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
Consider the following equation: 2x + 5 = 0
the root of the equation is `-5/2`. This can be represented on the number line as shown below:
While solving an equation, you must always keep the following points in mind.
The solution of a linear equation is not affected when:
- The same number is added to (or subtracted from) both sides of the equation.
- You multiply or divide both sides of the equation by the same non-zero number.
So, any equation that can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables.
Notes
Linear equation is an algebraic equation having a degree of 1. We can say `2x+5=0` is a linear equation, as the degree of the variable x is 1. But in a linear equation in two variables have two variables. E.g. `2x+5y-1=0` is in the form of a linear equation with two variables, x and y. `ax+by+c=0` is a standard form of linear equation in two variables, where a is the coefficient of x, b is the coefficient of y, and c is a constant term, a, b and c are real numbers, also a and b are not equal to zero.
In this chapter, we have to learn about the Pair of Linear Equations in Two Variables. Generally, A Pair of Linear Equations in Two Variables are written as `a_1x+b_1y+c_1=0` and `a_2x+b_2y+c_2=0.` Example `2x+9y+12=0` and `6x+1y+8=0`
To learn how to solve A Pair of Linear Equations in Two Variables, there are two methods: Graphical and Algebraic. We will study this in the following concepts.
