Topics
Number Systems
Algebra
Geometry
Trigonometry
Statistics and Probability
Coordinate Geometry
Mensuration
Internal Assessment
Real Numbers
Pair of Linear Equations in Two Variables
- Linear Equation 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 of a Quadratic Equation
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Application of Quadratic Equation
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 (Thales Theorem)
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity of Triangles
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity of Triangles
- 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
- Concept of Surface Area, Volume, and Capacity
- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures
definition
- Irrational numbers: Irrational numbers are numbers that cannot be written in the form `p/q`, where p and q are integers and q ≠ 0. Example, √2, √3, √15, π, 0.10110111011110.........etc.
notes
Irrational Numbers:
- Consider an isosceles right-angled triangle whose base and height are each 1 unit long. Using Pythagoras theorem, the hypotenuse can be seen having a length `sqrt(1^2 + 1^2) = sqrt2`. Greeks found that this `sqrt2` is neither a whole number nor an ordinary fraction. The belief of the relationship between points on the number line and all numbers were shattered! `sqrt2` was called an irrational number.
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Irrational numbers are numbers that cannot be written in the form `p/q`, where p and q are integers and q ≠ 0.
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Example, √2, √3, √15, π, 0.10110111011110.........etc.
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An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.
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Examples:
- Apart from √2, one can produce a number of examples for such irrational numbers. Here are a few: √5, √7, 2√3,......
- π, the ratio of the circumference of a circle to the diameter of that same circle is another example for an irrational number.
- e, also known as Euler’s number, is another common irrational number.
- The golden ratio, also known as the golden mean, or golden section, is a number often stumbled upon when taking the ratios of distances in simple geometric figures such as the pentagon, the pentagram, decagon, and dodecahedron, etc., it is an irrational number.
GOLDEN RATIO (1:1.6) The Golden Ratio has been heralded as the most beautiful ratio in art and architecture. Take a line segment and divide it into two smaller segments such that the ratio of the whole line segment (a + b) to segment a is the same as the ratio of segment a to the segment b. This gives the proportion `(a + b)/a = a/b`. Notice that 'a' is the geometric mean of a + b and b. |