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
Definition
- Quadratic Trinomial: An expression of the form `ax^2 + bx + c` is called a quadratic trinomial.
Formula
- (x + a)(x + b) = x2 + (a + b)x + ab
Notes
Factorisation of a Quadratic Trinomial:
An expression of the form `ax^2 + bx + c` is called a quadratic trinomial.
We know that `(x + a)(x + b) = x^2 + (a + b)x + ab.`
∴ the factors of `x^2 + (a + b)x + ab "are" (x + a) and (x + b).`
To find the factors of `x^2 + 5x + 6, "by comparing it with" x^2 + (a + b)x + ab.`
we get, a + b = 5 and ab = 6. So, let us find the factors of 6 whose sum is 5.
Then writing the trinomial in the form `x^2 + (a + b)x + ab`, find its factors.
`x^2 + 5x + 6 = x^2 + (3 + 2)x + 3 xx 2 ......[x^2 + (a + b)x + ab]`
`= x^2 + 3x + 2x + 2 xx 3` .......[Multiply (3 + 2) by x, make two groups of the four terms obtained.]
`= x(x + 3) + 2(x + 3)`
`= (x + 3)(x + 2)`
Factorise: 2x2 - 9x + 9.
First, we find the product of the coefficient of the square term and the constant term. Here the product is 2 × 9 = 18.
Now, find factors of 18 whose sum is -9, which is equal to the coefficient of the middle term.
2x2 - 9x + 9
= 2x2 - 6x - 3x + 9
= 2x(x - 3) - 3(x - 3)
= (x - 3)(2x - 3)
∴ 2x2 - 9x + 9 = (x - 3)(2x - 3)
Example
Factorise: 2x2 + 5x - 18
2x2 + 5x - 18
= 2x2 + 9x - 4x - 18
= x(2x + 9) - 2(2x + 9)
= (2x + 9)(x - 2)
Example
Factorise: x2 - 10x + 21.
x2 - 10x + 21
= x2 - 10x + 21
= x2 - 7x - 3x + 21
= x(x - 7) - 3(x - 7)
= (x - 7)(x - 3)
Example
Find the factors of 2y2 - 4y - 30.
2y2 - 4y - 30
= 2(y2 - 2y - 15) .....(taking out the common factor 2)
= 2(y2 - 5y + 3y - 15)
= 2[y(y - 5) + 3(y - 5)]
= 2(y - 5)(y + 3).
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