Topics
Rational and Irrational Numbers
Parallel Lines and Transversal
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
- Types of 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
- Basic Concept of Polynomial and its Degree
- Degree of Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Divide a Polynomial by a Binomial
Statistics
- Arithmetic Mean
- 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
- Criteria for Similarity of Triangles
- 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
- Chord
- Arcs Corresponding to the Chord of a Circle
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)3 = (a + b)(a2 + 2ab + b2)
(a + b)3 = a(a2 + 2ab + b2) + b(a2 + 2ab + b2)
(a + b)3 = a3 + 2a2b + ab2 + ba2 + 2ab2 + b3
(a + b)3 = a3 + 3a2b + 3ab2 + b3
∴ (a + b)3 = a3 + 3a2b + 3ab2 + b3.
Example
Expand: (x + 3)3.
(x + 3)3
We know that (a + b)3 = a3 + 3a2b + 3ab2 + b3.
In the given example, a = x and b = 3.
∴ `(x + 3)^3 = (x)^3 + 3 xx x^2 xx 3 + 3 xx x xx (3)^2 + (3)^3`.
∴ `(x + 3)^3 = x^3 + 9x^2 + 27x + 27`.
Example
Expand: (3x + 4y)3.
(3x + 4y)3
`= (3x)^3 + 3(3x)^2(4y) + 3(3x)(4y)^2 + (4y)^3`.
`= 27x^3 + 3 xx 9x^2 xx 4y + 3 xx 3x xx 16y^2 + 64y^3`.
`= 27x^3 + 108x^2y + 144xy^2 + 64y^3`
Example
Expand: (41)3
(41)3
= (40 + 1)3
= (40)3 + 3 × (40)2 × 1 + 3 × 40 × (1)2 + (1)3
= 64000 + 4800 + 120 + 1
= 68921
Example
Expand: `((2m)/n + n/(2m))^3`.
`((2m)/n + n/(2m))^3`
`= ((2m)/n)^3 + 3((2m)/n)^2(n/(2m)) + 3((2m)/n)(n/(2m))^2 + (n/(2m))^3`
`= (8m^3)/(n^3) + 3((4m^2)/(n^2))(n/(2m)) + 3((2m)/n)(n^2/(4m^2)) + ((n^3)/8m^3)`
`= ((8m^3)/(n^3)) + ((6m)/n) + (3n)/(2m) + (n^3)/(8m^3)`
Example
Simplify.
(p + q)3 + (p - q)3
(p + q)3 + (p - q)3
= p3 + 3p2q + 3pq2 + q3 + p3 - 3p2q + 3pq2 - q3
= 2p3 + 6pq2
