Topics
Number System
Rational Numbers
- Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Rational Numbers on a Number Line
- Inserting Rational Numbers Between Two Given Rational Numbers
- Method of Finding a Large Number of Rational Numbers Between Two Given Rational Numbers
Exponents
- Concept of Exponents
- Law of Exponents (For Integral Powers)
- Negative Integral Exponents
- More About Exponents
Squares and Square Root
- Concept of Square Roots
- Finding Square Root by Division Method
- Finding Square Root Through Prime Factorisation
- To Find the Square Root of a Number Which is Not a Perfect Square (Using Division Method)
- Properties of Square Numbers
Cubes and Cube Roots
Playing with Numbers
- Arranging the Objects in Rows and Columns
- Generalized Form of Numbers
- Some Interesting Properties
- Letters for Digits (Cryptarithms)
- Divisibility by 10
- Divisibility by 2
- Divisibility by 5
- Divisibility by 3
- Divisibility by 6
- Divisibility by 11
- Divisibility by 4
Sets
- Concept of Sets
- Representation of a Set
- Cardinal Number of a Set
- Types of Sets
- Subset
- Proper Subset
- Number of Subsets and Proper Subsets of a Given Set
- Super Set
- Universal Set
- Complement of a Set
- Set Operations
- Difference of Two Sets
- Distributive Laws
- Venn Diagrams
Ratio and Proportion
Percent and Percentage
Profit, Loss and Discount
- Concept of Discount
- Overhead Expenses
- To Find S.P., When C.P. and Gain (Or Loss) Percent Are Given
- To Find C.P., When S.P. and Gain (Or Loss) Percent Are Given
- Concept of Discount
- Computation of Tax
- Goods and Service Tax (Gst)
- Gst Comprises of
Interest
- Calculation of Interest
- Concept of Compound Interest
- To Find the Principle (P); the Rate Percent (R) and the Time
- Interest Compounded Half Yearly
- Applications of Compound Interest Formula
Direct and Inverse Variations
- Variations
- Types of Variation
- Direct Variation
- Inverse Variation
- Concept for Unitary Method (With Only Direct Variation Implied)
- Concept of Arrow Method
- Time and Work
Algebra
Algebraic Expressions
- Algebraic Expressions
- Degree of Polynomial
- Product , Factor and Coefficient
- Like and Unlike Terms
- Combining like Terms
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Polynomial
- Multiplying a Polynomial by a Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Simplification of Expressions
Identities
- Algebraic Identities
- Product of Sum and Difference of Two Terms
- Expansion Form of Numbers
- Important Formula of Expansion
- Cubes of Binomials
- Application of Formulae
Factorisation
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Factorisation by Difference of Two Squares
- Factorisation of Trinomials
- Factorising a Perfect Square Trinomial
- Factorising Completely
Linear Equations in One Variable
- Simple Linear Equations in One Variable
- Solving Linear Inequations
- Linear Equation in One Variable
- Equations Reducible to the Linear Form
Linear Inequations
- Introduction to linear equations in two variables
- Replacement Set and Solution Set
- Operation of Whole Numbers on Number Line
- Concept of Properties
Geometry
Understanding Shapes
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Concept of Polygons
- Sum of Angles of a Polynomial
- Sum of Exterior Angles of a Polynomial
- Regular Polynomial
- Concept of Quadrilaterals
Special Types of Quadrilaterals
Constructions
- Introduction of Constructions
- Construction of an Angle
- To Construct an Angle Equal to Given Angle
- To Draw the Bisector of a Given Angle
- Construction of an angle bisector of an Angles of 60°,30°,90° and 45°
- Construction of Bisector of a Line
- The Perpendicular Bisector
- Construction of Parallel Lines
- Constructing a Quadrilateral
- Construction of Parallelograms
- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Construction of Square
- Concept of Reflection Symmetry
Representing 3-D in 2-D
- 2dimensional Perspective of 3dimensional Objects
- Concept of Polyhedron
- Faces, Edges and Vertices
- Euler's Formula
- Concept of Polyhedron
- Nets for Building 3-d Shapes
Mensuration
Area of a Trapezium and a Polygon
Surface Area, Volume and Capacity
Data Handling (Statistics)
Data Handling
- Concept of Data Handling
- Collecting Data
- Frequency
- Raw Data, Arrayed Data and Frequency Distribution
- Constructing a Frequency Table
- Frequency Distribution Table
- Class Intervals and Class Limits
- Graphical Representation of Data
Probability
Notes
Divisibility by 11:
To check the divisibility of a number by 11, the rule is, find the difference between the sum of the digits at odd places (from the right)and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
|
Number |
Sum of the digits (at odd places) from the right |
Sum of the digit (at even places) from the right |
Difference |
|
308 |
8 + 3 = 11 |
0 |
11 – 0 = 11 |
|
1331 |
1 + 3 = 4 |
3 + 1 = 4 |
4 – 4 = 0 |
|
61809 |
9 + 8 + 6 = 23 |
0 + 1 = 1 |
23 – 1 = 22 |
We observe that in each case the difference is either 0 or divisible by 11. All these numbers are also divisible by 11.
For the number 5081, the difference of the digits is (5+ 8) – (1 + 0) = 12which is not divisible by 11. The number 5081 is also not divisible by 11.
More Examples:
Is the number 2547039 divisible by 11?
First, find the difference between the sum of its digits at odd and even places.
(Sum of digits at odd places) - (Sum of digits at even places)
= (9 + 0 + 4 + 2) - (3 + 7 + 5)
= 15 - 15 = 0
The number is 0, so the number 2547039 is divisible by 11.
Is the number 13165648 divisible by 11?
(Sum of digits at odd places) - (Sum of digits at even places)
= (8 + 6 + 6 + 3) - (4 + 5 + 1 + 1)
= 23 - 11 = 12
The number is 12, so the number 13165648 is not divisible by 11.
