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      Class 10 Maths

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      • Class 10 Maths
      CoursesClass 10MathsClass 10 Maths
      • 01. Real Numbers
        9
        • Lecture1.1
          Real Numbers and Fundamental Theorem of Arithmetic 59 min
        • Lecture1.2
          Divisibility and Euclid’s Division Lemma 49 min
        • Lecture1.3
          Finding HCF and LCM Using Fundamental Theorem of Arithmetic 02 hour
        • Lecture1.4
          Miscellaneous Questions 31 min
        • Lecture1.5
          Chapter Notes – Real Numbers
        • Lecture1.6
          NCERT Solutions – Real Numbers Exercise 1.1 – 1.4
        • Lecture1.7
          Revision Notes Real Numbers
        • Lecture1.8
          R S Aggarwal Real Numbers
        • Lecture1.9
          R D Sharma Real Numbers
      • 02. Polynomials
        11
        • Lecture2.1
          Introduction to Polynomials and Its Zeroes 44 min
        • Lecture2.2
          Finding Zeroes of Quadratic Polynomial 01 hour
        • Lecture2.3
          Relationship b/w the Zeroes and coefficients of a Polynomial 01 hour
        • Lecture2.4
          Cubic Polynomial 35 min
        • Lecture2.5
          Division and Division Algorithm for Polynomials 01 hour
        • Lecture2.6
          Geometrical Meaning of Zeroes of Polynomial 15 min
        • Lecture2.7
          Chapter Notes – Polynomials
        • Lecture2.8
          NCERT Solutions – Polynomials Exercise 2.1 – 2.4
        • Lecture2.9
          Revision Notes Polynomials
        • Lecture2.10
          R S Aggarwal Polynomials
        • Lecture2.11
          R D Sharma Polynomials
      • 03. Linear Equation
        9
        • Lecture3.1
          Solution of linear Equation in Two Variable by Graphical and Algebraic Methods 01 hour
        • Lecture3.2
          Some Basic Questions on Solving Pair of Linear Equations 01 hour
        • Lecture3.3
          Some Basics Problems on Numbers and Ages 40 min
        • Lecture3.4
          Problems on Money Matters, Time, Distance, Speed, Time and Work 01 hour
        • Lecture3.5
          Chapter Notes – Linear Equation
        • Lecture3.6
          NCERT Solutions – Linear Equation Exercise 3.1 – 3.7
        • Lecture3.7
          Revision Notes Linear Equation
        • Lecture3.8
          R S Aggarwal Linear Equation
        • Lecture3.9
          R D Sharma Linear Equation
      • 04. Quadratic Equation
        8
        • Lecture4.1
          Introduction and Finding The Roots of a Quadratic Equation 37 min
        • Lecture4.2
          Miscellaneous Questions 1 02 hour
        • Lecture4.3
          Miscellaneous Questions 2 02 hour
        • Lecture4.4
          Chapter Notes – Quadratic Equation
        • Lecture4.5
          NCERT Solutions – Quadratic Equation Exercise 4.1 – 4.4
        • Lecture4.6
          Revision Notes Quadratic Equation
        • Lecture4.7
          R S Aggarwal Quadratic Equation
        • Lecture4.8
          R D Sharma Quadratic Equation
      • 05. Arithmetic Progressions
        11
        • Lecture5.1
          Introduction to Arithmetic Progression 59 min
        • Lecture5.2
          Miscellaneous Questions Based on nth Term Formula of A.P. 55 min
        • Lecture5.3
          Middle term (s) of Finite A.P. and Arithmetic Mean 01 hour
        • Lecture5.4
          Selection of the Terms and Sum of nth Terms of A.P. 01 hour
        • Lecture5.5
          Miscellaneous Questions Based on Sum of nth Term of A.P. 01 hour
        • Lecture5.6
          Word Problems related to nth term and sum of nth terms of A.P. 01 hour
        • Lecture5.7
          Chapter Notes – Arithmetic Progressions
        • Lecture5.8
          NCERT Solutions – Arithmetic Progressions Exercise 5.1 – 5.4
        • Lecture5.9
          Revision Notes Arithmetic Progressions
        • Lecture5.10
          R S Aggarwal Arithmetic Progressions
        • Lecture5.11
          R D Sharma Arithmetic Progressions
      • 06. Some Applications of Trigonometry
        7
        • Lecture6.1
          Angle of Elevation and Depression; Problems involving Elevation and Double Elevation 01 hour
        • Lecture6.2
