Transcription

1

2

3

4

5

6

7

8 ADDITIONAL QUESTIONS FOR SELF EVALUATION 1. Write the direction cosines of the line parallel to Z-axis. (Ans 0,0,1) 2.Find the distance between the parallel planes. r.(2i-j+3k)=4 and r.(6i-3j+9k)+13=0 (Ans 25/3 14) 3.The Cartesian equation of the line is 3x+1= 6y-2=1-z. Find the direction ratios of the line (Ans (2,1,-6)) 4.Find the length and foot of the perpendicular from the point (2,-1,5) to the line (x- 11)/10 = (y+2)/-4 = (z+8)/-11. (ans (1,2,3) 14 ) 5.Write the intercept cut off by the plane 2x+y-z=5 on x axis (Ans x = 5/2) 6.Find the equation of a line passing through the point (1,2,3) and parallel to the planes x- y+2z=5 and 3x+ y+z=6. 7. Show that the lines r = -i-3j-5k+α(3i+5j+7k) and r = (2i+4j+6k) + β(i+3j+5k) intersect each other.

9 ANSWERS

10

11 LEVEL II

12

13 CHAPTER XII LINEAR PROGRAMMING LINEAR PROGRAMMING SCHEMATIC DIAGRAM Topic Concepts Degree of References Importance NCERT Book Vol. II Linear (i)lpp and its ** Articles 12.2 and Programming Formulation (ii)graphical method of ** Article Solving LPP (bounded Solved Ex. 1 to 5 unbounded solutions) Q. Nos 5 to 8 EX.12.1 (iii)diet Problem *** Q. Nos 1,2 and 9 Ex Solved Ex. 9 Q. Nos 2and3 Misc. Ex. (iv)manufacturing *** Solved Ex. 8 Q. Nos 3,4,5,6,7 of Ex.12.2 Solved EX.10 Q. Nos4 &10 Misc. Ex. (v)allocation Problem ** Solved Example 7Q. No 10 Ex.12.2, Q. No 5 &8 Misc. Ex. (vi)transportation * Solved EX.11 Q. Nos 6 &7 Misc. Ex. (vii)miscellaneous Problems ** Q. No 8 Ex SOME IMPORTANT RESULTS /CONCEPTS **Solving linear programming problem using Corner Point Method. The method comprises of the following steps: I.Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point. 2.Evaluate the objective function Z= ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points. 3.(i)When the feasible region is bounded, M and m are the maximum and minimum values of Z. (ii) in case, the feasible region is unbounded, we have: 4.(a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value. (b)similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value. (i)lpp and its Mathematical Formulation

14 LEVEL I I). Avinash has been given two lists of problems from his mathematics teacher with the instructions to submit not more than 100 of them correctly solved for getting assignment marks. The problems in the first list carry 10 marks each and those in the second list carry 5 marks each. He knows from past experience that he requires on an average of 4 minutes to solve a problem of 10 marks and 2 minutes to solve a problem of 5 marks. He has other subjects to worry about; he cannot devote more than 4 hours to his mathematics assignment. Formulate this problem as a linear programming problem to maximize his marks? What is the importance of time management for students? (ii)graphical method of solving LPP (bounded and unbounded solutions) LEVEL I Solve the following Linear Programming Problems graphically: 1) Minimize Z= - 3x+4y subject to x+2y 8, 3x+2y 12, x 0,y 0. 2) Maximize Z=5x+3y subject to 3x+5y I5, 5x+2y 10, x 0,y 0. 3) Minimize Z=3x+5y such that x+3y 3, x+y 2, x,y 0. (iii)diet Problem LEVEL ll 1) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2units/kg of vitamin A and 1 unit/kg of vitamin C, while food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs.5.00 per kg to purchase food I and Rs.7.00 per kg to purchase food II. Formulate this problem as a linear programming problem to minimize the cost of such mixture. Why should a person take balanced food? 2. Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat costs Rs. 20 per kg and rice costs Rs.20 per kg. The minimum daily requirements of protein and carbohydrates for an average child are 50 gm and 200 gm respectively. In what quantities, should wheat and rice be mixed in the daily diet to provide the minimum daily requirements of protein and carbohydrates at minimum cost? Which type of food an average child should consume? (iv) Manufacturing Problem LEVEL ll ILLUSTRATIVE EXAMPLE

15 A company manufactures two types of sweaters, type A and type B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make atmost 300 sweaters and can spend atmost Rs a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs. 200 on each unit of type A but considering the difficulties of a common man the company charges a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve the problem for maximum profit.(cbse Sample Paper 2014). Ans: let the company manufactures sweaters of type A = x, and that of type B = y daily LPP is to maximise P = 200x + 20y subject to the constraints: x+y x + 120y x y 100 x 0, y 0 Vertices of the feasible region are A (100, 0), B (175, 75), C (150, 150) and D (0, 300) Maximum profit is at B So Maximum Profit = 200 (175) + 20 (75) = = Rs A company manufactures two articles A and B. There are two departments through which these articles are processed: (i ) assembling and (ii) finishing departments. The maximum capacity of the assembling department is 60 hours a week and that of the finishing department is 48 hours a week. The

