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
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
Division of Polynomials
Statistics
Equations in One Variable
Congruence of Triangles
Compound Interest
Area
Surface Area and Volume
Circle - Chord and Arc
Formula
- (a - b)3 = a3 - 3a2b + 3ab2 - b3.
Notes
Expansion of (a - b)3:
∴ (a - b)3 = (a - b)(a - b)(a - b) = (a - b)(a - b)2
= (a - b)(a2 - 2ab + b2)
= a(a2 - 2ab + b2) - b(a2 - 2ab + b2)
= a3 - 2a2b + ab2 - a2b + 2ab2 - b3
= a3 - 3a2b + 3ab2 - b3
∴ (a - b)3 = a3 - 3a2b + 3ab2 - b3.
Example
Expand (x - 2)3.
(a - b)3 = a3 - 3a2b + 3ab2 - b3
Here taking, a = x and b = 2,
`(x - 2)^3 = x^3 - 3 xx x^2 xx 2 + 3 xx x xx (2)^2 - (2)^3`
`(x - 2)^3 = x^3 - 6x^2 + 12x - 8`
Example
Expand (4p - 5q)3
(4p - 5q)3 = (4p)3 - 3(4p)2(5q) + 3(4p)(5q)2 - (5q)3.
(4p - 5q)3 = 64p3 - 240p2q + 300pq2 - 125q3.
Example
Find the cube of 99 using the expansion formula.
(99)3
= (100 - 1)3
= (100)3 - 3 × (100)2 × 1 + 3 × 100 × (1)2 - 13
= 1000000 - 30000 + 300 - 1
= 9,70,299
Example
Simplify.
(2x + 3y)3 - (2x - 3y)3
(2x + 3y)3 - (2x - 3y)3
= [(2x)3 + 3(2x)2(3y) + 3(2x)(3y)2 + (3y)3 ] - [(2x)3 - 3(2x)2(3y) + 3(2x)(3y)2 - (3y)3]
= (8x3 + 36x2y + 54xy2 + 27y3) - (8x3 - 36x2y + 54xy2 - 27y3)
= 8x3 +36x2y + 54xy2 + 27y3 - 8x3 + 36x2y - 54xy2 + 27y3
= 72x2y + 54y3