          Miscellaneous Problems Involving Double Elevation, Depression and Elevation 01 hour
        • Lecture6.3
          Miscellaneous Questions (Level-1, 2, 3, 4) 49 min
        • Lecture6.4
          Chapter Notes – Some Applications of Trigonometry
        • Lecture6.5
          NCERT Solutions – Some Applications of Trigonometry
        • Lecture6.6
          Revision Notes Some Applications of Trigonometry
        • Lecture6.7
          R D Sharma Some Applications of Trigonometry
      • 07. Coordinate Geometry
        17
        • Lecture7.1
          Introduction to Terms related to Coordinate Geometry 49 min
        • Lecture7.2
          Locating Coordinates of Point on the Axes 37 min
        • Lecture7.3
          Finding vertices of a Geometrical Fig. and Distance b/w Two Points 58 min
        • Lecture7.4
          Distance b/w Two Points Formula Based Examples 44 min
        • Lecture7.5
          Finding the type of Triangle and Quadrilateral 01 hour
        • Lecture7.6
          Find the Collinear or Non-collinear points and Missing Vertex of a Geometrical Figure 42 min
        • Lecture7.7
          Section Formula and Corollary 01 hour
        • Lecture7.8
          Finding the Section Ratio and Involving Equation of Line 53 min
        • Lecture7.9
          Proof Related to Mid-points and Centroid of a Triangle 59 min
        • Lecture7.10
          Finding Area of Triangle and Quadrilateral 53 min
        • Lecture7.11
          Collinearity 34 min
        • Lecture7.12
          Examples Based on If Areas are Given 34 min
        • Lecture7.13
          Chapter Notes – Coordinate Geometry
        • Lecture7.14
          NCERT Solutions – Coordinate Geometry Exercise 7.1 – 7.4
        • Lecture7.15
          Revision Notes Coordinate Geometry
        • Lecture7.16
          R S Aggarwal Coordinate Geometry
        • Lecture7.17
          R D Sharma Coordinate Geometry
      • 08. Triangles
        15
        • Lecture8.1
          Introduction, Similarity of Triangles, Properties of Triangles, Thales’s Theorem 52 min
        • Lecture8.2
          Questions Based on Thales’s Theorem, Converse Thales’s Theorem 01 hour
        • Lecture8.3
          Questions Based on Converse Thales’s Theorem, Questions Based On Mid-points and its Converse 59 min
        • Lecture8.4
          Questions Based on Internal Bisectors of an Angle, Similar Triangles, AAA criterion for Similarity 52 min
        • Lecture8.5
          SSS & SAS Criterion of Similarity, Question Based on Criteria for Similarity 02 hour
        • Lecture8.6
          Questions Based on Criteria for Similarity Conti. 01 hour
        • Lecture8.7
          Questions Based On Area of Similar Triangles 02 hour
        • Lecture8.8
          Questions Based On Area of Similar Triangles cont., Pythagoras Theorem Level-1 01 hour
        • Lecture8.9
          Pythagoras Theorem Level-1 cont. 01 hour
        • Lecture8.10
          Miscellaneous Questions 01 hour
        • Lecture8.11
          Chapter Notes – Triangles
        • Lecture8.12
          NCERT Solutions – Triangles Exercise 8.1 – 8.6
        • Lecture8.13
          Revision Notes Triangles
        • Lecture8.14
          R S Aggarwal Triangles
        • Lecture8.15
          R D Sharma Triangles
      • 09. Circles
        8
        • Lecture9.1
          Introduction to Circle and Tangent and Theorems-1, 2 01 hour
        • Lecture9.2
          Property of Chord of Circle and Theorem-3 52 min
        • Lecture9.3
          Theorem 4 and Angle in Semicircle 01 hour
        • Lecture9.4
          Chapter Notes – Circles
        • Lecture9.5
          NCERT Solutions – Circles Exercise
        • Lecture9.6
          Revision Notes Circles
        • Lecture9.7
          R S Aggarwal Circles
        • Lecture9.8
          R D Sharma Circles
      • 10. Areas Related to Circles
        10
        • Lecture10.1
          Area and Perimeter of Circle, Semicircle and Quarter circle 56 min
        • Lecture10.2
          Touching Circles and Example on Bending Wire, Wheel Rotation 58 min
        • Lecture10.3
          Sector of A Circle 51 min
        • Lecture10.4
          Segment of Circle and Area of Combination of Plane Figures 01 hour
        • Lecture10.5
