Academic year:

## Units and Topics

## Syllabus

1 Rational and Irrational Numbers

2 Compound Interest [Without Using Formula]

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

- Concept of Compound Interest

3 Compound Interest [Using Formula]

4 Expansions

- Algebraic Identities
( a + b )

^{2}= a^{2}+ 2ab + b^{2}. - Expansion of (a + b)3
- 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
- Methods of Solving Simultaneous Linear Equations by Cross Multiplication Method

5 Factorisation

6 Simultaneous (Linear) Equations [Including Problems]

7 Indices [Exponents]

- 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

8 Logarithms

- 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

9 Triangles [Congruency in Triangles]

- 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.

- Congruence of Triangles
- Criteria for Congruence of Triangles

10 Isosceles Triangles

11 Inequalities

- 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.

12 Mid-point and Its Converse [ Including Intercept Theorem]

- The Mid-point Theorem
- 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.

13 Pythagoras Theorem [Proof and Simple Applications with Converse]

- 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

14 Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

- 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

15 Construction of Polygons (Using Ruler and Compass Only)

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

- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Construction of Square
- 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°`

16 Area Theorems [Proof and Use]

- Concept of Area
- 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

17 Circle

- 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.
- Converse: 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

18 Statistics

- 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

19 Mean and Median (For Ungrouped Data Only)

20 Area and Perimeter of Plane Figures

21 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
- Right Circular Cylinder
- Hollow Cylinder

- Cost of an Article
- Cost = Rate x Quantity

- Cross Section of Solid Shapes
- 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.

22 Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and Their Reciprocals]

23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

24 Solution of Right Triangles [Simple 2-d Problems Involving One Right-angled Triangle]

25 Complementary Angles

26 Co-ordinate Geometry

27 Graphical Solution (Solution of Simultaneous Linear Equations, Graphically)

28 Distance Formula

29 Profit , Loss and Discount

30 Construction of Triangles

31 Changing the Subject of a Formula

32 Similarity

- Similarity
- Similar triangles
- Criteria of Similarity

- AA Criterion of similarity

- SAS Criterion of similarity

- SSS Criterion of similarity - Construction of similar triangles

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