## Units and Topics

## Syllabus

- Concept of Rational Numbers
- Properties of Rational Numbers
- Decimal Representation of Rational Numbers
- Concept of Irrational Numbers
**Decimal Representation of Rational Numbers in Terms of Termination or Non-terminating Recurring Decimals.** - Concept of Real Numbers
- Concept of Surds
- Rationalization of Surd of order 2
- Simplifying an expression by rationalization of the Denominator
- Absolute value

- Concept of Principal, Interest, Amount, and Simple Interest
- Simple Interest for one year
- Simple Interest for multiple years

- Concept of Compound Interest

- Algebraic Identities
( a + b )

^{2}= a^{2}+ 2ab + b^{2}. - Expansion of (A + B)3
- ( a + b ) ( a - b ) = a
^{2}- b^{2} - ( a + b )
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - ( a - b
^{)3}= a^{3}- 3a^{2}b + 3ab^{2}- b^{3} - a
^{3}+ b^{3}= ( a + b )^{3}- 3ab ( a + b ) - a
^{3}- b^{3}= ( a - b )^{3}+ 3ab ( a - b ) - ( a + `1/a` )
^{3}= a^{3}+ `1/(a^3)` + 3 ( a + `1/a` ) - ( a - `1/a` )
^{3}= a^{3}- `1/(a^3)` - 3 ( a - `1/a` ) - a
^{3}+ `1/(a^3)` = ( a + `1/a` )^{3}- 3 ( a + `1/a` ) - a
^{3}- `1/(a^3)` = ( a - `1/a` )^{3}- 3 ( a - `1/a` ) - a
^{3}+ b^{3}+ c^{3}= 3abc

- ( a + b ) ( a - b ) = a
- Expansion of Formula
**1. Expansion of ( x + a ) ( x + b ) :**- ( x + a ) ( x + b ) = x
^{2}+ ( a + b ) x + ab - ( x + a ) ( x - b ) = x
^{2}+ ( a - b ) x - ab - ( x - a ) ( x + b ) = x
^{2}- ( a - b ) x - ab - ( x - a ) ( x - b ) = x
^{2 }- ( a + b ) x + ab

**2. Expansion of ( a + b + c )**:^{2}- ( a + b + c )
^{2}= a^{2}+ b^{2}+ c^{2}+ 2 ( ab + bc + ca ) - ( a + b - c )
^{2}= a^{2}+ b^{2}+ c^{2}+ 2 ( ab - bc - ca ) - ( a - b + c )
^{2}= a^{2}+ b^{2}+ c^{2}- 2 ( ab + bc - ca ) - ( a - b - c )
^{2}= a^{2}+ b^{2}+ c^{2}- 2 ( ab - bc + ca )

- ( x + a ) ( x + b ) = x
- Special Product
- ( x + a ) ( x + b ) ( x + c ) = x
^{3}+ ( a + b + c ) x^{3}+ ( ab + bc + ca ) x + abc - ( a + b ) ( a
^{2}- ab + b^{2}) = a^{3}+ b^{3} - ( a - b ) ( a
^{2}+ ab + b^{2}) = a^{3}- b^{3} - ( a + b + c ) ( a
^{2}+ b2 + c^{2}- ab - bc - ca ) = a^{3}+ b^{3}+ c^{3}- 3abc

- ( x + a ) ( x + b ) ( x + c ) = x
- Elimination Method of Solving Simultaneous Equations

- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Method of Factorisation : Trinomial of the Form
Trinomial of the form ax

^{2}+ bx + c ( By splitting the middle term ) - Method of Factorisation : Difference of Two Squares
- Method of Factorisation : the Sum Or Difference of Two Cubes

- Elimination Method of Solving Simultaneous Equations
- Equations Reducible to Linear Equations
- Simultaneous Equations
Based on Numbers, Fractions, Two-digit numbers, ages, C.P. and S.P., Time and work, Miscellaneous Problem.

- Simple Linear Equations in One Variable
- Linear Equations in Two Variables

- Laws of Exponents
- Handling Positive, Fraction, Negative and Zero Indices
- ( a x b )
^{m}= a^{m}x b^{m}and `(a/b)^m = a^m/b^m` - If a ≠ 0 and n is a positive integer, then `nsqrta` = a
^{1/n} - `a^(m/n) = nsqrt (a^m) ; Where a ≠ 0. `
- For any non - zero number a,

`a^n = 1/( a^-n ) and a^(-n) = 1/(a^n)` - Any non - zero number raised to the power zero is always equal to unity ( i.e., 1)

