Sample Question Paper (With value based questions) Issued by CBSE for 2013 Examination

Transcription

1 ample Question Paper (With value based questions) Issued by C for 03 xamination Mathematics Class XII lue Print. No. Topics V L Total. (a) Relations and Functions () (b) Inverse Trigonometric Functions () () 0(). (a) Matrices () 6() (b) Determinants () () 3(5) 3. (a) Continuity and Differentiability () (3) (b) pplications of Derivatives 6() () (c) Integration (3) (d) pplication of Integrals 6() (e) Differential quations () 6(). (a) Vectors () () (b) 3-dimensional Geometry () () 6() 7(6) 5. Linear Programming 6() 6() 6. Probability () 6() 0() Total 0(0) 8() (7) 00(9) Note: Number of questions are given within brackets and marks outside the brackets. The Question Paper will include question(s) based on values to the extent of 5 marks.

2 C ample Question Paper 03 (with Value ased Questions) Time allowed: 3 hours Maximum marks: 00 General Instructions:. ll questions are compulsory.. The question paper consists of 9 questions divided into three ections, and C. ection comprises of 0 questions of one mark each; ection comprises of questions of four marks each; and ection C comprises of 7 questions of six marks each. 3. ll questions in ection are to be answered in one word, one sentence or as per the exact requirement of the question.. There is no overall choice. However, internal choice has been provided in questions of four marks each and questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. se of calculator is not permitted. You may ask for logarithmic tables, if required. CTIN Question numbers to 0 carry mark each.. sing principal values, write the value of cos 3 sin.. valuate tan cos sin. 0 0 x 3. Write the value of x y z, if 0 0 y. 0 0 z 0. If is a square matrix of order 3 such that adj = 5, find 5. cos sin Write the inverse of the matrix sin cos 6. The contentment obtained after eating x-units of a new dish at a trial function is given by the Function C(x) = x 3 + 6x + 5x + 3. If the marginal contentment is defined as rate of change of (x) with respect to the number of units consumed at an instant, then find the marginal contentment when three units of dish are consumed. d y d y dy 7. Write the degree of the differential equation 0. dx dx dx

3 C ample Question Paper (iii) 8. If a and b are two vectors of magnitude 3 and, respectively such that a b is a unit vector, 3 write the angle between a and b. 9. If a 7 i j k and b i 6 j 3 k, find the projection of a on b. 0. Write the distance between the parallel planes x y 3z and x y 3z 8. CTIN Question numbers from to carry marks each.. Prove that the function f : N N, defined by f( x) x x is one one but not onto.. how that sin [cot {cos(tan x)}] x x R x x x olve for x: 3 sin cos tan x x x Two schools and decided to award prizes to their students for three values: honesty (x), punctuality (y) and obedience (z). chool decided to award a total of `,000 for these three values to 5, and 3 students, respectively, while school decided to award `0,700 for these three values to, 3 and 5 students, respectively. If all the three prizes together amount to `,700 then (i) Represent the above situation by a matrix equation and form Linear equations by using matrix multiplication. (ii) Is it possible to solve the system of equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?. If x a( sin ) and y a( cos ), find d y. dx sin x d y 5. If y, show that ( x ) 3x dy y 0. x dx dx x ax b, 0 x 6. The function f( x) is defined as f( x) 3x, x ax 5b, x 8 find the values of a and b. R Differentiate tan x x x x with respect to cos x.. If f( x) is continuous on [0, 8],

4 (iv) Xam idea Mathematics XII 3 x x 7. valuate: dx. x x x valuate: e ( sin ) ( cos x) dx. 8. valuate: x dx. ( x )( x ) 9. valuate: log( tan x) dx, using properties of definite integrals. R 0 0. Let a i 5 j k, b i j 5 k and c 3 i j k. Find a vector d which is perpendicular to both a and b and satisfying d. c.. Find the distance between the point P(6, 5, 9) and the plane determined by the points (3,, ), (5,, ), and C(,, 6) R Find the equation of the perpendicular drawn from the point P(,, ) to the line x 5 y 3 z 6. lso, write down the coordinates of the foot of the perpendicular from 9 P to the line.. There is a group of 50 people who are patriotic out of which 0 believe in non violence. Two persons are selected at random out of them, write the probability distribution for the selected persons who are non violent. lso find the mean of the distribution. xplain the importance of non violence in patriotism. CTIN C Question numbers from 3 to 9 carry 6 marks each If 3, find. Hence solve the following system of equations: 3 3 x y 3z ; x 3y z and 3x 3y z x 7. Find the equations of tangent and normal to the curve y at the point where it ( x )( x 3) cuts the x-axis. R Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone. 5. Find the area of the region which is enclosed between the two circles x y and ( x ) y.