16 production of each article of A requires 4 hours in assembling and 2 hours in finishing and that of each unit of B requires 2 hours in assembling and 4 hours in finishing. If the profit is Rs. 6 for each unit of A and Rs. 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit. 2. A company sells two different products A and B. The two products are produced in a common production process which has a total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The demand in the market shows that the maximum number of units of A that can be sold is 70 and that for B is 125. Profit on each unit of A is Rs. 20 and that on B is Rs. 15. How many units of A and B should be produced to maximize the profit? Solve it graphically.which are the factors affecting the demand of a product in the market? LEVELIII 1. An NGO is helping the poor people of earthquake hit village by providing medicines. In order to do this, they set up a plant to prepare two medicines A and B. There is sufficient raw material available to make bottles of medicine A and bottles of medicine B but there are bottles into which either of the medicines can be put. Further it takes 3 hours to prepare enough material to fill 1000 bottles of medicine A and takes 1 hour to prepare enough material to fill 1000 bottles of medicine B. There are 66 hours available for the operation. If the bottle of medicine A is used for 8 patients and bottle of medicine B is used for 7 patients. How the NGO should plan its production to cover maximum patients? How can you help others in case of natural disasters? (v)allocationproblem LEVELII 1. Ramesh wants to invest at most Rs.70,000 in Bonds A and B.According to the rules, he has to invest at least Rs.10,000 in Bond A and at least Rs.30,000 in Bond B. lf the rate of interest on bond A is 8% per annum and the rate of interest on bond B is 10% per annum, how much money should he invest to earn maximum yearly income? Find also his maximum yearly income. Why investment is important for future life? 2. lf a class XII student aged 17 years, rides his motor cycle at 40km/hr, the petrol cost is Rs.2 per km. If he rides at a speed of 70km/hr, the petrol cost increases to Rs.7per km. He has Rs.100 to spend on petrol and wishes to cover the maximum distance within one hour. (i) Express this as an L.P.P and solve it graphically. (ii) What is the benefit of driving at an economical speed? (iii) Should a child below 18 years be allowed to drive a motorcycle? Give reasons.

17 LEVELIII 1. An aero plane can carry a maximum of 250 passengers. A profit of Rs 500 is made on each executive class ticket and a profit of Rs 350 is made on each economy class ticket. The airline reserves at least 25 seats for executive class. However, at least 3 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit? Suggest necessary preparations to be made before going on a trip? 2. A farmer has a supply of chemical fertilizers of type 'A' which contains 10% nitrogen and 6% phosphoric acid and type 'B' contains 5% of nitrogen and 10% of phosphoric acid. After soil testing, it is found that at least 7kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizers of type A and type B costs Rs 5 and Rs 8 per kilograms respectively. Using L.P.P, find out what quantity of each type of fertilizers should be bought to meet the requirement so that the cost is minimum. Solve the problem graphically. What are the side-effects of using excessive fertilizers? (vi) Transportation Problem LEVEL III ILLUSTRATIVE EXAMPLE Q-1Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table: How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost? From/To A B D E F

18 X 0, Y 0, and 100 X Y 0 60 X 0, 50 Y 0, and X + Y 60 0 X 60, Y 50, and X + Y 60 Total transportation cost Z is given by, Z= 6x + 3y +2.5(100 x y) + 4(60 x) + 2(50 y) + 3(x + y 60) = 6x + 3y x 2.5y x y +3x + 3y 180 = 2.5x + 1.5y +410 The given problem can be formulated as Minimize Z= 2.5x + 1.5y (1) subject to the constraints, X + Y 100 (2) X 60.(3) Y 50.(4) X + Y 60.(5) X, Y 0.(6)

19 Z=2.5x + 1.5y ) In point A (60, 0) Z= 2.5 x x Z= 560 2) In point B (60, 40) ( Checking by solving the two lines x + y = 100 and x=60 we get x = 60, y = 40). Z= 2.5 x x Z= 620 3) In point C (50, 50) (Checking by solving the two lines x + y = 100 and y = 50 we get x = 50, y = 50.) Z= 2.5 x x Z= 610

20 4) In point D(10,50) (Checking by solving the two lines x + y = 60 and y = 50 we get x = 10, y = 50).Z=2.5 x x = 510 The minimum value of Z is 510 at (10, 50). RESULT : Thus, the amount of grain transported from A to D = 10 quintals A to E = 50 quintals A to F =40 quintals B to D = 50 quintals B to E = 0 quintals B to F = 0 quintals respectively. The minimum cost is Rs A medicine company has factories at two places A and B. From these places, suppiy is to be made to each of its three agencies P, Q and R. The monthly requirement of these agencies are respectively 40, 40 and 50 packets of the medicines, While the production capacity of the factories at A and B are 60 and 70 packets are respectively. The transportation cost per packet from these factories to the agencies are given: Transportation cost per packet (in Rs.) From To A B P 5 4 Q 4 2 R 3 5 How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost. What should be the features of best location for a factory? CBSE PREVIOUS YEAR QUESTIONS LEVEL-II 1.A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs to invest and has space for at most 20 items. An electronic sewing machine costs him Rs and a manually operated sewing machine Rs He can sell an electronic sewing machine at a profit of Rs and a manually operated sewing machine at a profit of Rs Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit? Make it as a linear programming problem and then, solve it graphically. Keeping the rural background in mind justify the

