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1.Number System
- Numbers and its Classification, Rational Number and its Representation on the Number Line
- Decimal Representation of Rational Number and Conversion of Decimal Number into Rational Number
- Representation of Decimal Number on Number Line, Finding Irrational Numbers Between Given Two Numbers
- Representing Irrational Number on the Number Line
- Rationalisation
- Introduction to Exponents, Laws of Exponents, BODMAS Rule
- Integral Exponents of Real Numbers
- Integral Exponents of Real Numbers and Solving Equation for x
- Rational Exponents of Real Numbers
- Examples Based on Rational Exponents of Real Numbers
- Chapter Notes – Number System
- NCERT Solutions – Number System Exercise 1.1 – 1.6
- R D Sharma Solutions Number System
- Revision Notes – Number System
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2.Polynomials
- Introduction to Polynomials and Some Basics Questions
- Value of Polynomial, Zeroes of Polynomial and Relationship between Zeroes and Coefficient of Linear Equation
- Division of Polynomials and Division Algorithm and Remainder Theorem
- Factorising by Spliting Method
- Factorisation By Different Methods and Factor Theorem
- Factorisation of Polynomial by Using Algebric Identities
- Chapter Notes – Polynomials
- NCERT Solutions – Polynomials Exercise 2.1 – 2.5
- R S Aggarwal Solutions Polynomials
- Revision Notes Polynomials
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3.Coordinate Geometry
- Representation of Position on Number Line, Finding the location by Using the Axis
- Quadrant, Coordinates of Point on Axis, Distance from Axis
- Finding the Area of Triangle and Rectangle and Mirror Image of Point On Axis
- Chapter Notes – Coordinate Geometry
- NCERT Solutions – Coordinate Geometry Exercise 3.1 – 3.3
- R D Sharma Solutions Coordinate Geometry
- R S Aggarwal Solutions Coordinate Geometry
- Revision Notes Coordinate Geometry
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4.Linear Equations
- Introduction of Linear Equation in Two Variables and Graphical Representation
- Example to find the value of Constant and Graph of Linear Equation in Two Variables
- Equations of lines Parallel to the x-axis and y-axis
- Chapter Notes – Linear Equations
- NCERT Solutions – Linear Equations Exercise 4.1 – 4.4
- R D Sharma Solutions Linear Equations
- R S Aggarwal Solutions Linear Equations
- Revision Notes Linear Equations
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5.Euclid's Geometry
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6.Lines and Angles
- Introduction to Lines and Angles, Types of Angle and Axioms
- Vertically opposite Angles and Angle bisector
- Angle made by a transversal with two Lines and Parallel Lines
- Lines Parallel to same Line; Exterior angle of a triangle; Angle Sum property of a Triangle
- Angle Sum Property of Triangle Including Interior and Exterior Angles
- Chapter Notes – Lines and Angles
- NCERT Solutions – Lines and Angles Exercise 6.1 – 6.3
- R D Sharma Solutions Lines and Angles
- R S Aggarwal Solutions Lines and Angles
- Revision Notes Lines and Angles
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7.Triangles
- Introduction, Types of Triangle, Altitude & Median of Triangle, Congruences, SAS criterion of Congruence
- Questions Based on SAS Criterion
- ASA Criterion of Congruence and Its based Questions
- Questions Based on AAS and SSS Criterion
- RHS Criterion of Congruence and Its based Questions, Questions Based on Multiple Criterion
- Inequalities of Triangle
- Inequalities of Triangle cont., Distance Between a line and The Point, Miscellaneous Questions
- Chapter Notes – Triangles
- NCERT Solutions – Triangles Exercise 7.1 – 7.5
- R D Sharma Solutions Triangles
- Revision Notes Triangles
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8.Quadrilaterals
- Congruence of Triangles- SAS Congruence Criteria, ASA Congruence Criteria, SSS Congruence Criteria, RHS Congruence Criteria; Types of Quadrilateral- Parallelogram, Rectangle, Rhombus, Square, Trapezium, Kite
- Theorem 1; Properties of parallelogram- Theorem 2, Theorem 3;
- Theorem 4; Theorem 5;
- Conditions for a Quadrilateral to be a parallelogram- Theorem 6
- Theorem 8
- Theorem 9
- Theorem 9(Mid-point Theorem);
- Theorem 10
- Chapter Notes – Quadrilaterals
- NCERT Solutions – Quadrilaterals Exercise 8.1 8.2
- R D Sharma Solutions Quadrilaterals
- R S Aggarwal Solutions Quadrilaterals
- Revision Notes Quadrilaterals
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9.Area of Parallelogram
- Parallelograms on the Same Base and between the Same Parallels; A Diagonal of a parallelogram divides it into two triangles of Equal Area
- Questions based on parallelogram and Triangle are on same base & parallel lines and Area of Triangle
