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 the previous section, you have learnt one method of obtaining the roots of a quadratic equation. In this section, we shall study another method. To implement this method we have certain steps to follow. We will understand the steps with `2x^2+7x-9=0` this example-
1) Make the coefficient of ` x^2` as 1. For that we have to divide the whole equation with the coefficient of `x^2`. In the above example coefficient of `x^2` is 2 so will divide the equation with 2, we get, `x^2+"7x"/2-9/2=0`
2) After that take the constant to RHS. So we get `x^2+"7x"/2=9/2`.
3) Now take the square of the coefficient of x after multiplying with `1/2`, and then add that number to both the sides i.e add: `(1/2× "coefficient" "of" x)^2`
Coefficient of x here is `7/2, (1/2×7/2)^2 = (7/4)^2`
by adding `(7/4)^2` both the sides we get, `x^2+"7x"/2+(7/4)^2 = 9/2+(7/4)^2`
4) Use `(a+b)^2` or `(a-b)^2`
`x^2+"7x"/2+(7/4)^2 = 9/2+(7/4)^2`
By obervation we get, `c= 9/2 + 49/16`
`(x+7/4)^2= (72+49)/16`
`(x+7/4)^2= 121/16`
`x+7/4 = + or - sqrt 121/16`
`x+7/4 = + or - 11/4`
`x+7/4 = 11/4 or -11/4`
`x=11/4-7/4 or -11/4-7/4`
`x= 4/4 or -18/4`
`x= 1 or -9/2`
Thus, the roots of `2x^2+7x-9=0` are `1 or -9/2`
We can solve more examples using complete square method to understand easily.
