Topics
Rational and Irrational Numbers
Parallel Lines and Transversal
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Properties of Parallel Lines
- Corresponding Angle Theorem
- Alternate Angles Theorems
- Interior Angle Theorem
- To Draw a Line Parallel to the Given Line Through a Point Outside the Given Line Using Set-square.
- To Draw a Parallel Line to a Given Line at a Given Distance.
Indices and Cube Root
- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Meaning of Numbers with Rational Indices
- Concept of Cube Number
- Concept of Cube Root
- Cube Root Through Prime Factorisation Method
Altitudes and Medians of a Triangle
Expansion Formulae
Factorisation of Algebraic Expressions
Variation
- Direct Variation
- Inverse Variation
- Time, Work, Speed
Quadrilateral : Constructions and Types
- Constructing a Quadrilateral
- 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
- Properties of Rectangle
- Properties of a Square
- Properties of Rhombus
- Properties of a Parallelogram
- Properties of Trapezium
- Properties of Kite
Discount and Commission
- Concept of Discount
- Commission
- Rebate
Division of Polynomials
- Polynomials
- Degree of Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Divide a Polynomial by a Binomial
Statistics
- Arithmetic Mean - Raw Data
- Subdivided Bar Graph
- Percentage Bar Graph
Equations in One Variable
- Solution of Equations in One Variable
- Word Problems of Equation in One Variable
Congruence of Triangles
- Congruence of Triangles
- Criteria for Congruence of Triangles
- SAS Congruence Criterion
- SSS Congruence Criterion
- ASA Congruence Criterion
- AAS (Or SAA) Test
- RHS Congruence Criterion
Compound Interest
Area
- Area of a Parallelogram
- Area of a Rhombus
- Area of Trapezium
- Area of a Triangle
- Area of Figures Having Irregular Shape
- Circumference of a Circle
- Area of Circle
Surface Area and Volume
Circle - Chord and Arc
- Properties of Chord of a Circle
- Arcs Corresponding to the Chord of a Circle
Formula
- a3 - b3 = ( a - b )( a2 + ab + b2)
Notes
Factorisation using Identity a3 - b3 = (a - b)(a2 + ab + b2):
(a - b)3 = a3 - 3a2b + 3ab2 - b3 = a3 - b3 - 3ab(a - b)
Now, a3 - b3 - 3ab(a - b) = (a - b)3
∴ a3 - b3 = (a - b)3 + 3ab(a - b)
a3 - b3 = [(a - b)(a - b)2 + 3ab(a - b)]
a3 - b3 = (a - b)[(a - b)2 + 3ab]
a3 - b3 = (a - b)(a2 - 2ab + b2 + 3ab)
a3 - b3 = (a - b)(a2 + ab + b2)
∴ a3 - b3 = (a - b)(a2 + ab + b2).
Example
Factorise: x3 - 8y3
x3 - 8y3 = x3 - (2y)3
∴ x3 - 8y3 = x3 - (2y)3
x3 - 8y3 = (x - 2y)(x2 + 2xy + 4y2)
Example
Factorise: 27p3 - 125q3.
27p3 - 125q3
= (3p)3 - (5q)3
= (3p - 5q) (9p2 + 15pq +25q2)
Example
Factorise: 54p3 - 250q3.
54p3 - 250q3.
= 2[27p3 - 125q3]
= 2[(3p)3 - (5q)3]
= 2(3p - 5q)(9p2 + 15pq + 25q2)
Example
Factorise: `a^3 - 1/(a^3)`
`a^3 - 1/(a^3)`
`= (a - 1/a)(a^2 + 1 + 1/(a^2))`
Example
Simplify: (a - b)3 - (a3 - b3)
(a - b)3 - (a3 - b3)
= a3 - 3a2b + 3ab2 - b3 - a3 + b3
= 3a2b + 3ab2.
Example
Simplify: (2x + 3y)3 - (2x - 3y)3
Using the formula a3 - b3 = (a - b)(a2 + ab + b2)
∴ (2x + 3y)3 - (2x - 3y)3
= [(2x + 3y) - (2x - 3y)][(2x + 3y)2 + (2x + 3y)(2x - 3y) + (2x - 3y)2]
= [2x + 3y - 2x + 3y][4x2 + 12xy + 9y2 + 4x2 - 9y2 + 4x2 - 12xy + 9y2]
= 6y(12x2 + 9y2)
= 72x2y + 54y3