21 'values' to be promoted for the selection of the manually operated machine (CBSE sample paper 2013). 2. A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week? (CBSE 2014) LEVEL III If a young man drives his scooter at 25 kmph, he has to spend Rs 2 per kilometer on petrol. If he drives the scooter at a speed of 40 kmph, it produces more pollution and increases his expenditure on petrol to Rs 5 per km. He has a maximum of Rs 100 to spend on petrol and wishes to travel a maximum distance in 1 hour time with less pollution. Express this problem as an LPP and solve it graphically. What value do you find here? [CBSE 2013 C (DB)] LEVEL-II I A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs to invest and has space for at most 20 items. An electronic sewing machine costs him Rs and a manually operated sewing machine Rs He can sell an Electronic Sewing Machine at a profit of Rs and a manually operated sewing machine at a profit of Rs Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a linear programming problem and then, solve it graphically. Keeping the rural background in mind justify the 'values' to be promoted for the selection of the manually operated machine Questions for self evaluation l. Solve the following linear programming problem graphically: maximize Z =x - 7y+ 190 subject to the constraints x + y 8, x5, y5, x+y4, x0, y0. 2. Solve the following linear programming problem graphically: Maximize z=3x+5y subject to the constraints x+ y2, x+3y3, x0, y0. 3. Kelloggis a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains, 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kilogram and rice costs Rs. 4 per

22 kilogram. 4. A shopkeeper deals only in two items- tables and chairs. He has Rs. 6,000 to invest and a space to store at most 20 pieces. A table costs him Rs. 400 and a chair Rs He can sell a table at a profit of Rs. 25 and a chair at a profit of Rs. 40. Assume that he can sell all items that he buys. Using linear programming formulate the problem for maximum profit and solve it graphically. What would be your criteria to select a good piece of furniture? 5. A small firm manufactures items A and B. The total number of items A and B it can manufacture a day is at most 24 items. A takes one hour to make while item B takes only half an hour. The maximum time available per day is 16 hours. If the profit on one unit of item A be Rs. 300 and one unit of item B be Rs. 160, how many of each type of item be produced to maximize the profit? Solve the problem graphically. A firm has 2 types of machines. Machine A operates on electricity, Machine B operates on coal. Which machine would you prefer? 6. A chemist requires 10, 12 and 12 units of chemicals A, Band C respectively for his analysis. A liquid product contains 5, 2 and 1 units of A, Band C respectively and it costs Rs. 3 per jar. A dry product contains 1, 2 and 4 units of A. Band C per carton and costs Rs. 2 per carton. How many of each should he purchase in order to minimize the cost and meet the requirement? 7. A person wants to invest at most Rs. 18,000 in Bonds A and E. According to the rules, he has to invest at least Rs. 4,000 in Bond A and at least Rs. 5,000 in Bond B. If the rate of interest on bond A is 9% per annum and the rate of interest on bond B is 11 %per annum, how much money should he invest to earn maximum yearly income? Explain the importance of investment for future life? 8. Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively by stitching uniform. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to stitch at least 60 shirts and 32 pants at a minimum labour cost. What should be the features of uniform of a student?

23 ANSWERS LINEAR PROGRAMMING (i)lpp and its Mathematical Formulation LEVEL l 1. X+y: 100 4x+2y 240 Z=10 x+5y Students who divide the time for each subject per day according to their need don't feel burden of any subject before the examination (ii) Graphical method of solving LPP (bounded and unbounded solutions) I. Minimum z= - 12 at (4.0). 2. Maximum Z= 235 at 20, Minimum Z=7 at (3/2, 1/2) (iii)diet Problem LEVELII I. Minimum cost = Rs at x = 2, Y = 4. Balanced diet keeps fit, healthy and disease free life for a person 2. Minimum cost = Rs.6 at x = 400 and y = 200 Qualities of food are a) It should not contain more fats b) It should not contain more carbohydrates c) It should contain enough fiber, vitamin etc (iv)manufacturing Problem LEVELII 1). Maximum profit is Rs.120 when 12 units of A and 6 units of B are produced 2). For maximum profit, 25 units of product A and125 units of product B are produced and sold. The factors affecting the demand of a product in the market are a) Quality of the product b) Timely supply of the product c) Customer's satisfaction LEVEL III bottles of medicine A and bottles of medicine B and they can cover patients. We should not get panic and should not create panic in case of natural disaster. We must have the helpline numbers of government agencies and NGO working in case of natural disaster. (v)allocation Problem

24 LEVEL-II Maximum annual income =Rs. 6,200 on investment of Rs. 40,000 on Bond A and Rs.30, 000 on Bond B. We save money with a purpose of making use of it when we face any kind of financial crisis in our life. We will also be to able to achieve our goals of life if we have enough investment. Max. Z = x + y. Subject to constraints: x/40 + y/701, 2x + 7y100; x, yo. Here x & y represents the distance travelled by the boy at speed of 40km/hr&70km/h respectively. (i) x= 1560/41km, y = 140/41km. (ii) It saves petrol. It saves money. (iii) No, because according to the law driving license is issued when a person is above the 18 years of age. LEVEL-III 1) For maximum profit, 62executive class tickets and 188 economy class ticket should be sold. 1) Plan the trip 2) Check the journey tickets 3) Check the weather forecast 4) Do not take too much of cash 3.Type A fertilizers = 50 kg, Type B = 40 kg. Minimum cost = RS.570. Side effects: Excessive use of fertilizers can spoil the quality of crop also it may cause infertility of land. (vi)transportation Problem LEVEL-III I. Minimum transportation cost is Rs. 400 when 10, 0 and 50 packets are transported from factory at A and 30, 40 and 0 packets are transported from factory at B to the agencies at P, Q and R respectively. The location for a factory should have the following features 1) enough transport facility 2) enough natural resources 3) enough water 4) availability of electricity 5) availability of labours CBSE PREVIOUS YEAR QUESTIONS LEVEL-II 1. Max. Z = Rs.392. No. of electronic machines = 8 and no. of manually operated machines = 12. Keeping the 'save environment' factor in mind the manually operated machine should be promoted so that maximum use of man power and thereby leading to minimum use of energy resources providing more opportunities for employment in the rural areas (CBSE sample paper 2013) 2. Max profit = Rs 1680 when 12 pieces of type A and 6 pieces of type B are manufactured per week (CBSE 2014) 3. Max distance = 30 Km. at (50/3, 40/3) value save natural resources / our earth [CBSE 2013 C(DB)]