- ( a x b )
- Simplification of Expressions
- Solving Exponential Equations

- Introduction of Logarithms
- Interchanging Logarithmic and Exponential Forms
- Laws of Logarithm
- Product Law
`log_a (mxxn) = log_a (m) + log_a (n)`

- Quotient Law
`log_a (m/n) = log_a (m) - log_a (n)`

- Power Law
`log_a (m)^n = nlog_a (m)`

- Product Law
- Expansion of Expressions with the Help of Laws of Logarithm
- More About Logarithm

- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Relation Between Sides and Angles of Triangle
- If all the sides of a triangle are of different lengths, its angles are also of different measures in such a way that, the greater side has greater angle opposite to it.
- If all the angles of a triangle have different measures, its sides are also of different lengths in such a way that, the greater angle has greater side opposite to it.
- If any two sides of a triangle are equal, the angles opposite to them are also equal. Conversely, if any two angles of a triangle are equal, the sides opposite to them are also equal.
- If all the sides of a triangle are equal, all its angles are also equal. Conversely, if all the angles of a triangle are equal, all its sides are also equal.

- If all the sides of a triangle are of different lengths, its angles are also of different measures in such a way that, the greater side has greater angle opposite to it.
- Important Terms of Triangle
**Median :**The median of a triangle, corresponding to any side, is the line joining the mid-point of that side with the opposite vertex.**Centroid :**The point of intersection of the medians is called the centroid of the triangle.**Altitude :**An altitude of a triangle, corresponding to any side, is the length of the perpendicular drawn from the opposite vertex to that side.**Orthocentre :**The point of intersection of the altitudes of a triangle is called the orthocentre.**Corollary 1 :**If one side of a triangle is produced, the exterior angle so formed is greater than each of the interior opposite angles.**Corollary 2 :**A triangle cannot have more than one right angle.**Corollary 3 :**A triangle cannot have more than one obtuse angle.**Corollary 4 :**In a right angled triangle, the sum of the other two angles ( acute angles ) is 90°.**Corollary 5 :**In every triangle, at least two angles are acute.**Corollary 6 :**If two angles of a traingle are equal to two angles of any other triangle, each to each, then the third angles of both the triangles are also equal.

- Concept of Congruency of Triangles
- Two triangles are said to be congruent to each other, if on placing one over the other, they exactly coincide.
- In fact, Two triangle are congruent, If all the angles and all the sides of one triangle are equal to the corresponding angles and the corresponding sides of the other triangle each to each.
- The symbol ≅ is read as "is congruent to"
- In Congruent triangles, the sides and the angles that coincide by superposition are called corresponding sides and corresponding angles.
- Corresponding parts of Congruent Triangles are also Congruent.

Abbreviated as :**C.P.C.T.C.**

- Condition for Congruency of Triangle
- If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, the triangles are congruent.
**Abbreviated as :**S.A.S. - If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, the triangles are congruent.
**Abbreviated as :**A.S.A. - If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, the triangles are congruent.
**Abbreviated as :**A.A.S. - If three sides of one triangle are equal to three sides of the other triangle, each to each, the triangles are congruent.
**Abbreviated as :**S.S.S. - Two right-angled triangles are congruent, if the hypotenuse and one side of one triangle are equal to the hypotenuse and corresponding side of the other triangle.
**Abbreviated as :**R.H.S.

- If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, the triangles are congruent.

- Inequalities in a Triangle
- Of All the Lines, that Can Be Drawn to a Given Straight Line from a Given Point Outside It, the Perpendicular is the Shortest.
**Corollary 1 :**The sum of the lengths of any two sides of a triangle is always greater than the third side.**Corollary 2 :**The difference between the lengths of any two sides of a triangle is always less than the third side.

- Of All the Lines, that Can Be Drawn to a Given Straight Line from a Given Point Outside It, the Perpendicular is the Shortest.

- The Mid-point Theorem
**Theorem of midpoints of two sides of a triangle :**The line segment joining the mid-points of any two sides of a triangle is parallel to the third side, and is equal to half of it.**Converse of midpoint theroem :**The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.

- Equal Intercept Theorem
- If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

- Pythagoras Theorem
- Regular Polygon
- If all the sides and all the angles of a polygon are equal, it is called a regular polygon.
- Sum of interior angles of an 'n' sided polygon ( whether it is regular or not) = ( 2n - 4 )rt. angles and sum of its exterior angles = 4 right angles = 360°
- At each vertex of every polygon, Exterior angle + Interior angle = 180°.
- Each interior angle of a regular polygon = `[( 2n - 4 ) "rt. angles"]/[n] = [( 2n - 4 ) xx 90°]/n`
- Each exterior

- Introduction of Rectilinear Figures
- Rectilinear means along a straight line or in a straight line or forming a straight line.
- A plane figure bounded by straight lines is called a rectilinear figure.
- A closed plane figure, bounded by at least three line segments, is called a polygon.

