Topics
Rational Numbers
- Rational Numbers
- Closure Property of Rational Numbers
- Commutative Property of Rational Numbers
- Associative Property of Rational Numbers
- Distributive Property of Multiplication Over Addition for Rational Numbers
- Identity of Addition and Multiplication of Rational Numbers
- Negative Or Additive Inverse of Rational Numbers
- Concept of Reciprocal or Multiplicative Inverse
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
- Variable of Equation
- Concept of Equation
- Expressions with Variables
- Balancing an Equation
- The Solution of an Equation
- Linear Equation in One Variable
- Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Solving Equations Having the Variable on Both Sides
- Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
- Reducing Equations to Simpler Form
- Equations Reducible to the Linear Form
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Concept of Polygons
- Classification of Polygons
- Properties of a Quadrilateral
- Interior Angles of a Polygon
- Exterior Angles of a Polygon and Its Property
- Concept of Quadrilaterals
- Properties of Trapezium
- Properties of Kite
- Properties of a Parallelogram
- Properties of Rhombus
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The adjacent angles in a parallelogram are supplementary.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Properties of Rectangle
- Properties of a Square
- Property: The diagonals of a square are perpendicular bisectors of each other.
Practical Geometry
- Introduction to Geometric Tool
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Some Special Cases
Data Handling
- Concept of Data Handling
- Interpretation of a Pictograph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Organisation of Data
- Frequency Distribution Table
- Graphical Representation of Data as Histograms
- Concept of Pie Graph (Or a Circle-graph)
- Interpretation of Pie Diagram
- Chance and Probability - Chance
- Basic Ideas of Probability
Squares and Square Roots
- Concept of Square Number
- Properties of Square Numbers
- Some More Interesting Patterns of Square Number
- Finding the Square of a Number
- Concept of Square Roots
- Finding Square Root Through Repeated Subtraction
- Finding Square Root Through Prime Factorisation
- Finding Square Root by Division Method
- Square Root of Decimal Numbers
- Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
- Concept of Ratio
- Basic Concept of Percentage
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Sales Tax, Value Added Tax, and Good and Services Tax
- Calculation of Interest
- Concept of Compound Interest
- Deducing a Formula for Compound Interest
- Rate Compounded Annually Or Half Yearly (Semi Annually)
- Applications of Compound Interest Formula
Algebraic Expressions and Identities
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Like and Unlike Terms
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Visualizing Solid Shapes
Exponents and Powers
Direct and Inverse Proportions
Factorization
- Factors and Multiples
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Concept of Find the Error
Introduction to Graphs
- Concept of Bar Graph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Concept of Double Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Concept of Pie Graph (Or a Circle-graph)
- Graphical Representation of Data as Histograms
- Concept of a Line Graph
- Linear Graphs
- Some Application of Linear Graphs
Playing with Numbers
Notes
Multiplying a Monomial by a Monomial:
1. Multiplying two monomials:
While multiplying two monomials
-
Coefficient of product = Coefficient of first monomial × Coefficient of the second monomial
-
Algebraic factor of product = Algebraic factor of first monomial × Algebraic factor of the second monomial.
1) 5x × 3y
= 5 × x × 3 × y
= 5 × 3 × x × y
= 15xy.
2) 5x × (- 4xyz)
= (5 × - 4) × (x × xyz)
= -20 × (x × x × yz)
= -20 x2yz.
2. Multiplying three or more monomials:
It is clear that we first multiply the first two monomials and then multiply the resulting monomial by the third monomial.
1) 4xy × 5x2y2 × 6x3y3
= (4xy × 5x2y2) × 6x3y3
= 20x3y3 × 6x3y3
= 120x3y3 × x3y3
= 120(x3 × x3) × (y3 × y3)
= 120 x6 × y6
= 120 x6y6.
2) 2x × 5y × 7z
= (2x × 5y) × 7z
= 10xy × 7z
= 70xyz.
= 70xyz.
Example
Solve: ( -3x2 ) × ( -4xy)
( -3x2 ) × ( -4xy)
= (-3) × (-4) × x2 × x × y
= 12x3y
Example
Solve: (-12x) × 3y2
(-12x) × 3y2
= - 12 × 3 × x × y × y
= - 36xy2.
Example
Find the volume of each rectangular box with given length, breadth, and height.
| Length | breadth | height | |
| (i) | 2ax | 3by | 5cz |
| (ii) | m2n | n2p | p2m |
| (iii) | 2q | 4q2 | 8q3 |
Volume = length × breadth × height
Hence, for
(i) Volume = (2ax) × (3by) × (5cz) = 2 × 3 × 5 × (ax) × (by) × (cz) = 30abcxyz.
(ii) Volume = m2n × n2p × p2m = (m2 × m) × (n × n2) × (p × p2) = m3n3p3.
(iii) Volume = 2q × 4q2 × 8q3 = 2 × 4 × 8 × q × q2 × q3 = 64q6.
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