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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
NCERT Solutions – Probability Exercise 15.1
Exercise 15.1
Q.1 In a cricket match, batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.
Sol.
Since in a cricket match ,a batswoman hits a boundary 6 times out of 30 balls she plays i.e., she missed the boundary 30 – 6 = 24 times out of 30 balls.
Therefore, the probability that a batswoman does not hit a boundary.
=NumberofnothittingaboundaryTotalnumberoftrials(i.e.,balls)
=2430=45
Q.2 1500 families with 2 children were selected randomly, and the following data were recorded :
Compute the probability of a family, chosen at random having
(i) 2 girls (ii) 1 girl (iii) No girl
Also check whether the sum of these probabilities is 1.
Sol.
Let E0,E1andE2 be the event of getting no girl, 1girl and 2 girls.
(i) Therefore P(E2) = Probability of a family having 2 girls
=Numberoffamilieshaving2girlsTotalnumberofgirls
=4751500=1960
(ii) Therefore P(E1) = Probability of a family having 1 girl
=Numberoffamilieshaving1girlsTotalnumberoffamilies
=8141500=407750
(iii) Therefore P(E0) = Probability of a family having no girls
=NumberoffamilieshavingnogirlsTotalnumberoffamilies
=2111500
Therefore Sum of probabilities = P(E0)+P(E1)+P(E2)
=2111500+407750+1960
=211+814+4751500=15001500=1
Q.3 Refer to Example 5, section 14.4 Chapter 14. Find the probability that a student of the class was born in August.
Sol.
Clearly from the histograph six students were born in the month of August out of 40 students of a particular section of class – IX.
Probability that a student of the class was born in August
=NumberofstudentsborninAugustTotalnumberofstudents
=640=320
Q.4 Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes :
If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.
Sol.
Since three coins are tossed 200 times, so the total number of trials is 200.
Probability of getting 2 heads
=No.ofoutcomeshaving2headsTotalno.oftrials
=72200
=925
Q.5 An organisation selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family.
The information gathered is listed in the table below :
Suppose a family is chosen. Find the probability that the family chosen is
(i) earning Rs 10000-13000 per month and owning exactly 2 vehicles.
(ii) earning Rs 16000 or more per month and owning exactly 1 vehicles.
(iii) earning less than Rs 7000 per month and does not own any vehicle.
(iv) earning Rs 13000 -16000 per month and owning more than 2 vehicles.
(v) owning not more than 1 vehicle
Sol.
The total number of families = 2400
(i) Number of families earning Rs 10000 – 13000 per month and owning exactly 2 vehicles = 29.
Therefore, P (families earning Rs 10000 – 13000 per month and owning exactly 2 vehicles)
=292400
(ii) Number of families earning Rs 16000 or more per month and owning exactly 1 vehicle = 579.
Therefore P (Families earning Rs 16000 or more per month and owning exactly 1 vehicle)
=5792400
=193800
(iii) Number of families earning less than Rs. 7000 per month and does not own any vehicle = 10
Therefore, P (Families earning less than Rs 7000 per month and does not own any vehicle)
=102400=1240
(iv) Number of families earning Rs 13000 – 16000 per month and owning more than 2 vehicles = 25
Therefore, P (Families earning Rs 13000 – 16000 per month and owning more than two vehicles)
=252400=196
(v) Number of families owning not more than 1 vehicle
= Families having no vehicle + Families having 1 vehicle
= (10 + 0 + 1 + 2 + 1) + (160 + 305 + 535 + 469 + 579)
= 14 + 2048 = 2062
P (Families owning not more than 1 vehicle)
=20622400=10311200
Q.6 Refer to Table 14.7 Chapter 14.
(i) Find the probability that a student obtained less than 20% in the mathematics test.
(ii) Find the probability that a student obtained marks 60 or above.
Sol.
Total number of students in mathematics is 90.
(i) Clearly, from the given table, the number of student who obtained less than 20% marks in the mathematics test = 7.
P (a student obtaining less than 20% marks) = 790
(ii) Clearly, from the given table, number of students who obtained marks 60 or above.
= (students in 60 – 70) + (students above 70)
= 15 + 8 = 23
Therefore, P (a student obtaining marks 60 and above) = 2390
Q.7 To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table.
Find the probability that a student chosen at random
(i) likes statistics, (ii) does not like it.
Sol.
The total number of students = 200
(i) P (a student likes statistics)
=NumberofstudentswholikestatisticsTotalnumberofstudents
=135200=2740
(ii) P (a student does not like statistics)
=NumberofstudentswhodoesnotlikestatisticsTotalnumberofstudents
=65200=1340
Q.8 Refer to Q.2, Ex 14.2. What is the empirical probability that an engineer lives :
(i) less than 7 km from her place of work?
(ii) more than or equal to 7 km from her place of work?
(iii) within 12km from her place of work?
Sol.
Total number of engineers = 40
(i) Number of engineers living less than 7 km from their place of work = 9
Therefore P (an engineer lives less than 7 km from her place of work)
=940
(ii) Number of engineers living more than or equal to 7km from her place of work = 31
Therefore P (an engineer lives less than or equal to 7 km from her place of work)
=3140
(iii) Number of engineer lives within 12 km from her place of work = 0
Therefore P (an engineer lives with 12 km from her place of work) = 040=0
Q.9 Activity : Note the frequency of two- wheeler, three – wheeler and four – wheeler going past during a time interval, in front of your school gate.
Find the probability that any one vehicle out of the total vehicles you have observed is a two wheeler.
Sol. Activity problem : Collect the data and find the desired probability.
Q.10 Activity : Ask all the students in your class to write a 3-digit number. Choose any student from the room at random. What is the probability that the number written by her/him is divisible by 3? Remember that a number is divisible by 3, if the sum of its digits is divisible by 3.
Sol. Activity problem : Do as directed and find the desired probability.
Q.11 Eleven bags of wheat flour, each marked 5 kg , actually contained the following weights of flour (in kg) :
4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00
Find the probability that any of these bags chosen at random contains more than 5kg of flour.
Sol.
Total number of wheat bags = 11
Number of bags having more than 5 kg = 7
Therefore P(a bag contains more than 5kg ) = 711
Q.12 In Q.5 Exercise 14.2 you were asked to prepare a frequency distribution table, regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days. Using this table, find the probability of the concentration of sulphur dioxide in the interval 0.12 – 0.16 on any of these days.
Sol.
Total number of days = 30
Concentration of sulphur dioxide in 0.12 – 0.16 on any day = 2
Therefore , required probability =230=115
Q.13 In Q. 1, Exercise 14.2, you were asked to prepare a frequency distribution table regarding the blood group of 30 students of a class. Use this table to determine the probability that a student of this class, selected at random, has blood group AB.
Sol.
Total number of students = 30
Number of students having blood group AB = 3
Therefore Required probability =330=110