          Questions Based on Area of Combination of Plane Figures 01 hour
        • Lecture10.6
          Chapter Notes – Areas Related to Circles
        • Lecture10.7
          NCERT Solutions – Areas Related to Circles
        • Lecture10.8
          Revision Notes Areas Related to Circles
        • Lecture10.9
          R S Aggarwal Areas Related to Circles
        • Lecture10.10
          R D Sharma Areas Related to Circles
      • 11. Introduction to Trigonometry
        7
        • Lecture11.1
          Introduction, Trigonometric Ratios, Sums when Sides of Triangle are given, Miscellaneous Questions 01 hour
        • Lecture11.2
          Trigonometric Ratios for Some Specific Angles, Trigonometric Ratio for 30 & 60, Value Table, Word Problems 48 min
        • Lecture11.3
          Complimentary angles in Inverse Multiplication, Miscellaneous Problems 01 hour
        • Lecture11.4
          Miscellaneous Questions 01 hour
        • Lecture11.5
          Chapter Notes – Introduction to Trigonometry
        • Lecture11.6
          NCERT Solutions – Introduction to Trigonometry
        • Lecture11.7
          Revision Notes Introduction to Trigonometry
      • 12. Surface Areas and Volumes
        9
        • Lecture12.1
          Introduction, Surface Area of Cube, Cuboid, Cylinder, Hollow cylinder, Cone, Sphere, Type-1 : Surface Area and Volume of A Solid 01 hour
        • Lecture12.2
          Volume Related Questions, Based On Embarkment 48 min
        • Lecture12.3
          Based on the Rate of Flowing, Frustum of Cone (A) Surface Area (B) Volume 52 min
        • Lecture12.4
          Quantity, Percentage, Very Short Answer Types Questions 44 min
        • Lecture12.5
          Chapter Notes – Surface Areas and Volumes
        • Lecture12.6
          NCERT Solutions – Surface Areas and Volumes
        • Lecture12.7
          Revision Notes Surface Areas and Volumes
        • Lecture12.8
          R S Aggarwal Surface Areas and Volumes
        • Lecture12.9
          R D Sharma Surface Areas and Volumes
      • 13. Statistics
        12
        • Lecture13.1
          Introduction and Mean of Ungrouped Data 46 min
        • Lecture13.2
          Mean of Grouped Data 58 min
        • Lecture13.3
          Median of Ungrouped and grouped Data 01 hour
        • Lecture13.4
          Mode of Ungrouped and Grouped Data 49 min
        • Lecture13.5
          Mean, Median and Mode & Cumulative Frequency Curve-Less Than Type 53 min
        • Lecture13.6
          Cumulative Frequency Curve (Ogive)-More Than Type 52 min
        • Lecture13.7
          Miscellaneous Questions 38 min
        • Lecture13.8
          Chapter Notes – Statistics
        • Lecture13.9
          NCERT Solutions – Statistics
        • Lecture13.10
          Revision Notes Statistics
        • Lecture13.11
          R S Aggarwal Statistics
        • Lecture13.12
          R D Sharma Statistics
      • 14. Probability
        9
        • Lecture14.1
          Activities, Understanding term Probability, Experiments 54 min
        • Lecture14.2
          Calculations 43 min
        • Lecture14.3
          Event, Favorable Outcomes, Probability, Some important experiments- Tossing a coin, Rolling dice, Drawing a card 01 hour
        • Lecture14.4
          Complement of an event, Elementary Events, Equally Likely events, Some problems of Probability 58 min
        • Lecture14.5
          Chapter Notes – Probability
        • Lecture14.6
          NCERT Solutions – Probability
        • Lecture14.7
          Revision Notes Probability
        • Lecture14.8
          R S Aggarwal Probability
        • Lecture14.9
          R D Sharma Probability
      • 15. Construction
        7
        • Lecture15.1
          Introduction, Construction to Divide a Line Segment in Certain Ratio, Construction of Similar Triangles 44 min
        • Lecture15.2
          Construction of Different Angles, Construction of Angle Bisector 41 min
        • Lecture15.3
          Construction of Tangent of Circle at a Given Point 35 min
        • Lecture15.4
          Chapter Notes – Construction
        • Lecture15.5
          NCERT Solutions – Construction
        • Lecture15.6
          R S Aggarwal Construction
        • Lecture15.7
          R D Sharma Construction

