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      Class 9 Maths

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      • Class 09
      • Class 9 Maths
      CoursesClass 09MathsClass 9 Maths
      • 1.Number System
        14
        • Lecture1.1
          Numbers and its Classification, Rational Number and its Representation on the Number Line 50 min
        • Lecture1.2
          Decimal Representation of Rational Number and Conversion of Decimal Number into Rational Number 01 hour
        • Lecture1.3
          Representation of Decimal Number on Number Line, Finding Irrational Numbers Between Given Two Numbers 46 min
        • Lecture1.4
          Representing Irrational Number on the Number Line 41 min
        • Lecture1.5
          Rationalisation 54 min
        • Lecture1.6
          Introduction to Exponents, Laws of Exponents, BODMAS Rule 41 min
        • Lecture1.7
          Integral Exponents of Real Numbers 57 min
        • Lecture1.8
          Integral Exponents of Real Numbers and Solving Equation for x 01 hour
        • Lecture1.9
          Rational Exponents of Real Numbers 52 min
        • Lecture1.10
          Examples Based on Rational Exponents of Real Numbers 30 min
        • Lecture1.11
          Chapter Notes – Number System
        • Lecture1.12
          NCERT Solutions – Number System Exercise 1.1 – 1.6
        • Lecture1.13
          R D Sharma Solutions Number System
        • Lecture1.14
          Revision Notes – Number System
      • 2.Polynomials
        10
        • Lecture2.1
          Introduction to Polynomials and Some Basics Questions 58 min
        • Lecture2.2
          Value of Polynomial, Zeroes of Polynomial and Relationship between Zeroes and Coefficient of Linear Equation 01 hour
        • Lecture2.3
          Division of Polynomials and Division Algorithm and Remainder Theorem 51 min
        • Lecture2.4
          Factorising by Spliting Method 01 hour
        • Lecture2.5
          Factorisation By Different Methods and Factor Theorem 27 min
        • Lecture2.6
          Factorisation of Polynomial by Using Algebric Identities 01 hour
        • Lecture2.7
          Chapter Notes – Polynomials
        • Lecture2.8
          NCERT Solutions – Polynomials Exercise 2.1 – 2.5
        • Lecture2.9
          R S Aggarwal Solutions Polynomials
        • Lecture2.10
          Revision Notes Polynomials
      • 3.Coordinate Geometry
        8
        • Lecture3.1
          Representation of Position on Number Line, Finding the location by Using the Axis 01 hour
        • Lecture3.2
          Quadrant, Coordinates of Point on Axis, Distance from Axis 01 hour
        • Lecture3.3
          Finding the Area of Triangle and Rectangle and Mirror Image of Point On Axis 57 min
        • Lecture3.4
          Chapter Notes – Coordinate Geometry
        • Lecture3.5
          NCERT Solutions – Coordinate Geometry Exercise 3.1 – 3.3
        • Lecture3.6
          R D Sharma Solutions Coordinate Geometry
        • Lecture3.7
          R S Aggarwal Solutions Coordinate Geometry
        • Lecture3.8
          Revision Notes Coordinate Geometry
      • 4.Linear Equations
        8
        • Lecture4.1
          Introduction of Linear Equation in Two Variables and Graphical Representation 01 hour
        • Lecture4.2
          Example to find the value of Constant and Graph of Linear Equation in Two Variables 57 min
        • Lecture4.3
          Equations of lines Parallel to the x-axis and y-axis 46 min
        • Lecture4.4
          Chapter Notes – Linear Equations
        • Lecture4.5
          NCERT Solutions – Linear Equations Exercise 4.1 – 4.4
        • Lecture4.6
          R D Sharma Solutions Linear Equations
        • Lecture4.7
          R S Aggarwal Solutions Linear Equations
        • Lecture4.8
          Revision Notes Linear Equations
      • 5.Euclid's Geometry
        7
        • Lecture5.1
          Euclid’s Definitions and Euclid’s Axioms 32 min
        • Lecture5.2
          Eulcid’s Postulates and Playfair’s Axiom 37 min
        • Lecture5.3
          Chapter Notes – Euclid’s Geometry
        • Lecture5.4
          NCERT Solutions – Euclid’s Geometry Exercise 5.1 5.2
        • Lecture5.5
          R D Sharma Solutions Euclid’s Geometry
        • Lecture5.6
          R S Aggarwal Solutions Euclid’s Geometry
        • Lecture5.7
          Revision Notes Euclid’s Geometry
      • 6.Lines and Angles
        10
        • Lecture6.1
          Introduction to Lines and Angles, Types of Angle and Axioms 01 hour
        • Lecture6.2
          Vertically opposite Angles and Angle bisector 46 min
        • Lecture6.3
          Angle made by a transversal with two Lines and Parallel Lines 01 hour
        • Lecture6.4
          Lines Parallel to same Line; Exterior angle of a triangle; Angle Sum property of a Triangle 52 min
        • Lecture6.5
          Angle Sum Property of Triangle Including Interior and Exterior Angles 01 hour
        • Lecture6.6