- Triangles on the same Base and between the same Parallels
- Miscellaneous Questions Based on Triangles on the same Base and between the same Parallels
- Sums Based on Median of a Triangle
- Two Triangles on the same Base and equal Area lie b/w the same parallel
- Summary of All Concepts and Miscellaneous Questions
- Chapter Notes – Area of Parallelogram
- NCERT Solutions – Area of Parallelogram Exercise 9.1 – 9.4
- R D Sharma Solutions Area of Parallelogram
- Revision Notes Area of Parallelogram
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10.Constructions
- Construction of Perpendicular Bisector of a Line Segment, Different Angles and Angle bisector
- Construction of Different types of Triangle
- Chapter Notes – Constructions
- NCERT Solutions – Constructions Exercise
- R D Sharma Solutions Constructions
- R S Aggarwal Solutions Constructions
- Revision Notes Constructions
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11.Circles
- Introduction, Important Terms related to circle, Theorem-1, 2, 3 and 4
- Problems Solving
- Problems Based on Parallel Chords, Common Chord of Circles
- Problems Based on Concentric Circles, Triangle in Circle
- Equal Chords and Their Distance, Theorem-5, 6, 7, 8
- Circle Through Points, Theorem-9, Arc of a Circle, Theorem-10
- Chapter Notes – Circles
- NCERT Solutions – Circles Exercise
- R D Sharma Solutions Circles
- R S Aggarwal Solutions Circles
- Revision Notes Circles
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12.Heron's Formula
- Introduction of Heron’s Formula; Area of Different Triangles
- Area of Triangles Using Heron’s Formula (Type-1 level-1 and 2)
- Area of Quadrilaterals Using Heron’s Formula (Type-2 level-1)
- Area of Quadrilaterals Using Heron’s Formula (Type-2 level-2)
- Chapter Notes – Heron’s Formula
- NCERT Solutions – Heron’s Formula Exercise 12.1 12.2
- R D Sharma Solutions Heron’s Formula
- Revision Notes Heron’s Formula
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13.Surface Area and Volume
- Introduction, Surface Area of Cube, Cuboid, Cylinder, Hollow cylinder, Cone, Sphere
- Type-1: Surface Area and Volume of Cube and Cuboid
- Level-2, Question based on Cube and Cuboid, Level-1 Questions based on Cylinder
- Level-2 Questions based on Cylinder, Type-3 Surface Area and Volume of Right Circular Cone (Level-1)
- Level-2 Questions based on Right Circular Cone, Level-1& 2 Questions Based on Sphere & Hemisphere
- Type-5 Short Answer Types Questions,
- Chapter Notes – Surface Area and Volume of a Cuboid and Cube
- Chapter Notes – Surface Area and Volume of A Right Circular Cylinder
- Chapter Notes – Surface Area ad Volume of A Right Circular Cone
- Chapter Notes – Surface Area ad Volume of A Sphere
- NCERT Solutions – Surface Area and Volume Exercise 13.1 – 13.9
- R D Sharma Solutions Surface Areas And Volumes of A cuboid and A cube
- R D Sharma Solutions Surface Areas And Volumes of A Right Circular Cylinder
- R D Sharma Solutions Surface Areas And Volumes of A Right Circular Cone
- R D Sharma Solutions Surface Areas And Volumes of A Sphere
- Revision Notes Surface Areas And Volumes
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14.Statistics
- Data; Presentation of Data; Understanding Basic terms;
- Questions;
- Bar Graph, Histogram, Frequency Polygon;
- Histogram for Grouped Frequency Distribution (Class interval in Inclusive form), Frequency Polygon, Measures of Central Tendency,
- Examples Based on Mean of Ungrouped Data
- Direct Method of Finding Mean of Ungrouped Data, Median of Cont. Freq. Distribution (Grouped Data)
- Mean; Mean for an ungrouped freq. Dist. Table
- Chapter Notes – Statistics – Tabular Representation of Statistical Data
- Chapter Notes – Statistics – Graphical Representation of Statistical Data
- Chapter Notes – Statistics – Measures of Central Tendency
- NCERT Solutions – Statistics Exercise 14.1 – 14.4
- R D Sharma Solutions Tabular Representation of Statistical Data
- R D Sharma Solutions Graphical Representation of Statistical Data
- R S Aggarwal Solutions Statistics
- Revision Notes Statistics
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15.Probability
- Activities; Understanding term Probability; Experiments;
- Event; Probability; Empirical Probability- Tossing a coin;
- Rolling a dice; Miscellaneous Example;
- Chapter Notes – Probability
- NCERT Solutions – Probability Exercise 15.1
- R D Sharma Solutions Probability
- R S Aggarwal Solutions Probability
- Revision Notes Probability
Chapter Notes – Statistics – Graphical Representation of Statistical Data
(1) Following are the methods of graphical representation of data:
(i) Bar graphs:
For Example: The population of Delhi state in different census years is as given below:
Represent the above information with the help of a bar graph.