5 C ample Question Paper (v) 6. Find the particular solution of the differential equation: ( x sin y) dy (tan y) dx 0: given that y 0 when x Find the vector and Cartesian equations of the plane containing the two lines r ( i j 3 k ) ( i j 5 k ) and r ( 3i 3j k ) ( 3i j 5k ). 8. dealer in rural area wishes to purchase a number of sewing machines. He has only `5, to invest and has space for at most 0 items. n electronic sewing machine costs him ` and a manually operated sewing machine `0.00. He can sell an lectronic ewing Machine at a profit of `.00 and a manually operated sewing machine at a profit of `8.00. ssuming 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 solve it graphically. Keeping the rural background in mind justify the values to be promoted for the selection of the manually operated machine. 9. In answering a question on a MCQ test with choices per question, a student knows the answer, guesses or copies the answer. Let be the probability that he knows the answer, be the probability that he guesses and that he copies it. ssuming that a student, who copies the answer, will be correct with the probability 3, what is the probability that the student knows the answer, given that he answered it correctly? rjun does not know the answer to one of the questions in the test. The evaluation process has negative marking. Which value would rjun violate if he resorts to unfair means? How would an act like the above hamper his character development in the coming years? R n insurance company insured 000 cyclists, 000 scooter drivers and 6000 motorbike drivers. The probability of an accident involving a cyclist, scooter driver and a motorbike driver are 0.0, 0.03 and 0.5 respectively. ne of the insured persons meets with an accident. What is the probability that he is a scooter driver? Which mode of transport would you suggest to a student and why? olutions CTIN 7. cos 3 sin tan cos sin tan cos 3 tan = tan ( )

6 (vi) Xam idea Mathematics XII 3. Given 0 0 x 0 0 y 0 0 z 0 x y z 0 x, y, z 0 x y z 0 0. We know that adj = n and cos sin 5. Let = sin cos Then, = cos sin cos and adj sin cos sin sin cos 6. M( X) C ( X) 3x x 5 sin cos M( X)( at X 3) = 68 units a b = a b sin n a b = a b sin 3 sin 3 sin or 6 9. Projection of a on b = b. a 6 8 b Given parallel planes are x y 3z...(i) and x y 3z 8...(ii) Let P(x, y, z) be a point on plane (i) x y 3z Now the distance 'd' from P( x, y, z ) to plane (ii) is given as x y 3z 8 d 9 8 d. f( x) x x CTIN Let x, y N such that f( x) f( y)

7 C ample Question Paper (vii) x x y y x y x y 0 ( x y)( x y ) 0 ( x y) 0 [s x y 0 any x, y N] x y f is one-one function Clearly f( x) x x 3 for x N / ut f( x) does not assume values and f: N N is not onto function. cos(tan ) cos x cos x x cot sin sin x x ( x ) sincot sin sin x R x x x x x x Let x tan / LH = 3 sin tan tan tan cos tan tan tan tan = 3 sin (sin ) cos (cos ) tan (tan ) = 3 8 tan x tan tan x x 3 6 x x (i) The given situation can be represented as follows: 3 5 y 0700 z 700 or 5x y 3z 000 x 3y 5z 0700 x y z (ii) Let 3 5 +

8 (viii) Xam idea Mathematics XII 5( ) ( ) 3( ) exists, so equations have a unique solution. / (iii) ny answer of three values with proper reasoning which will be considered correct. [For example, I prefer the value "punctuality" for rewards, because a punctual students can study better.]. x a( sin ) dx a( cos ) a sin d y a( cos ) dy a.sin a sin cos d dy a sin.cos cot dx a sin / d y d cosec.. cosec dx dx sin a sin a 5. y sin x x. y sin x x / Differentiating both sides w.r.t x, we get dy y( x) x dx x x dy ( x ) xy 0 dx / Differentiating again w.r.t x, we get ( d y ) x. 0 y x dy dx dx dx d y ( x ) 3x y 0 dx dx 6. lim f( x) lim x ax b a b x x lim f( x) lim ( 3x ) 8 x x s f is continuous at x a b 8 a b...(i) imilarly as f is continuous at x, lim f( x) lim f( x) x x lim ( 3x ) lim ( ax 5 b), 8a 5b...(ii) x x olving (i) and (ii), we get a 3, b