        NCERT Solutions – Probability

        Q.1     Complete the statements:
        (i) Probability of event E + Probability of event ‘not E’ = _______

                   (ii) The probability of an event that cannot happen is _______. Such an event is called                       _______
                   (iii) The probability of an event that is certain to happen is _______ . Such an event is called            _______
                   (iv) The sum of the probabilities of all the elementary events of an experiment is _______
                   (v) The probability of an event is greater than or equal to _______ and less than or equal to  _______
        Sol.       (i) 1          (ii) 0, impossible event               (iii) 1, sure or certain event             (iv) 1                   (v) 0, 1

         


        Q.2     Which of the following experiments have equally likely outcomes? Explain.
                   (i) A driver attempts to start a car. The car starts or does not start.
                   (ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
                   (iii) A trial is made to answer a true-false question. The answer is right or wrong.
                   (iv) A baby is born. It is a boy or a girl.
        Sol.       (i) In the experiment “A driver attempts to start a car. The car starts or does not start”, we are not justified to assume that each outcome is as likely to occur as the other. Thus, the experiment has no equally likely outcomes.

                     (ii) In the experiment “A player attempts to shoot a basket ball. She/he shoots or misses the shot”, we are not justified to assume that each outcome is as likely to occur as the other. Thus, the experiment has no equally likely outcomes.

                     (iii) In the experiment “A trial is made to answer a true-false question. The answer is right or wrong. We know, in advance, that the result can lead in one of the two possible ways — either right or wrong. We can reasonably assume that each outcome, right or wrong, is likely to occur as the other. Thus, the outcomes right or wrong, are equally likely.

                      (iv) In the experiment “A baby is born. It is a boy or a girl”. We know, in advance, that the outcome can lead in one of two possible outcome — either a boy or a girl. We are justified to assume that each outcome, boy or girl, is likely to occur as the other. Thus, the outcomes boy or girl, are equally likely.


        Q.3      Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
        Sol.       The tossing of a coin is considered to be a fair way of deciding which team should get the ball at the beginning of a football game as we know that the tossing of the coin only land in one of two possible ways — either head up or tail up. It can reasonably be assumed that each outcome, head or tail, is as likely to occur as the other, i.e., the outcomes head and tail are equally likely, So, the result of the tossing of a coin is completely unpredictable.


        Q.4     Which of the following cannot be the probability of an even?
                    (A) 23                        (B) – 1.5                       (C) 15%                     (D) 0.7
        Sol.        Since the probability of an event E is a number P(E) such that
        0 ≤P(E) ≤1
        Therefore, –1.5 cannot be the probability of an event.
        Therefore, (B) is the correct answer.


        Q.5      IF P(F) = 0.05, what is the probability of ‘not E’?
        Sol.
                Since P(E) + P (not – E) = 1
        P (not – E) = 1 – P(E)  = 1 – 0.05 = 0.95


        Q.6      A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out
                    (i) an orange flavoured candy ?                           (ii) a lemon flavoured candy ?
        Sol.       (i) Consider the event related to the experiment of taking out of an orange flavoured candy from a bag containing only lemon flavoured candies. Since no outcome gives an orange flavoured candy, therefore, it is an impossible event so its probability is 0.
                      (ii) Consider the event of taking a lemon flavoured candy out of a bag containing only lemon flavoured candies. This event is a certain event so its probability is 1.


        Q.7     It is given that in a group of 3 students, the probability of 2 student not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?
        Sol.         Let E be the event of having the same birthday
        i.e, P(E) = 0.992
        But P(E) + P(E¯) = 1
        P(E¯) = 1 – P(E)
        1 – 0.992 = 0.008
        Hence, the probability that the 2 students have the same birthday is 0.008.


        Q.8       A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is :    (i) red?                 (ii) not red?
        Sol.          There are 3 + 5 = 8 balls in a bag. Out of these 8 balls, one can be chosen in 8 ways.
        Therefore, Total number of elementary events = 8
                         (i) Since the bag contains 3 red balls, therefore, one red ball can be drawn in 3 ways.
        Therefore, Favourable number of elementary events = 3
        Hence, P (getting a red ball ) = 38
        (ii) Since the bag contains 5 black balls along with 3 red ball, therefore, one black (not red) ball can be drawn in 5 ways.
        Therefore, Favourable number of elementary events = 5
        Hence, P(not getting a red ball) =58.


        Q.9     A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be  
                   (i) red ?                 (ii) white ?                        (iii) not green ?
        Sol.       Total number of marbles in the box = 5 (Red) + 8 (White) + 4 (Green) = 17
        Therefore,Total number of elementary events = 17
        (i) There are 5 red marbles in the box
        Therefore, Favourable number of elementary events = 5
        Hence, P (getting a red marble) = 517
                      (ii) There are 8 white marbles in the box.
        Therefore, Favourable number of elementary events = 8
        Hence, P (getting a white marble) = 817
                      (iii) There are 5 + 8 = 13 marbles which are not green in the box.
        Therefore, Favourable number of elementary events = 13
        Hence, P (not getting a green marble) = 1317


        Q.10     A piggy bank contains hundred 50 p coins, fifty Re 1 coins, twenty Rs 2 coins and ten Rs 5 coins, If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a 50 p coin ? (ii) will not be a Rs 5 coin ?
        Sol.        Total number of coins in a piggy bank   = 100 + 50 + 20 + 10 = 180
                       Therefore, Total number of elementary events = 180
                       (i) There are one hundred 50 paise coins in the piggy bank.
                       Therefore, Favourable number of elementary events = 100
                       Hence, P(falling out of a 50p coin) = 100180=59
                       (ii) There are 100 + 50 + 20,i.e., 170 coins other than Rs 5 coin.
                       Therefore, Favourable number of elementary events  = 170
                       Hence, P(falling out of a coin other than Rs 5 coin)
                       170180=1718


        Q.11     Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fishes and 8 females fishes (see fig.). What is the probability that the fish taken out is a male fish?
        img4.............
        Sol.       Total number of fish in the tank = 5 + 8 = 13
        Total number of elementary events = 13
                      There are 5 male fishes in the tank.
                      Therefore,Favourable number of elementary events = 5
                       Hence, P(taking out a male fish) = 513


        Q.12     A game of chance consists of spinning an arrow which comes to rest pointing at one of the number 1, 2, 3, 4, 5, 6, 7, 8, (see Fig.), and these are equally likely outcomes. What is the probability that it will point at

        1
                     (i) 8 ?                                                 
        (ii) an odd number ?                      
                     
        (iii) a number greater than 2 ?        (iv) a number less than 9?
        Sol.          Out of 8 numbers, an arrow can point any of the numbers in 8 ways.
                        Therefore, Total number of elementary events = 8
                         (i) There is only one ‘8’ on the spinning plant
                         Therefore, Favourable number of outcomes = 1
                         Hence, P(arrow points at 8) = 18
                         (ii) There are 4 odd numbers (viz. 1, 3, 5 and 7)
                         Therefore, Favourable number of outcomes = 4
                          Hence, P(arrow points at an odd number) = 48=12
                         (iii) There are 6 numbers greater than 2 (viz. 3, 4, 5, 6, 7 and 8)
                         Therefore, Favourable number of outcomes = 6
                         Hence, P(arrow points at a number > 2) = 68=34
                         (iv) There are 8 numbers less than 9 ( viz. 1,2,3,…8)
                         Therefore, Favourable number of outcomes = 8
                         Hence, P(arrow points at a number < 9) = 88 = 1.


        Q.13     A die is thrown once. Find the probability of getting
                     (i) a prime number;               (ii) a number lying between 2 and 6;
                     (iii) an odd number.
        Sol.         In a single throw of a die we can get any one of the six numbers 1, 2, …..,6 marked on its six faces. Therefore, the total number of elementary events associated with the random experiment of throwing a die is 6.
                        (i) Let A denotes the event “Getting a prime number”.Clearly, event A occurs if we obtain any one of 2, 3, 5 as an outcome. Therefore, Favourable number of elementary events = 3 Hence, P(A) = 36=12.
                        (ii) Let A denotes the event “Getting a number lying between 2 and 6”. Clearly, event A occurs if we obtain any one of 3, 4, 5 as an outcome. Therefore, Favourable number of elementary events
                         = 3Hence P(A) = 36=12
                       (iii) Let A denotes the event “Getting an odd number”. Clearly, event A occurs if we obtain any one of 1, 3, 5, as and outcome. Therefore, Favourable number of elementary events
                        = 3 Hence, P(A) = 36=12