Solution: To represent the given data by a pictograph follow the below steps:
Step 1: Draw a horizontal and vertical line.
Step 2: Mark 5 point on horizontal line at equal distance for census year.
Step 3: Erect rectangles of the same width at marked points.
Step 4: The height of the rectangles are proportional to the numerical values of the given population.
(ii) Histogram:
For Example: Construct a histogram for the following data:
Solution: To represent the given data by a histogram follow the below steps:
Step 1: Draw a horizontal and vertical line.
Step 2: Taking class interval on horizontal line and corresponding frequencies on vertical line.
Step 3: Construct rectangles to obtain the histogram of given frequency distribution as shown in fig.
(iii) Frequency polygon:
For Example: The following table gives the distribution of IQ’s (intelligence quotients) of 60 pupils of class V in a school:
Draw a frequency polygon for the above data.
Solution: To represent the given data by frequency polygon follow the below steps:
Step 1: We need to find class mark.
We know, Class Mark = (Upper limit + Lower limit)/2.
| Expenditure | Class Mark | No. of workers |
| 62.5 – 69.5 | 66 | 1 |
| 69.5 – 76.5 | 73 | 3 |
| 76.5 – 83.5 | 80 | 5 |
| 83.5 – 90.5 | 87 | 15 |
| 90.5 – 97.5 | 94 | 12 |
| 97.5 – 104.5 | 101 | 10 |
| 104.5 – 111.5 | 108 | 6 |
| 111.5 – 118.5 | 115 | 4 |
| 118.5 – 125.5 | 122 | 3 |
| 125.5 – 132.5 | 129 | 1 |
Step 2: Draw a horizontal and vertical line.
Step 3: Taking class interval on horizontal line and corresponding frequencies on vertical line.
Step 4: Plot (66, 1), (73, 3), (80, 5), (87, 15), (94, 12), (101, 10), (108, 6), (115, 4), (122, 3), (129, 1)
Step 5: Join the plotted points.
Step 6: The end point (66, 1) and (129, 1) are joined to the midpoint (59, 0) and (136, 0) respectively of imagined class intervals to obtain the frequency polygon.
(2) A bar graph is a pictorial representation of the numerical data by a number of bars (rectangles) of uniform width erected horizontally or vertically with equal spacing between them. Each rectangle or bar represents only one value of the numerical data and so there are as many bars as the number of values in the numerical data. The height or length of a bar indicates on a suitable scale the corresponding value of the numerical data.
For Example: The following table shows the daily production of T.V. sets in an industry for 7 days of a week:
Represent the above information by a pictograph.
Solution: To represent the given data by a pictograph follow the below steps:
Step 1: Draw a horizontal and vertical line.
Step 2: Mark 7 days on horizontal line at equal distance.
Step 3: Erect rectangles of the same width at marked points.
Step 4: The height of the rectangles are proportional to the numerical values of the given data of no of TV sets.
(3) A histogram or frequency histogram is a graphical representation of a frequency distribution in the form of rectangles with class intervals as bases and heights proportional to corresponding frequencies such that there is no gap between any two successive rectangles.
For Example: Construct a histogram for the following data:
Solution: To represent the given data by a histogram follow the below steps:
Step 1: Draw a horizontal and vertical line.
Step 2: Taking class interval on horizontal line and corresponding frequencies on vertical line.
Step 3: Construct rectangles to obtain the histogram of given frequency distribution as shown in fig.
(4)A frequency polygon of a given frequency distribution is another method of representing frequency distribution graphically.
For Example: Draw, in the same diagram, a histogram and a frequency polygon to represent the following data which shows the monthly cost of living index of a city in a period of 2 years:
Solution: To represent the given data by a histogram follow the below steps:
Step 1: Draw a horizontal and vertical line.
Step 2: Taking class interval on horizontal line and corresponding frequencies on vertical line.
Step 3: Construct rectangles to obtain the histogram of given frequency distribution as shown in fig.
Step 4: Obtain the midpoints of the upper horizontal side of each rectangle.
Step 5: Join these midpoints of the adjacent rectangles of the histogram by line segment.
Step 6: Obtain the midpoints of two class intervals of zero frequency i.e. on X axis, one adjacent to the first, on its left and one adjacent to the last, on its right.
These class intervals are known as imagined class intervals.
Step 7: Complete the polygon by joining the midpoints of first and last intervals to the midpoints of imagined class interval adjacent to them.