9 C ample Question Paper (ix) R x x y tan, Let x cos x x cos cos cos tan tan cos cos sin cos sin tan tan tan tan tan y cos x : Let z cos x dy y z dz 7. 3 x x x I = dx x dx x ( x )( x ) x Let x x x 3 + =, =, ( )( ) x 3 dx dx x I = x dx x x C x 3 log( ) log( ) + x R x x x e ( sin cos ) x x e sin x dx cos x dx sin x x sin cos x x x = e dx e x cosec cot dx sin / x x x e dx e x cot cosec dx / x cot. e x e x x x x cosec. dx e cosec dx e x e x x cot dx e cosec cosec dx C x x e cot C 8. I = x dx ( x )( x ) dt Let x t, x dx dt I = ( t )( t )

10 (x) Xam idea Mathematics XII Let + = 0 ( t )( t ) t t on solving, we get, dt dt I = t t C t log log t log x log x C / 9. Let I = log( tan x) dx or 0 log tan tan 0 x dx = log x tan x dx 0 I = log log log( tan )] tan x dx x dx (i)...(ii) dding (i) and (ii), we get I = log dx log 0 I = 8 0. Let d xi yj zk log s d a and d b d. a 0 and d. b 0 d. a 0 x 5y z 0 and d. b 0 x y 5z 0...(i) d. c 3x y z...(ii) / olving (i) and (ii), we get x 7, y 7, z 7 d i 7j 7k /. The equation of plane passing through (3,, ), (5,, ) and (,, 6) is x 3 y z x 3 y z or ( x 3)( ) ( y )( 6) ( z )( ) 0 x 6y z 76 or 3x y 3z 9...(i) / Length of from (6, 5, 9) to (i) is

11 C ample Question Paper (xi) R ny point on the line x 5 y 3 z 6 is ( 5, 3; 9 6) 9 P (,, ) x + 5 Q = y + 3 = z 6 9 Let (,, ) is the foot of the perpendicular drawn from point P(,, ) to the given line x 5 y 3 z 6 (i) 9 ince, (, ) be on line (i) ( say) 9 5, 3, 9 6 PQ ( 5 ) i ( 3 ) j ( 9 6 ) k ( 7) i ( 7) j ( 9 7) k PQ line (i) ( 7). ( 7). ( 9 7) The pt. Q is (,, 3) / quation of PQ is x y z and foot of is (,, ) 6 3. Let X denote the number of non-violent persons out of selected two. X can take values 0,, non-violent 0: Violent patriotism: 30 / P(X = 0) = / P(X = ) = / P(X = ) = /

12 (xii) Xam idea Mathematics XII Probability distribution is X 0 P(x) Mean = Importance: In order to have a peaceful environment both the values are required patriotism and non-violence because of patriotism with violence could be very dangerous. CTIN C 3 3. The given matrix is 3, = = exists dj = 5 8, The given system of equations can be written as X = 3 x Where = 3, X = y, 3 3 z X x 3, y, z /. The given curve cuts the x-axis at x = 7, and y = 0 / x 7 dy ( x 5x 6) ( x 7)( x 5) y x 5x 6 dx ( x 5x 6) dy ( ) ( 0) ( at x 7) dx ( ) quation of tangent to the curve at (7, 0) is y 0 ( x 7) or x 0y quation of normal to the curve at (7, 0) is y 0 0( x 7) 0x y 0 0

13 C ample Question Paper (xiii) R Let x and r be radius of base of cylinder and cone respectively Let C = x, V ~ D V D h ( r x ) V h D D r Let be the curved surface area of cylinder. xh x h r x h ( ) rx x r r ' ' ' [ ] h d h d s h r x h' ( ), 0 dx r dx r is maximum C D d 0 r dx x r x r is is maximum when x, i.e., when radius of base of cylinder is half the radius of base of cone. 5. n solving the equations of the two circles, we get points of intersection as, D, 3 3 and x + y = Y, ( 3 ) (x ) + y = X X Y D, ( 3 ) rea of shaded region = (rea C) ( x ) dx x dx 0 ( x ) x x ( x ) sin x sin x 0