25 Questions for self evaluation 1) Minimum 155 at (0, 5) 2) Minimum value is 5 at(3/2, I /2) 3) Maximum is Rs 4.60 at (0.6, 0.4) 4) Maximum is Rs.800 at(0, 20) The criteria which we have to take into consideration for selecting a good piece of furniture are a) durability b) cost effectiveness c) attractive d) occupy minimum area 5). 8 items of type A and 16 items of type B I would prefer machine A because machine B is not eco-friendly 6. 1 jar of liquid and 5 cartons of dry product. 7. Rs.4,000 in Bond A and Rs.14,000 in Bond B. We save money with a purpose of making use of it when we face any kind of financial crisis in our life. We will also be to able to achieve our goals of life if we have enough investment. 8. Minimum cost Rs.1350 at (5, 3) The uniform of a student should be a) well pressed b) neat and tidy c) properly stitched d) shoe must be polished ADDITIONAL IMPORTANT QUESTIONS: 1. A manufacturer makes two types of cups A and B. Three machines are required to manufacture the cups and time in minutes required by each is as given below : Types of Cup Machines I II III A B Each machine is available for a maximum period of 6 hours per day. If the profit on each cup A is 75 paise and on B is 50 paise. Find how many cups of each type should be manufactured to maximize the profit per day. [ Ans : Cup A: 15, Cup B: 30 ] 2. A catering agency has two kitchens to prepare food at two places A and B. From these places, mid-day meal is to be supplied to three different schools situated at P, Q, R. The monthly requirement of these schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchens A and B are 60 and 70 packets per month respectively. The transportation cost per packet from the kitchens to the school is given below:

26 Transportation Cost per packet (in Rs.) To FROM A B P 5 4 Q 4 2 R 3 5 How many packets from each kitchen should be transported to schools so that the the cost of transportation is minimum? Also find the minimum cost. [ Ans : Min cost = Rs. 400] 3. Every gram of wheat provides 0.1 gm of proteins and 0.25 gram of carbohydrates. The corresponding values for rice are 0.05 gram and 0.5 gram respectively. Wheat costs Rs 4 per kg. and rice Rs 6 per kg. The minimum daily requirements of protein and carbohydrates for an average child are 50 grams and 200 grams respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of protein and carbohydrates at minimum cost. Frame an L.P.P and solve it graphically. [ Ans : wheat = 400 gm and rice = 200 gm ]

27 CHAPTER XIII

28

29

30

31

32 ADDITIONAL IMPORTANT QUESTIONS: 1. There are three coins.one is a two-headed coin (having head on both faces),another is a biased coin that comes up heads75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random &tossed, and it shows heads What is the probability that it was the two-headed coin? [Ans :4/9] 2. In a bolt factory, three machines A, B, and C manufactures 25,35 and 40 percent of the total bolts manufactured. Of their outputs, 5, 4 and 2 percent are defective respectively. A bolt is drawn at random and is found defective. Find the probability that it was manufactured by either machine A or C. [ Ans : 41/69] 3. Coloured balls are distributed in three bags as shown in the following table: Colour of the ball Bag Black White Red I II III A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they came from bag I?[ Ans : 231/551] 4. A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white?[ Ans : 3/5]

33 5. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable, and hence find the mean of the distribution. [Ans: X P(X) 1/15 2/15 3/15 4/15 5/15 Mean=14/3 6.In a game,a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as when he gets a six.find the expected value of the amount he wins/loses. [ Ans: 11/216 ] 7. Two balls are drawn one by one with replacement from a bag containing 4 red and 6 black balls. Find the probability distribution of number of red balls. [Ans: X: P(X) : 9/25 12/25 4/25 ] 8. Find the probability distribution of the number of doublets in three throws of a pair of dice. [ Ans : X : P(X) : 125/216 75/216 15/216 1/216 ] VALUE BASED QUESTIONS 1. In a school, 30% of the student has 100% attendance. Previous year result report tells that 70% of all students having 100% attendance attain A grade and 10% of remaining students attain A grade in their annual examination. At the end of the year, One student is chosen at random and he has an A grade. What is the probability that the student has 100% attendance? Also state the factors which affect the result of a student in the examination. [Ans.45 3/4 Factors :-(i) Regular study (ii) Hard work (iii) Good memory (iv) Well time management (v) Writing skills] 2. A company has two plants of scooter manufacturing. Plant I manufacture 70% Scooter and plant II manufactures 30%. At plant I 80% of the scooter s are maintaining pollution norms and in plant II 90% of the scooter maintaining Pollution norms. A Scooter is chosen at random and is found to be fit on pollution norms. What is the probability that it has come from plant II. What is importance of pollution norms for a vehicle? [ Ans: 27/53, Pollution free environment minimize the health problems in the human being.] 3. In a group of students, 200 attend coaching classes, 400 students attend school regularly and 600 students study themselves with help of peers. The probability that a student will succeed in life who attend coaching

34 classes, attend school regularly and study themselves with help of peers are 0.1, 0.2 and 0.5 respectively. One student is selected who succeeded in life, what is the probability that he study himself with help of peers. What type of study can be considered for the success in life and why? [Ans:0.75self studies with the help of peers is best as through it students can get the knowledge in depth of each concept. But students should be regular in school and if they feel need they could join different classes]. 4. A clever student used a biased coin so that the head is 3 times as likely to occur as tail. If the coin tossed twice find the probability distribution and mean of numbers of tails. Is this a good tendency? Justify your answer. [ Ans: X : P(X) : 9/16 6/16 1/16 Mean = ½. No, it may be good once or twice but not forever. Honesty pays in a long run. ]

35

36 SYLLABUS

37

38

39

40

41 SAMPLEPAPER BLUE PRINT S.No. Topics VSA(1) SA(1) LA (6) Total 1.(a) (b) 2.(a) (b). Relations&Functions 4(1) 4(1) 10(2) InverseTrigonometricFunctions 6(1) Matrices 1(2) 6(1) 2(2) 13(5) Determinants 1(1) 4(1) 6(1) 11(3) 3(a) Continuity&Differentiability 4(2) 8(2) (c) (e) ApplicationsofDerivatives Integrals 4(1) 4(3) 6(1) 10(2) 12(3) 44(10) 4.(a) (b) Applicationof Integrals 6(1) 6(1) DifferentialEquation 8(2) Vectors 1(2) 4(1) 6(3) Three DimensionalGeometry 1(1) 4(1) 6(1) 11(3) 17(6) 5. LinearProgramming 6(1) 6(1) 6. Probability 4(1) 6(1) 10(2) Total 6(6) 52(13) 42(7) 100(26)

42 140

43 141

44 142

45 143

46 144

47 145

48 146

49 147

50 148

51 149

52 150

53 151

54 152

55 153

56 154

57 155

58 156

59 157

60 158

61 IITJEE

62 160

63 161

64 162

65 163

66 164

67 165

68 166

69 167

70 168

71

72

73 VALUE BASED PROBLEMS MATHEMATICS CLASS-XII RELATIONS AND FUNCTIONS Q.4 Let A be the set of all students of class XII in a school and R be the relation, having the same sex on A, and then prove that R is an equivalence relation. Do you think, coeducation may be helpful in child development and why? Q.5 Consider a relation R in the set A of people in a colony, defined as arb, if and only if a and b are members of a joint family. Is R an equivalence relation? Staying with Grandparents in a joint family imbibes the moral values in us. Can you elicit two such values? Q.6 Let R be a relation defined as R : { (x,y) : x and y study in the same class} Show that R is an Equivalence Relation. If x is a brilliant student and y is a slow learner and x helps y in his studies. What quality does x possess? Q.7 Let L be the set of all lines in a plane and R be the relation in L defined by R = { (L 1, L 2 ): L 1 is parallel to L 2 } Show that R is an Equivalence Relation. L 1 represents the ideologies of Gandhi, L 2 represents ideologies of NetajiSubhash Chandra Bose. Even though their ideologies ran on parallel tracks both had the common goal to achieve independence for India. Which common value did they both exhibit?

74 MATRICES & DETERMINANTS Q.1. Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and newspaper s envelope as carry bags. it is found that the shopkeepers A, B, C are using (20,30,40), (30,40,20,), (40,20,30) polythene, handmade bags and newspapers envelopes respectively. The shopkeepers A, B, C spent Rs.250, Rs.220 & Rs.200 on these carry bags respectively.find the cost of each carry bags using matrices. Keeping in mind the social & environmental conditions, which shopkeeper is better? & why? Q.2 In a Legislative assembly election, a political party hired a public relation firm to promote its candidate in three ways; telephone, house calls and letters. The numbers of contacts of each type in three cities A, B & C are (500, 1000, and 5000), (3000, 1000, 10000) and (2000, 1500, 4000), respectively. The party paid Rs. 3700, Rs.7200, and Rs.4300 in cities A, B & C respectively. Find the costs per contact using matrix method. Keeping in mind the economic condition of the country, which way of promotion is better in your view? Q.3 A trust fund has Rs. 30,000 is to be invested in two different types of bonds. The first bond pays 5% interest per annum which will be given to orphanage and second bond pays7% interest per annum which will be given to an N.G.O. cancer aid society. Using matrix multiplication, determine how to divide Rs 30,000 among two types of Bonds if the trust fund obtains an annual total interest of Rs What are the values reflected in the question. Q.4 Using matrix method solve the following system of equations x + 2y + z = 7 x y + z =4 x + 3y +2z = 10 If X represents the no. of persons who take food at home. Y represents the no. of parsons who take junk food in market and z represent the no. of persons who take food at hotel. Which way of taking food you prefer and way? Q.5 A school has to reward the students participating in co-curricular activities (Category I) and with 100% attendance (Category II) brave students (Category III) in a function. The sum of the numbers of all the three category students is 6. If we multiply the number of category III by 2 and added to the number of category I to the result, we get 7. By adding second and third category would to three times the first category we get 12.Form the matrix equation and solve it. Q.6 F for keeping Fit X people believes in morning walk, Y people believe in yoga and Z people join Gym. Total no of people are 70.further 20% 30% and 40% people are suffering from any disease who believe in morning walk, yoga and GYM respectively. Total no. of such people is 21. If morning walk cost Rs.0 Yoga cost Rs.500/month and GYM cost Rs.400/ month and total expenditure is Rs (i) Formulate a matrix problem.

75 (ii) Calculate the no. of each type of people. (iii)why exercise is important for health? Q.7. An amount of Rs. 600 crores is spent by the government in three schemes. Scheme A is for saving girl child from the cruel parents who don t want girl child and get the abortion before her birth. Scheme B is for saving of newlywed girls from death due to dowry. Scheme C is planning for good health for senior citizen. Now twice the amount spent on Scheme C together with amount spent on Scheme A is Rs 700 crores. And three times the amount spent on Scheme A together with amount spent on Scheme B and Scheme C is Rs 1200 crores. Find the amount spent on each Scheme using matrices? What is the importance of saving girl child from the cruel parents who don t want girl child and get the abortion before her birth? Q.8. There are three families. First family consists of 2 male members, 4 female members and 3 children. Second family consists of 3 male members, 3 female members and 2 children. Third family consists of 2 male members, 2 female members and 5 children. Male member earns Rs 500 per day and spends Rs 300 per day. Female member earns Rs 400 per day and spends Rs 250 per day child member spends Rs 40 per day. Find the money each family saves per day using matrices? What is the necessity of saving in the family? CONTINUITY AND DIFFERENTIABILITY Q.1. A car driver is driving a car on the dangerous path given by m N Find the dangerous point (point of discontinuity) on the path. Whether the driver should pass that point or not? Justify your answers. APPLICATION OF DERIVATIVES Q.1 A car parking company has 500 subscribers and collects fixed charges of Rs.300 per subscriber per month. The company proposes to increase the monthly subscription and it is believed that for every increase of Re.1, one subscriber will discontinue the service. What increase will bring maximum income of the company? What values are driven by this problem? Q.2. Check whether the function + is strictly increasing or strictly decreasing or none of both on. Should the nature of a man be like this function? Justify your answers. Q.3. If, when denotes the number of hours worked and denotes the amount (in Rupees) earned. Then find the value of (in interval) for which the income remains increasing? Explain the importance of earning in life? Q.4. If performance of the students y depends on the number of hours x of hard work done per day is given by the relation. Find the number of hours, the students work to have the best performance.

76 Hours of hard work are necessary for success Justify. Q.5. A farmer wants to construct a circular well and a square garden in his field. He wants to keep sum of their perimeters fixed. Then prove that the sum of their areas is least when the side of square garden is double the radius of the circular well. Do you think good planning can save energy, time and money? Q.6. Profit function of a company is given as where x is the number of units produced. What is the maximum profit of the company? Company feels its social responsibility and decided to contribute 10% of his profit for the orphanage. What is the amount contributed by the company for the charity? Justify that every company should do it. Q.7. In a competition a brave child tries to inflate a huge spherical balloon bearing slogans against child labour at the rate of 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when its radius is 15cm. Also write any three values/life skill reflected in this question. Q.8. Q.9. In a kite festival, a kite is at a height of 120m and 130m string is out. If the kite is moving horizontally at the rate of 5.2m/sec, find the rate at which the string is being pulled out at that instant. How a festival enhance national integration. An expensive square piece of golden color board of side 24 centimeters. is to be made into a box without top by cutting a square from each corner and folding the flaps to form a box. What should be the side of the square piece to be cut from each corner of the board to hold maximum volume and minimize the wastage? What is the importance of minimizing the wastage in utilizing the resources? Q.10. A student is given card board of area 27 square centimeters. He wishes to form a box with square base to have maximum capacity and no wastage of the board. What are the dimensions of the box so formed? Do you agree that students don t utilize the resources properly? Justify. INTEGRATION Q.1 Evaluate,, Discuss the importance of integration (unity) in life. APPLICATIONS OF INTEGRALS Q.1. A farmer has a piece of land. He wishes to divide equally in his two sons to maintain peace and harmony in the family. If his land is denoted by area bounded by curve and and to divide the area equally he draws a line what is the value of a? What is the importance of equality among the people? Q.2. A circular Olympic gold medal has a radius 2cm and taking the centre at the origin, Find its area by method of integration. What is the importance of Olympic Games for a sportsman and why? Olympic game is a supreme platform for a sportsman. In Olympic Games all countries of the world participate and try their best and make their country proud. Q.3. A poor deceased farmer has agriculture land bounded by the curve y=, between x = 0 and x=2 π. He has two sons. Now they want to distribute this land in three parts (As

77 already partitioned).find the area of each part. Which parts should be given to the farmer & why? Justify your answer. Q.4 If a triangular field is bounded by the lines x+2y = 2, y-x = 1 and 2x+y = 7Using integration compute the area of the field (i) If in each square unit area 4 trees may be planted. Find the number of trees can be planted In the field. (ii) Why plantation of trees is necessary? Q.5 A parking lot in an IT company has an area bounded by the curve y= 4-x 2 and the lines y=0 and y = 3 divides the area in to two equal parts out of which the greater area is allotted for car owners who practice carpooling. Find this area using integration. Write any two benefits of carpooling. Ans. Fuel saving, Less pollution Q.6 Find the area of the region enclosed by the curve y= x 2 and the lines x=0, y=1 and y=4. A farmer plans to construct an electrical fence around this bounded region to protect his crop. But his son rejects this idea and wants wooden fence to be erected. Who would you favour? Mention two values demonstrated by the son. Ans. Concern for animals, kind hearted, not being cruel, bold DIFFERENTIAL EQUATIONS 1. Solve the differential equation (x+ 2y 2 )y =y. Given that when x= 2, y=1. If x denotes the % of people who are polite and y denote the % of people who are intelligent. Find x when y=2%. A polite child is always liked by all in society. Do you agree? Justify. 2. y + = 0 where x denotes the percentage of population living in a city and y denotes the area for living a healthy life of population.find the particular solution when x=100, y=1. Is higher density of population is harmful? Justify your VECTORS & 3-DIMENSIONAL GEOMETRY Q.1. considering the earth as a plane having equation, A monument is standing vertically such that its peak is at the point (1, 2, -3). Find the height of the monument. How can we save our monument? Q.2. Let the point p (5, 9, 3) lies on the top of QutubMinar, Delhi. Find the image of the point on the line = = Do you think that the conservation of monuments is important and why? Q.3 Two bikers are running at the Speed more than allowed speed on the road along the Lines = and = Using Shortest distance formula check whether they meet to an accident or not? While driving should driver maintain the speed limit as allowed. Justify?

78 LINEAR PROGRAMMING PROBLEMS Q.1. A dietician wishes to mix two types of food in such a way that the vitamin content of the mixture contain at least 8 unit of vitamin A and 10 unit of vitamin C. Food I contains 2unit/kg of vitamin A and 1unit/kg of vitamin C, while food II contains I unit/kg of vitamin A and 2unit/kg of vitamin C. It cost Rs.5.00 per kg to purchase food I and Rs.7.00 per kg to produce food II. Determine the minimum cost of the mixture. Formulate the LPP and solve it. Why a person should take balanced food? Q.2. A farmer has a supply of chemical fertilizers of type A which contains 10% nitrogen and 6% phosphoric acid and type B contains 5% of nitrogen and 10% of phosphoric acid. After soil testing it is found that at least 7kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizers of type A and type B cost Rs.5 and Rs.8 per kilograms respectively. Using L.P.P, find how many Kgs. of each type of fertilizers should be bought to meet the requirement and cost be minimum solve the problem graphically. What are the side effects of using excessive fertilizers? Q.3 If a class XII student aged 17 years, rides his motor cycle at 40km/hr, the petrol cost is Rs. 2 per km. If he rides at a speed of 70km/hr, the petrol cost increases Rs.7per km. He has Rs.100 to spend on petrol and wishes to cover the maximum distance within one hour. 1. Express this as an L.P.P and solve graphically. 2. What is benefit of driving at an economical speed? 3. Should a child below 18years be allowed to drive a motorcycle? Give reasons. Q.4. Vikas has been given two lists of problems from his mathematics teacher with the instructions to submit not more than 100 of them correctly solved for marks. The problems in the first list are worth 10 marks each and those in the second list are worth 5 marks each. Vikas knows from past experience that he requires on an average of 4 minutes to solve a problem of 10 marks and 2 minutes to solve a problem of 5 marks. He has other subjects to worry about; he cannot devote more than 4 hours to his mathematics assignment. With reference to manage his time in best possible way how many problems from each list shall he do to maximize his marks? What is the importance of time management for students? Q.5. An NGO is helping the poor people of earthquake hit village by providing medicines. In order to do this they set up a plant to prepare two medicines A and B. There is sufficient raw material available to make bottles of medicine A and bottles of medicine B but there are bottles into which either of the medicine can be put. Further it takes 3 hours to prepare enough material to fill 1000 bottles of medicine A and takes 1 hour to prepare enough material to fill 1000 bottles of medicine B and there are 66 hours available for the operation. If the bottle of medicine A is used for 8 patients and bottle of medicine B is used for 7 patients. How the NGO should plan his production to cover maximum patients? How can you help others in case of natural disaster? Q.6 A retired person has Rs. 70,000 to invest in two types of bonds. First type of bond yields 10% per annum. As per norms he has to invest minimum to Rs. 10,000 in first type and not more than Rs. 30,000 in second type. How should he plan his investment so as to get maximum return after one year of investment? What values have to be inculcated by a person for a peaceful retired life.

