Geometry
Trigonometry
Statistics and Probability
Coordinate Geometry
Mensuration
Internal Assessment
Real Numbers
Pair of Linear Equations in Two Variables
- Linear Equations in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient
Arithmetic Progressions
Quadratic Equations
- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Quadratic Equations Examples and Solutions
Polynomials
Circles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
Triangles
- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem Or Thales Theorem
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity Triangle Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity
- Ratio of Sides of Triangle
Constructions
Heights and Distances
Trigonometric Identities
Introduction to Trigonometry
Probability
Statistics
Lines (In Two-dimensions)
Areas Related to Circles
Surface Areas and Volumes
notes
Around 300 BC a philosopher known as Euclid of Alexandria understood that all numbers could be split into these two distinct categories. He realized that any number can be divided down over and over until you reach a group of smallest equal numbers. And by definition these smallest numbers are always prime numbers. All numbers are built out of smaller primes.
Take any composite number ex: 125 and break it down, it will always be left with prime numbers i.e 5 × 5 × 5. Euclid knew that every number could be expressed using a group of smaller primes. No matter what number you choose, it can always be built with an addition of smaller primes. This is the root of his discovery, known as the fundamental theorem of arithmetic. Every possible number has one, and only one prime factorization.
Every composite number can be written as the product of powers of primes. For example 4 is composite numbe and it can be split into 2 × 2. Similarly 9, 18, 21, 33, 55 could be split into 3 × 3, 3 × 3 × 2, 7 × 3, 11 × 3, 11 × 5 respectively.
Let's consider a example 4n, where n is a natural number. Check whether there is any value of for which 4n ends with the digit zero.If 4n has to end with digit zero then its factors must be 5 × 2. Therefore 4n = k × 5 where k is a constant.
4n = (2 × 2)n i.e. 2 × 2 × 2 × 2 ×......×n
This does not have 5 in the prime factors. Therefore, 4n doesn't end with 0.
Let us learn how to find LCM and HCF by the prime factorisation method. For example what is the LCF and HCF of 96 and 404 by prime factorisation.
96 = 2 × 2 × 2 ×2 × 3 Therefore 96 = 25 × 3
404 = 2 × 2 × 101 Hence 404 = 22 × 101
While finding HCF we will take the lowest common prime factor whereas to find LCM the highest prime factors are considered
HCF = 22 = 4
LCM= 25 × 3 × 101 = 9696
description
- Significance of the Fundamental Theorem of Arithmetic