14 (xiv) Xam idea Mathematics XII 3 3 sin sin ( ) sin ( ) sin The given differential equation can be written as dx (cot y) x cos y dy I.F. = e cot y dy log sin y e sin y 3 sq. units The solution is x sin y = sin y cos y dy C sin y dy C or x sin y cos y C It is given that y 0, when x 0 C 0 C x sin y ( cos y) sin y x sin y is the reqd. solution 7. Here a i j 3k and a 3i 3j k b i j 5k and b 3i j 5k i j k n b b 5 0i 0j 8k 3 5 Vector equation of the required plane is ( r a). n 0 or r. n a. n or r.( 0 i 0 j 8 k ) ( i j 3 k ).( 0 i 0 j 8 k ) r.( 0 i 5 j k ) 37 The cartesian equation of plane is 0x 5y z uppose number of electronic operated machine = x and number of manually operated sewing machines = y /

15 C ample Question Paper (xv) Y 8 (0, ) 0 (0, 0) 6 P(8, ) 8 (6, 0) X X Y 3x + y = 8 x + y = 0 x y 0...(i) and, 360x 0y 5760 or 3x y 8...(ii) x 0, y 0 To maximise Z x 8y Corners of feasible region are ( 0, 0), P( 8, ), ( 6, 0) Z / Z P Z 35 Z is maximum at x 8 and y The dealer should invest in 8 electric and manually operated machines. Keeping the save environment factor in mind the manually operated machine should be promoted so that energy could be saved. 9. Let be the event that he knows the answer, be the event that he guesses and C be the event that he copies. / Then P( ), P( ) and P( C) / Let X be the event that he has answered correctly. lso, P X P X, and P X C 3

16 (xvi) Xam idea Mathematics XII Thus, required probability = P P X P( ) X P X P( ) P X P ( ) P X P( C) C If rjun copies the answer, he will be violating the value of honesty in his character. He should not guess the answer as well as that may fetch him negative marking for a wrong guess. He should accept the question the way it is and leave it unanswered as cheating may get him marks in this exam but this habit may not let him develop an integrity of character in the long run. Let the events defined are : Person chosen is a cyclist : Person chosen is a scooter-driver R 3 : Person chosen is a motorbike driver / : Person meets with an accident / P( ) = 6, P( ) = 3, P( 3) = P = 0.0, P = 0.03, P = 0.5, P = Required 3 P P. P( ) P P P. ( ). P ( ) P. P( 3 ) uggestion: Cycle should be promoted as it is good for / (i) Health / (ii) No pollution / (iii) aves energy( no petrol) /

17 Value ased Questions Mathematics XII. veryone wants to be a perfect ideal human being. Let us assume that dishonesty is one of the factors that affects our perfectness and perfectness has an inverse square relationship with dishonesty. For any value x of level of dishonesty we have a unique value y of perfection. (i) Write down the equation that relates y with x. (ii) Is this relationship from x X ( 0, ) to y ( 0, ), forms a function? (iii) For what level of dishonesty one can achieve th level of perfection? (iv) What will be the change in level of perfection when the level of dishonesty changes from to? ol. (i) y x 0, x (ii) Yes (iii) When y, we have x x (iv) When x, y 6 x x, but x can not be ve When x, y Change in level of perfection = = trust fund has `30,000 that is to be invested in two different types of bonds. The first bond pays 5% p.a. interest which will be given to orphanage and second bond pays 7% interest p.a. which will be given to financial benefits of the trust. sing matrix multiplication, determine how to divide `30,000 among two types of bonds if the trust fund obtains an annual total interest of `800. (i) What are the values reflected in the question? (ii) Why is it required to help orphan children? ol. Let `x be invested in Ist bond, then `30,000 x will be invested in IInd bond. Total interest = `800

18 (xviii) Xam idea Mathematics XII V L D Q T I N ol. 5 Now, [ x x] 00 [ 800] x 7 ( x) x x = x = x 5000 For investment in IInd bond, amount = = o, equal amount is invested in both of the bonds. (i) Values reflected are helping poor and needy children. