Topics
Rational and Irrational Numbers
- Rational Numbers
- Properties of Rational Numbers
- Decimal Representation of Rational Numbers
- Concept of Irrational Numbers
- Concept of Real Numbers
- Surds
- Rationalisation of Surds
- Simplifying an Expression by Rationalization of the Denominator
Compound Interest [Without Using Formula]
- Calculation of Interest
- Concept of Compound Interest
- Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
- Concept of Compound Interest
Compound Interest [Using Formula]
- Concept of Compound Interest
- Inverse Formula
- Miscellaneous Problem
- When the Interest is Compounded Half Yearly
- When the Time is Not an Exact Number of Years and the Interest is Compounded Yearly
- Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
Expansions
- Algebraic Identities
- Expansion of (a + b)3
- Expansion of Formula
- Special Product
- Methods of Solving Simultaneous Linear Equations by Cross Multiplication Method
Factorisation
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Factorisation of a Quadratic Trinomial by Splitting the Middle Term
- Method of Factorisation : Difference of Two Squares
- Method of Factorisation : the Sum Or Difference of Two Cubes
Simultaneous (Linear) Equations [Including Problems]
- Methods of Solving Simultaneous Linear Equations by Elimination Method
- Methods of Solving Simultaneous Linear Equations by Elimination Method
- Method of Elimination by Equating Coefficients
- Equations Reducible to Linear Equations
- Simultaneous method
- Methods of Solving Simultaneous Linear Equations by Cross Multiplication Method
- Simple Linear Equations in One Variable
- Introduction to linear equations in two variables
Indices [Exponents]
- Laws of Exponents
- Handling Positive, Fraction, Negative and Zero Indices
- Simplification of Expressions
- Solving Exponential Equations
Logarithms
- Introduction of Logarithms
- Interchanging Logarithmic and Exponential Forms
- Logarithmic to Exponential
- Exponential to Logarithmic
- Laws of Logarithm
- Product Law
- Quotient Law
- Power Law
- Expansion of Expressions with the Help of Laws of Logarithm
- More About Logarithm
Triangles [Congruency in Triangles]
- Concept of Triangles
- Relation Between Sides and Angles of Triangle
- Important Terms of Triangle
- Congruence of Triangles
- Criteria for Congruence of Triangles
Isosceles Triangles
- Classification of Triangles based on Sides- Equilateral, Isosceles, Scalene
- Isosceles Triangles Theorem
- Converse of Isosceles Triangle Theorem
Inequalities
- Inequalities in a Triangle
- If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.
- If Two Angles of a Triangle Are Unequal, the Greater Angle Has the Greater Side Opposite to It.
- Of All the Lines, that Can Be Drawn to a Given Straight Line from a Given Point Outside It, the Perpendicular is the Shortest.
Mid-point and Its Converse [ Including Intercept Theorem]
- Theorem of Midpoints of Two Sides of a Triangle
- Equal Intercept Theorem
Pythagoras Theorem [Proof and Simple Applications with Converse]
Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
- Introduction of Rectilinear Figures
- Names of Polygons
- Concept of Quadrilaterals
- Types of Quadrilaterals
- Diagonal Properties of Different Kinds of Parallelograms
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Property: The diagonals of a square are perpendicular bisectors of each other.
Construction of Polygons (Using Ruler and Compass Only)
- Constructing a Quadrilateral
- Construction of Parallelograms
- Construction of Trapezium
- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Construction of Square
- To Construct a Regular Hexagon
Area Theorems [Proof and Use]
- Concept of Area
- Figures Between the Same Parallels
- Triangles with the Same Vertex and Bases Along the Same Line
Circle
- Concept of Circle
- Arc, Segment, Sector
- Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord
- Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)
- Theorem: Equal chords of a circle are equidistant from the centre.
- Theorem : The Chords of a Circle Which Are Equidistant from the Centre Are Equal.
- Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line
- Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse
Statistics
- Concepts of Statistics
- Variable of Equation
- Tabulation of Data
- Frequency
- Frequency Distribution Table
- Frequency Distribution Table
- Class Intervals and Class Limits
- Cumulative Frequency Table
- Graphical Representation of Data
- Graphical Representation of Continuous Frequency Distribution
Mean and Median (For Ungrouped Data Only)
- Mean of Ungrouped Data
- Properties of Mean
- Concept of Median
Area and Perimeter of Plane Figures
Solids [Surface Area and Volume of 3-d Solids]
- Introduction of Solids
- Surface Area of a Cuboid
- Surface Area of a Cube
- Surface Area of Cylinder
- Cost of an Article
- Cross Section of Solid Shapes
- Flow of Water ( or any other liquid )
Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and Their Reciprocals]
- Concept of Perpendicular, Base, and Hypotenuse in a Right Triangle
- Notation of Angles
- Trigonometric Ratios and Its Reciprocal
- Reciprocal Relations
Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]
- Trigonometric Ratios of Some Special Angles
- Trigonometric Ratios of Some Special Angles
- Trigonometric Equation Problem and Solution
- Trigonometric Ratios of Some Special Angles
Solution of Right Triangles [Simple 2-d Problems Involving One Right-angled Triangle]
- Solution of Right Triangles
Complementary Angles
- Complementary Angles
- Trigonometric Ratios of Complementary Angles
- Complementary Angles for Sine ( Sin ) and Cosine ( Cos )
- Complimentary Angles for Tangent ( Tan ) and Contangency ( Cot )
- Complimentary Angles for Secant ( Sec ) and Cosecant ( Cosec )
Co-ordinate Geometry
- Coordinate Geometry
- Dependent and Independent Variables
- Ordered Pair
- Cartesian Coordinate System
- Co-ordinates of Points
- Quadrants and Sign Convention
- Plotting of Points
- Graph
- Graphs of Linear Equations
- Inclination and Slope
- Y-intercept
- Finding the Slope and the Y-intercept of a Given Line
Graphical Solution (Solution of Simultaneous Linear Equations, Graphically)
Distance Formula
- Distance Formula
- Distance Formula
- Circumcentre of a Triangle
Profit , Loss and Discount
- Concept of Discount
- To Find C.P., When S.P. and Gain (Or Loss) Percent Are Given
- To Find S.P., When C.P. and Gain (Or Loss) Percent Are Given
- Profit or Loss as a Percentage
- Concept of Discount
- Overhead Expenses
Construction of Triangles
- Construction of Triangles
- Construct Isosceles Triangle
Changing the Subject of a Formula
- Changing the Subject of a Formula
Similarity
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
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