79 Q.7 A company manufactures two types of stickers A: SAVE ENVIRONMENT and B: BE COURTEOUS. Type A requires five minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There 3 hours and 20 minutes available for cutting and 4 hours available for assembling in a day. He earns a profit of Rs. 50 on each type A and Rs. 60 on each type B. How stickers of each type should company manufacture in a day of each type should company manufacture in a day to maximize profit? Give your views about SAVE ENVIRONMENT and BE COURTEOUS Q.8 Suppose every gram of wheat produces 0.1 g of protein and 0.25 g of carbohydrates and corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat cost Rupees 25 and rice Rs.100 per kilogram. The minimum daily requirements of proteins and carbohydrates for an average man are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in daily diet to provide minimum daily requirements or proteins of carbohydrates at minimum cost, assuming that wheat and rice are to be taken in a diet? What is your opinion about healthy diet? PROBABILITY Q.1 Probability of winning when batting coach A and bowling coach B working independently are ½ and ⅓ respectively. If both try for the win independently find the probability that there is a win. Will the independently working may be effective? And why? Q.2. Q.3. A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike and 0.32 that the construction job will be completed on time if there is strike. Determine the probability that the construction job will be completed on time. What values are driven by this question? A clever student used a biased coin so that the head is 3 times as likely to occur as tail. If the coin tossed twice find the probability distribution and mean of numbers of tails. Is this a good tendency? Justify your answer. Q.4 A man is known to speak truth 5 out of 6 times. He draws a ball from the bag containing 4 white and 6 black balls and reports that it is white. Find the probability that it is actually white? Do you think that speaking truth is always good? Q.5 A drunkard man takes a step forward with probability 0.6 and takes a step backward with probability 0.4. He takes 9 steps in all. Find the probability that he is just one step away from the initial point. Do you think drinking habit can ruin one s family life? Q.6. If group A contains the students who try to solve the problem by knowledge, Group B contains the students who guess to solve the problem Group C contains the students who give answer by cheating. If n (A) = 20, n (B) = 15, n(c) = 10, 2 Students are selected at random. Find the probability that they are from group c. Do you think that cheating habit spoils the career?

80 Q.7 In a school, 30% of the student has 100% attendance. Previous year result report tells that 70% of all students having 100% attendance attain A grade and 10% of remaining students attain A grade in their annual examination. At the end of the year, One student is chosen at random and he has an A grade. What is the probability that the student has 100% attendance? Also state the factors which affect the result of a student in the examination. Q.8 A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is six. Find the probability that it is actually a six. Write any three benefits of speaking the truth. Q.9. There are 20 People in a group. Out of them 7 people are non vegetarian, 2 people are selected randomly. Write the probability distribution of non vegetarian people. Explain whether you would like to be vegetarian or non- vegetarian and why? Also keeping life of animals in mind how would you promote a person to be vegetarian? Q.10 Two third of the students in a class are sincere about their study and rest are careless Probability of passing in examination are 0.7 and 0.2 for sincere and careless studentsrespectively, A Student is chosen and is found to be passed what is the probability that he/she was sincere. Explain the importance of sincerity for a student. Q.11. A company has two plants of scooter manufacturing. Plant I manufacture 70% Scooter and plant II manufactures 30%. At plant I 80% of the scooter s are maintaining pollution norms and in plant II 90% of the scooter maintaining Pollution norms. A Scooter is chosen at random and is found to be fit on pollution norms. What is the probability that it has come from plant II. What is importance of pollution norms for a vehicle? Q. 12 A chairman is biased so that he selects his relatives for a job 3 times as likely as others. If there are 3 posts for a job. Find the probability distribution for selection of persons other than their relatives. If the chairman is biased than which value of life will be demolished? Q.13 A manufacturer has three machine operators A (skilled) B (Semi- skilled) and C (nonskilled).the first operator A Produces 1% defective items where as the other two operators B and C produces 5% and 7 % defective items respectively. A is on the job for 50% of time B in the job for 30% of the time and C is on the job for 20 % of the time. A defective item is produced what is the probability that it was produced by B? What is the value of skill? Q.14 In a group of 100 families, 30 families like male child, 25 families like female child and 45 families feel both children are equal. If two families are selected at random out of 100 families, find the probability distribution of the number of families feel both children are equal. What is the importance in the society to develop the feeling that both children are equal? Q.15 In a group of 200 people, 50% believe in that anger and violence will ruin the country, 30% do not believe in that anger and violence will ruin the country and 20% are not sure about anything. If 3 people are selected at random find the probability that 2 people believe and 1 does not believe that anger and violence will ruin the country. How do you consider that anger

81 and violence will ruin the country? Q.16 In a group of students, 200 attend coaching classes, 400 students attend school regularly and 600 students study themselves with help of peers. The probability that a student will succeed in life who attend coaching classes, attend school regularly and study themselves with help of peers are 0.1, 0.2 and 0.5 respectively. One student is selected who succeeded in life, what is the probability that he study himself with help of peers. What type of study can be considered for the success in life and why? Q.17 Ramesh appears for an interview for two posts A and B for which selection is independent.the probability of his selection for post A is 1/6 and for post B is 1/7. He prepared well for the two posts by getting all the possible information. What is the probability that he is selected for at least one of the post? Which values in life he is representing? Q.18 Past experience shows that 80% of operations performed by a doctor are successful. If he performs 4 operations in a day, what is the probability that at least three operations will be successful? Which values are reflected by the doctor? Q.19 A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise it is rejected. Find the probability that the box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale. Q.20 In answering a multiple choice question test with four choices per question, a student knows the answer, guesses or copies the answer. If ½ be the probability that he knows the answer, 1/4 be the probability he guesses and ¼ that he copies it.assuming that a student who copies the answer will be correct with the probability 3/4, what is the probability that the student knows the answer given that he answered it correctly? Mehul does not know the answer to one of the question in the test. The evaluation process has negative marking. Which value would Mehul violate if he restores to unfair means? Q.21 In a class, having 60% boys, 5% of the boys and 10% of the girls have an IQ of more than 150. A student is selected at random and found to have an IQ of more than 150. Find the probability that the selected student is a boy. It has been seen that students with not high IQ have also performed well. What values have been inculcated by the student? **********************