          Chapter Notes – Lines and Angles
        • Lecture6.7
          NCERT Solutions – Lines and Angles Exercise 6.1 – 6.3
        • Lecture6.8
          R D Sharma Solutions Lines and Angles
        • Lecture6.9
          R S Aggarwal Solutions Lines and Angles
        • Lecture6.10
          Revision Notes Lines and Angles
      • 7.Triangles
        11
        • Lecture7.1
          Introduction, Types of Triangle, Altitude & Median of Triangle, Congruences, SAS criterion of Congruence 42 min
        • Lecture7.2
          Questions Based on SAS Criterion 01 hour
        • Lecture7.3
          ASA Criterion of Congruence and Its based Questions 42 min
        • Lecture7.4
          Questions Based on AAS and SSS Criterion 29 min
        • Lecture7.5
          RHS Criterion of Congruence and Its based Questions, Questions Based on Multiple Criterion 01 hour
        • Lecture7.6
          Inequalities of Triangle 42 min
        • Lecture7.7
          Inequalities of Triangle cont., Distance Between a line and The Point, Miscellaneous Questions 48 min
        • Lecture7.8
          Chapter Notes – Triangles
        • Lecture7.9
          NCERT Solutions – Triangles Exercise 7.1 – 7.5
        • Lecture7.10
          R D Sharma Solutions Triangles
        • Lecture7.11
          Revision Notes Triangles
      • 8.Quadrilaterals
        13
        • Lecture8.1
          Congruence of Triangles- SAS Congruence Criteria, ASA Congruence Criteria, SSS Congruence Criteria, RHS Congruence Criteria; Types of Quadrilateral- Parallelogram, Rectangle, Rhombus, Square, Trapezium, Kite 01 hour
        • Lecture8.2
          Theorem 1; Properties of parallelogram- Theorem 2, Theorem 3; 01 hour
        • Lecture8.3
          Theorem 4; Theorem 5; 24 min
        • Lecture8.4
          Conditions for a Quadrilateral to be a parallelogram- Theorem 6 01 hour
        • Lecture8.5
          Theorem 8 01 hour
        • Lecture8.6
          Theorem 9 51 min
        • Lecture8.7
          Theorem 9(Mid-point Theorem); 01 hour
        • Lecture8.8
          Theorem 10 27 min
        • Lecture8.9
          Chapter Notes – Quadrilaterals
        • Lecture8.10
          NCERT Solutions – Quadrilaterals Exercise 8.1 8.2
        • Lecture8.11
          R D Sharma Solutions Quadrilaterals
        • Lecture8.12
          R S Aggarwal Solutions Quadrilaterals
        • Lecture8.13
          Revision Notes Quadrilaterals
      • 9.Area of Parallelogram
        11
        • Lecture9.1
          Parallelograms on the Same Base and between the Same Parallels; A Diagonal of a parallelogram divides it into two triangles of Equal Area 57 min
        • Lecture9.2
          Questions based on parallelogram and Triangle are on same base & parallel lines and Area of Triangle 01 hour
        • Lecture9.3
          Triangles on the same Base and between the same Parallels 56 min
        • Lecture9.4
          Miscellaneous Questions Based on Triangles on the same Base and between the same Parallels 54 min
        • Lecture9.5
          Sums Based on Median of a Triangle 52 min
        • Lecture9.6
          Two Triangles on the same Base and equal Area lie b/w the same parallel 01 hour
        • Lecture9.7
          Summary of All Concepts and Miscellaneous Questions 56 min
        • Lecture9.8
          Chapter Notes – Area of Parallelogram
        • Lecture9.9
          NCERT Solutions – Area of Parallelogram Exercise 9.1 – 9.4
        • Lecture9.10
          R D Sharma Solutions Area of Parallelogram
        • Lecture9.11
          Revision Notes Area of Parallelogram
      • 10.Constructions
        7
        • Lecture10.1
          Construction of Perpendicular Bisector of a Line Segment, Different Angles and Angle bisector 42 min
        • Lecture10.2
          Construction of Different types of Triangle 43 min
        • Lecture10.3
          Chapter Notes – Constructions
        • Lecture10.4
          NCERT Solutions – Constructions Exercise
        • Lecture10.5
          R D Sharma Solutions Constructions
        • Lecture10.6
          R S Aggarwal Solutions Constructions
        • Lecture10.7
          Revision Notes Constructions
      • 11.Circles
        11
        • Lecture11.1
          Introduction, Important Terms related to circle, Theorem-1, 2, 3 and 4 01 hour
        • Lecture11.2
          Problems Solving 48 min
        • Lecture11.3
          Problems Based on Parallel Chords, Common Chord of Circles 01 hour
        • Lecture11.4
          Problems Based on Concentric Circles, Triangle in Circle 41 min
        • Lecture11.5
          Equal Chords and Their Distance, Theorem-5, 6, 7, 8 01 hour
        • Lecture11.6
          Circle Through Points, Theorem-9, Arc of a Circle, Theorem-10 01 hour
        • Lecture11.7
          Chapter Notes – Circles
        • Lecture11.8
          NCERT Solutions – Circles Exercise
        • Lecture11.9
          R D Sharma Solutions Circles
        • Lecture11.10
          R S Aggarwal Solutions Circles
