Topics
Number Systems
Real Numbers
Algebra
Polynomials
Pair of Linear Equations in Two Variables
- Introduction to linear equations in two variables
- Graphical Method
- 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
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
Arithmetic Progressions
Coordinate Geometry
Lines (In Two-dimensions)
Constructions
- Division of a Line Segment
- Construction of Tangents to a Circle
- Constructions Examples and Solutions
Geometry
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
- Concept of Angle Bisector
- Similarity of Triangles
- Ratio of Sides of Triangle
Circles
Trigonometry
Introduction to Trigonometry
- Trigonometry
- Trigonometry
- Trigonometric Ratios
- Trigonometric Ratios and Its Reciprocal
- Trigonometric Ratios of Some Special Angles
- Trigonometric Ratios of Complementary Angles
- Trigonometric Identities
- Proof of Existence
- Relationships Between the Ratios
Trigonometric Identities
Some Applications of Trigonometry
Mensuration
Areas Related to Circles
- Perimeter and Area of a Circle - A Review
- Areas of Sector and Segment of a Circle
- Areas of Combinations of Plane Figures
- Circumference of a Circle
- Area of Circle
Surface Areas and Volumes
- 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
Statistics and Probability
Statistics
Probability
Internal Assessment
Notes
In this method, there are three conditions.
(1) Condition- `a_1/a_2` is not equal to `b_1/b_2`
Example- `2x+9y+12=0` and `6x+1y+8=0`
`2/6` is not equal to `9/1`
If we represent these equations on a graph, then the lines of this equation will intersect each other at some point.
In this condition, we can conclude
- We get an intersecting line.
- Such a type of pair of linear equations with two variables where `a_1/a_2` is not equal to `b_1/b_2` have only one solution, i.e. unique solution.
- This type of equation is called a Consistent equation.
(2) Condition- `a_1/a_2`= `b_1/b_2`= `c_1/c_2`
Example- `2x+4y+8=0` and `6x+12y+24=0`
`2/6=1/3, 4/12=1/3, 8/24=1/3` i.e `1/3=1/3=1/3`
In the graphical representation of these equations, the lines will coincide.
Thus, the conclusion in this condition is
- We get Consistent lines.
- Such a type of pair of linear equations with two variables where `a_1/a_2=b_1/b_2=c_1/c_2` have infinitely many solutions.
- This equation is also called a Consistent equation.
(3) Condition- `a_1/a_2= b_1/b_2` is not equal to `c_1/c_2`
Example- `2x+3y+4=0` and `4x+6y+7=0`
`2/4=1/2, 3/6=1/2, 4/7` i.e `1/2=1/2` is not equal to `4/7`
The graphical representation of these equations will result in parallel lines.
Here is the conclusion
- We will get parallel.
- Such a type of pair of linear equations with two variables where `a_1/a_2= b_1/b_2` is not equal to `c_1/c_2` have no solution.
- This type of Linear equation, which doesn't give any solution, is called an Inconsistent equation.
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