Topics
Number Systems
Real Numbers
Algebra
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
Geometry
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
Trigonometry
Heights and Distances
Trigonometric Identities
Introduction to Trigonometry
Statistics and Probability
Probability
Statistics
Coordinate Geometry
Lines (In Two-dimensions)
Mensuration
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
Internal Assessment
- general term of the AP
Notes
nth term of an AP is also known as last term or general term.
Let a be the first term of an AP and d be the common difference then a standard form of an AP will be a, a+d, a+2d, a+3d,............. and so on upto nth term.
In an AP terms can also be written as `a_1, a_2, a_3, a_4,........, a_n`
`a_1= a+ (1-1)d`
`a_2= a+d= a+ (2-1)d`
`a_3= a+2d= a+ (3-1)d`
`a_4= a+3d= a+ (4-1)d`
And so on...
So if we generalize this pattern we get
`a_n= a+ (n-1)d`
Example- In an AP 2, 7, 12, ........... Then find a20
`a= 2, n=20, d= a_2-a_1= 7-2= 5`
`a_20= 2+ (20-1)5`
= `2+ 19(5)`
= `2+ 95`
`a_20= 97`
Now, what will do if we are asked find nth term form the end? So, we will use a different approch.
Let a be the first term of an AP and d be the common difference then a standard form of an AP will be `a, a+d, a+2d, a+3d, ......, l ` where `l` is a last term.
If the last term is `l` then the term before l will be `l-d`, and if the second last term is `l-d` then `l-d-d` i.e `l-2d` will be the term before it and so on.
First term from the end`= l= l- (1-1)d`
Second term from the end`= l-d= l- (2-1)d`
Third term form the end`= l-2d= l- (3-1)d`
And so on...
So if we generalize this pattern we get
nth term from the end= `l- (n-1)d`
Example- 4, 9, 14, ....., 254 Find the 10th term from the end
`a= 4, n=10, d= a_2-a_1= 9-4= 5, l=254`
nth term from the end=` l- (n-1)d`
10th term from the end= `254- (10-1)5`
= `254- (9)5`
= `254-45`
10th term from the end= `209`