        Q.14     One card is drawn from a well – shuffled deck of 52 cards. Find the probability of getting
                      (i) a king of red colour                                       (ii) a face card
                      (iii) a red face card                                              (iv) the jack of hearts
                      (v) a spade                                                             (vi) the queen of diamonds
        Sol.           Out of 52 cards, one card can be drawn in 52 ways. So, total number of elementary events = 52
                          (i) There are two suits of red cards viz. diamond and heart. Each suit contains one king.
        Therefore, Favourable number of elementary events = 2 × 1 = 2
        Hence, P(a king of red colour) = 252=126                 

                        (ii) In a deck of 52 cards: kings, queens and jacks are called face cards. Thus, there are 12 face cards. So, one face card can be chosen in 12 ways. Therefore, Favourable number of elementary events = 12 
                         Hence, P(a face card) = 1252=313

                         (iii) There are two suits of red cards viz. diamond and heart. Each suit contains 3 face cards.
                         Therefore, Favourable number of elementary events = 2 × 3 = 6
                          Hence, P(a red dace card) = 652=326

                          (iv)There is only one jack of hearts
                          Therefore, Favourable number of elementary events = 1
                          Hence, P(the jack of hearts) = 152

                          (v) There are 13 cards of spade.
                          Therefore, Favourable number of elementary events = 13
                          Hence, P(a spade) = 1352=14

                          (vi) There is only one queen of diamonds.
                          Therefore, Favourable number of elementary events = 1
                          Hence, P(the queen of diamonds) = 152.


        Q.15     Five cards – the ten, jack, queen, king and ace of diamonds, are well – shuffled with their face downwards. One card is then picked up at random.
                     (i) What is the probability that the card is the queen?
                     (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?
        Sol.         Out of five cards — the ten, jack, queen, king and ace of diamonds, one card can be drawn in 5 ways. So, total number of elementary events = 5
                        (i) There is only one queen.
        Therefore, Favourable number of elementary events = 1
        Hence, P(the queen) = 15

                        (ii) After keeping the queen card aside, we are left with 4 cards. So total number of elementary events now = 4.
                        (a) There is only one ace
                        Therefore,Favourable number of elementary events = 1
                        Hence, P(an ace) = 14
                        (b) There is no card as queen.
                        Therefore,Favourable number of elementary events = 0
                        Hence, P(the queen) = 04 = 0.


        Q.16     12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.
        Sol.            Out of 132 + 12, i.e., 144 pens, one pen can be chosen in 144 ways.
        Total number of elementary events = 144
        There are 132 non-defective i.e., good pens out of which one pen can be chosen in 132 ways.
         Favourable number of elementary events = 132
        Hence, P(getting a good pen) = 132144=1112


        Q.17      (i) A lot of 20 bulbs contains 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
                    (ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?
        Sol.           (i) Out of 20 bulbs one bulb can be chosen in 20 ways.
        Therefore,Total number of elementary events = 20
        There are 4 defective bulbs out of which one bulb can be chosen in 4 ways.
        Hence, P(getting a defective bulb) = 420=15

        (ii) On drawing a non-defective bulb out of 20 bulbs we are left with 19 bulbs including 4 defective bulbs.
        Therefore,Total number of elementary events = 19
        There are 19 – 4 = 15 non-defective bulbs out of which one bulb can be drawn in 15 ways.
        Therefore, Favourable number of elementary events = 15
        Hence, P(getting a non – defective bulb) =1519


        Q.18       A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two- digit number (ii) a perfect square number (iii) a number divisible by 5.
        Sol.            There are 90 discs bearing numbers 1 to 90 in the box of which one disc can be drawn in 90 ways .
                           Therefore, Total number of elementary events = 90
                           (i) There are 90 – 9, i.e., 81 discs bearing a two – digit number in the box of which one disc can be drawn in
                           81 ways.
        Therefore, Favourable number of elementary events = 81
        Hence, P(getting a disc bearing a two-digit number) = 8190=910.

