#### 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]

- Concept of Principal, Interest, Amount, and Simple 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 Linear Equations
- Methods of Solving Simultaneous Linear Equations by Cross Multiplication Method
- Simple Linear Equations in One Variable
- Linear Equation 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 - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Relation Between Sides and Angles of Triangle
- Important Terms of Triangle
- Congruence of Triangles
- Criteria for Congruence of Triangles

##### Isosceles Triangles

- Isosceles Triangles
- 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 - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- 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 - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- 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
- 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)

- Graph of a Linear Equation in Two Variables
- Methods of Solving Simultaneous Linear Equations by Graphical Method

##### 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) = x
^{2}+ (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:** 2x^{2} - 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.

2x^{2} - 9x + 9

= 2x^{2} - 6x - 3x + 9

= 2x(x - 3) - 3(x - 3)

= (x - 3)(2x - 3)

∴ 2x^{2} - 9x + 9 = (x - 3)(2x - 3)

## Example

**Factorise:** 2x^{2} + 5x - 18

2x^{2} + 5x - 18

= 2x^{2} + 9x - 4x - 18

= x(2x + 9) - 2(2x + 9)

= (2x + 9)(x - 2)

## Example

**Factorise:** x^{2} - 10x + 21.

x^{2} - 10x + 21

= x^{2} - 10x + 21

= x^{2} - 7x - 3x + 21

= x(x - 7) - 3(x - 7)

= (x - 7)(x - 3)

## Example

Find the factors of 2y^{2} - 4y - 30.

2y^{2} - 4y - 30

= 2(y^{2} - 2y - 15) .....(taking out the common factor 2)

= 2(y^{2} - 5y + 3y - 15)

= 2[y(y - 5) + 3(y - 5)]

= 2(y - 5)(y + 3).