        • Lecture11.11
          Revision Notes Circles
      • 12.Heron's Formula
        8
        • Lecture12.1
          Introduction of Heron’s Formula; Area of Different Triangles 01 hour
        • Lecture12.2
          Area of Triangles Using Heron’s Formula (Type-1 level-1 and 2) 53 min
        • Lecture12.3
          Area of Quadrilaterals Using Heron’s Formula (Type-2 level-1) 32 min
        • Lecture12.4
          Area of Quadrilaterals Using Heron’s Formula (Type-2 level-2) 38 min
        • Lecture12.5
          Chapter Notes – Heron’s Formula
        • Lecture12.6
          NCERT Solutions – Heron’s Formula Exercise 12.1 12.2
        • Lecture12.7
          R D Sharma Solutions Heron’s Formula
        • Lecture12.8
          Revision Notes Heron’s Formula
      • 13.Surface Area and Volume
        16
        • Lecture13.1
          Introduction, Surface Area of Cube, Cuboid, Cylinder, Hollow cylinder, Cone, Sphere 40 min
        • Lecture13.2
          Type-1: Surface Area and Volume of Cube and Cuboid 27 min
        • Lecture13.3
          Level-2, Question based on Cube and Cuboid, Level-1 Questions based on Cylinder 28 min
        • Lecture13.4
          Level-2 Questions based on Cylinder, Type-3 Surface Area and Volume of Right Circular Cone (Level-1) 25 min
        • Lecture13.5
          Level-2 Questions based on Right Circular Cone, Level-1& 2 Questions Based on Sphere & Hemisphere 27 min
        • Lecture13.6
          Type-5 Short Answer Types Questions, 12 min
        • Lecture13.7
          Chapter Notes – Surface Area and Volume of a Cuboid and Cube
        • Lecture13.8
          Chapter Notes – Surface Area and Volume of A Right Circular Cylinder
        • Lecture13.9
          Chapter Notes – Surface Area ad Volume of A Right Circular Cone
        • Lecture13.10
          Chapter Notes – Surface Area ad Volume of A Sphere
        • Lecture13.11
          NCERT Solutions – Surface Area and Volume Exercise 13.1 – 13.9
        • Lecture13.12
          R D Sharma Solutions Surface Areas And Volumes of A cuboid and A cube
        • Lecture13.13
          R D Sharma Solutions Surface Areas And Volumes of A Right Circular Cylinder
        • Lecture13.14
          R D Sharma Solutions Surface Areas And Volumes of A Right Circular Cone
        • Lecture13.15
          R D Sharma Solutions Surface Areas And Volumes of A Sphere
        • Lecture13.16
          Revision Notes Surface Areas And Volumes
      • 14.Statistics
        15
        • Lecture14.1
          Data; Presentation of Data; Understanding Basic terms; 01 hour
        • Lecture14.2
          Questions; 52 min
        • Lecture14.3
          Bar Graph, Histogram, Frequency Polygon; 01 hour
        • Lecture14.4
          Histogram for Grouped Frequency Distribution (Class interval in Inclusive form), Frequency Polygon, Measures of Central Tendency, 01 hour
        • Lecture14.5
          Examples Based on Mean of Ungrouped Data 56 min
        • Lecture14.6
          Direct Method of Finding Mean of Ungrouped Data, Median of Cont. Freq. Distribution (Grouped Data) 01 hour
        • Lecture14.7
          Mean; Mean for an ungrouped freq. Dist. Table 25 min
        • Lecture14.8
          Chapter Notes – Statistics – Tabular Representation of Statistical Data
        • Lecture14.9
          Chapter Notes – Statistics – Graphical Representation of Statistical Data
        • Lecture14.10
          Chapter Notes – Statistics – Measures of Central Tendency
        • Lecture14.11
          NCERT Solutions – Statistics Exercise 14.1 – 14.4
        • Lecture14.12
          R D Sharma Solutions Tabular Representation of Statistical Data
        • Lecture14.13
          R D Sharma Solutions Graphical Representation of Statistical Data
        • Lecture14.14
          R S Aggarwal Solutions Statistics
        • Lecture14.15
          Revision Notes Statistics
      • 15.Probability
        8
        • Lecture15.1
          Activities; Understanding term Probability; Experiments; 44 min
        • Lecture15.2
          Event; Probability; Empirical Probability- Tossing a coin; 31 min
        • Lecture15.3
          Rolling a dice; Miscellaneous Example; 27 min
        • Lecture15.4
          Chapter Notes – Probability
        • Lecture15.5
          NCERT Solutions – Probability Exercise 15.1
        • Lecture15.6
          R D Sharma Solutions Probability
        • Lecture15.7
          R S Aggarwal Solutions Probability
        • Lecture15.8
          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 :
        1
        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 :
        statics1
        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 :
        3
        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.

        4
        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

        Prev Chapter Notes – Probability
        Next R D Sharma Solutions Probability

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