- Names of Polygons
- A Polygon is named by the number of sides in it,

No. of sides 3 4 5 6 7 8 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon **Concave Polygon :**If at least one angle of a polygon is greater than 180°, the polygon is called a convex polygon.**Convex Polygon :**If each angle of a polygon is less than 180°, it is called a concave polygon.

- A Polygon is named by the number of sides in it,
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Types of Quadrilaterals
- Diagonal Properties of Different Kinds of Parallelograms
- Properties of a Parallelogram
- Properties of Rhombus
- Properties of Rectangle
- Properties of a Square

- Construction of an Angle
- Construction of Quadrilateral
- To construct a quadrilateral, whose four sides and one angle are given.
- To construct a quadrilateral, whose three sides and two consecutive angles are given.
- To construct a quadrilateral, whose four sides and one diagonal are given.
- To construct a quadrilateral, whose three sides and two diagonals are given.

- Construction of Rhombus
- To construct a rhombus, whose diagonal are given.

- Construction of Square
- To construct a square whose diagonal is given.

- Construction of Quadrilateral
- Construction of Parallelograms
- To construct a parallelogram, whose two consecutive sides and the included angle are given.
- To construct a parallelogram, whose one side and both the diagonals are given.
- To construct a parallelogram, whose two consecutive sides and one diagonal are given.
- To construct a parallelogram, whose two diagonal and included angle are given.
- To construct a parallelogram, whose two adjacent sides and height are given.

- Construction of Trapezium
- To construct a trapezium ABCD, whose four sides are given.

- Construction of Rectangles
- To construct a rectangle whose adjacent sides are given.
- To construct a rectangle, whose one side and one diagonal are given.

- To Construct a Regular Hexagon
**Method 1 :**Each interior angle of a regular hexagon is 120° and its opposite sides are parallel.The length of the side of a regular hexagon is equal to the radius of its circumcircle.

Method 2 :**Method 3 :**The angle subtended by each side of a regular hexagon at the centre of its circumcircle is `(360°)/6 = 60°`

- Introduction of Area Theorems
- Area of a triangle = `1/2` x base x height
- Area of a rectangle = length x breadth
- Area of a parallelogram = base x height

- Figures Between the Same Parallels
- Parallelograms on the same base and between the same parallels are equal in area.
**Corollary :**The area of a parallelogram is equal to the area of a rectangle on the same base and between the same parallels.- The area of a triangle is half that of a parallelogram on the same base and between the same parallels.
- Triangles on the same base and between the same parallels are equal in area.
**Corollaries :**Parallelograms on equal bases and between the same parallels are equal in area.

1.

2. Area of a triangle is half the area of the parallelogram if both are on equal bases and between the same parallels.

3. Two triangles are equal in area if they are on the equal bases and between the same parallels.

- Triangles with the Same Vertex and Bases Along the Same Line

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior.
- 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)
- Chord Properties - Equal Chords Are Equidistant from the Center
- Chord Properties - Chords Equidistant from the Center Are Equal (Without Proof)
- 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

- Concepts of Statistics
- Variable
- Continuous Variable
- Discrete Variable
- Raw and Arrayed Data

- Tabulation of Data
- Frequency
- Frequency Distribution Table
- Ungrouped Frequency Distribution Table:
- Grouped Frequency Distribution Table:

- Inclusive Frequency Distribution

- Exclusive Frequency Distribution

- Class Intervals and Class Limits
- Cumulative Frequency Table
- Less than Cumulative frequency less than the upper class limit
- Cumulative frequency more than or equal to the lower class limit

- Graphical Representation of Data
- Bar graph
- Pie graph
- Histogram

- Graphical Representation of Continuous Frequency Distribution
- Histogram
- Frequency Polygon

- Introduction of Area and Perimeter of Plane Figures
- Perimeter
- Area

- Perimeter of Triangles
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Area of a General Quadrilateral
- Some Special Types of Quadrilaterals
- Rectangle
- Square
- Rhombus
- Trapezium

- Circumference of a Circle
- Area of Circle

- Introduction of Solids
- Surface Area of a Cuboid
- Surface Area of a Cube
- Surface Area of Cylinder
- Right Circular Cylinder
- Hollow Cylinder

- Cost of an Article
- Cost = Rate x Quantity

- Cross Section
- Volume = Area of cross - section x length
- Surface area ( excluding cross - section ) = Perimeter of cross - section x length

- Flow of Water ( or any other liquid )
- The volume of water that flows in unit time = Area of cross-section x speed of flow of water.