                         (ii) Those numbers from 1 to 90 which are perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, i.e., squares of 1, 2, 3, 4, 5, 6, 7, 8, and 9 respectively. Therefore, there are 9 discs marked with numbers which are perfect squares.
                         Therefore, Favourable number of elementary events = 9
                         Hence, P (getting a disc marked with a number which is a perfect square)
                         = 990=110

                         (iii) Those numbers from 1 to 90 which are divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, and 90. They are 18 in number.
                         Therefore, there are 18 discs marked with the numbers which are divisible by 5.
                         Therefore, Favourable number of elementary events = 18
                         Hence, P (getting a disc marked with a number which is divisible by 5)
                         = 1890=15


        Q.19      A child has a die whose six faces show the letters as given below:

        2
        The die is thrown once. What is the probability of getting

                       (i) A ?                                    (ii) D ?
        Sol.            In a single throw of a die we can get any one of the six letters A, B, C, D, E, A marked on its faces. Therefore, the total numbers of elementary events associated with the random experiment of throwing a die is 6.
                          (i) Let E denotes the event “Getting a letter A” Clearly, event E occurs on the two faces.
        Therefore, Favourable number of elementary events = 2
        Therefore, P(E) = 26=13

        (ii) Let E denote the event “Getting a letter D”. Clearly event E occurs on the two face.
        Therefore, Favourable number of elementary events = 1
        Therefore, P(E) = 16


         

        Q.20     Suppose you drop a die at random on the rectangular region shown in the figure given on the next page. What is the probability that it will land inside the circle with diameter 1 m?

        3
        Sol.          Total area of the figure i.e., rectangle
                         = 3m × 2m = 6m2
                         Area of the circle = πr2 = π(12m)2 = π4m2
                         Probability (die to land inside the circle)
                        = π/46 = π24

        Q.21      A lot consists of 144 ball pens of which 20 are defective and the other are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that
                      (i) She will buy it?                                         (ii) She will not buy it?
        Sol.          There is a lot of 144 ball pens. Out of these 144 ball pens 20 are defective ball pens.
        = 144 – 20 = 124
        (i) P(she will buy) = P(a non – defective pen)
                        = 124144=3136
                        (ii) P(she will not buy)
                        = P(a defective pen)
                        = 20144=536

        Q.22     Refer to Example 13.
        (i) Complete the following table:

        4
                       (ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability 111. Do you agree with this argument? Justify your answer.
        Sol.             Elementary events associated to the random experiment of throwing two dice are —
        (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
                           (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
                           (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
                           (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
                           (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
                           (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
                           Therefore,Total number of elementary events = 6 × 6 = 36
                           Let A be the event of getting the sum as 3.
                           The elementary events favourable to event A are (1, 2) and (2, 1)
                           Clearly, favourable number of elementary events = 2
                           Hence, required probability = 236
                           Let A be the event of getting the sum as 4.
                           The elementary events favourable to event A are (1, 3), (3, 1) and (2, 2)
                           Clearly, favourable number of elementary events = 3
                           Hence, required probability = 336
                           Let A be the event of getting the sum as 5.
                           The elementary events favourable to event A are (1, 4), (4, 1), (2, 3) and (3, 4)
                           Clearly, favourable number of elementary events = 4
                           Hence, required probability = 436 = 19
                           Let A be the event of getting the sum as 6.
                           The elementary events favourable to event A are (1, 5), (5,1), (2, 4), (4, 2) and (3, 3)
                           Clearly, favourable number of elementary events = 5
                           Hence, required probability = 536
                           Let A be the event of getting the sum as 7.
                           The elementary events favoourable to event A are (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)
                           Clearly, favourable number of elementary events = 6
                           Hence, required probability = 636 =16
                           Let A be the event of getting the sum as 9.
                           The elementary events favourable to event A are (3, 6), (6, 3), (4, 5) and (5, 4)
                           Clearly, favourable number of elementary events = 4
                           Hence, required probability = 436 =19
                           Let A be the event of getting the sum as 10.
                           The elementary events favourable to event A are (4, 6), (6, 4), (5, 5)
                           Clearly, favourable number of elementary event = 3
                           Hence, required probability = 336 =112
                           Let A be the event of getting the sum as 11.
                           The elementary events favourable to event A are (5, 6), (6, 5)
                           Clearly, favourable number of elementary events = 2
                           Hence, required probability = 236 =118
                           Thus, the complete table is as under:

        5

        (ii) I do not agree with the argument given here. Justification has already been given in part (i).


         

        Q.1      Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?
        Sol.         Elementary events associated to the random experiment of visiting a particular shop in the same week (Tuesday to Saturday) by two customers Shyam and Ekta are —
        (T, T), (T, W), (T, TH), (T, F), (T, S)
        (W, T), (W, W), (W, TH), (W, F), (W, S)
        (TH, T), (TH, W), (TH, TH), (TH, F), (TH, S)
        (F, T), (F, W), (F, TH), (F, F), (F, S)
        (S, T), (S, W), (S, TH), (S, F), (S, S)
        where T = Tuesday, W = Wednesday, Th = Thursday, F = Friday and S = Saturday.
        Therefore, Total number of elementary events = 5 × 5 = 25
                       (i) Let A be the event of visiting a particular shop on the same day by two customers.
                       Then, elementary events favourable to A are (T, T), (W, W), (TH, TH), (F, F) and (S, S)
        Therefore, Favourable number of elementary events = 5
        Hence, required probability = 525=15
                       (ii) Let A be the event of visiting a particular shop by two customer on consecutive days. Then, elementary even favourable to A are (T, W), (W, T), (W, TH), (TH, W) (TH, F) (F, TH) (S, F) (F, S).
        Therefore, Favourable number of elementary events = 8
        Hence, the required probability = 825
                       (iii) Let A be the event of visiting a particular shop by two customers on different days. Then, elementary events favourable to A are excluding (T, T), (W, W), (TH, TH), (F, F) and (S, S). So, these are 25 – 5 = 20 in number
        Therefore, Favourable number of elementary events = 20
        Hence, required probability = 2025=45. 

        Q.2     A die is numbered in such a way that its faces show the numbers 1, 2, 2, 3, 3, 6. It is thrown two time and the total score in two throws is noted. Complete the following table which gives a few values of the total score on the two throws:

        6

                   What is the probability that the total score is
                   (i) even?                      (ii) 6?                         (iii) at least 6?
        Sol.       Complete table is as under :
        Number in first throw

        7

        Clearly total number of elementary events
        = 6× 6 =36
                     (i) Let A be the event of getting total score even. Then, elementary events favourable to A are 2, 4, 4, 4, 4, 8, 4, 4, 8, 4, 6, 6, 4, 6, 6, 8, 8 and 12
        Therefore, Favourable number of elementary events = 18
        Hence, P(A) = 836=12
                     (ii) Let A be the event of getting total score 6. Then, elementary events favourable to A are 6, 6, 6 and 6.
                      Therefore, Favourable number of elementary events = 4
                      Hence, P(A) = 436=19
                     (iii) Let A be the event of the total score is at least 6. Then, elementary events favourable to A are 7, 8, 8, 6, 6, 9, 6, 6, 9, 7, 8, 8, 9, 9 and 12  Therefore, Favourable number of elementary events = 15
        Hence, P(A) = 1536=512.


        Q.3      A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball double that of a red ball, determine the number of blue balls in the bag.
        Sol.         Let there be x blue balls in the bag.
        Therefore, Total number of balls in the bag = 5 + x
        Now, p1 = Probability of drawing a blue ball
        = x5+x
        p2 = Probability of drawing a red ball
        =  55+x
        But it is given that p1 = 2p2
        ⇒x5+x=2×55+x
                       ⇒x=10
        Hence, there are 10 blue balls in the bag.


        Q.4      A box contains 12 balls out of which x are black. If one ball is drawn at random from the box what is the probability that it will be a black ball ? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x.
        Sol.       There are 12 balls in the box. Out of these 12 balls, one can be chosen in 12 ways.
        Therefore, Total number of elementary events = 12
        There are x black balls out of which one can be chosen in x ways.
        Therefore, Favourable number of elementary events = x
        Hence, p1 = P (getting a black ball) = x12
        If 6 more black balls are put in the box, then
        Total number of balls in the box = 12 + 6 = 18
        Number of black balls in the box = x + 6
        Therefore, p2 = P (getting a white ball) = x+618
        It is given that p2=2p1
        ⇒x+618=2×x12
                      ⇒x+618=x6
        ⇒ x + 6 = 3x
                      ⇒ 2x = 6
                      ⇒ x = 3

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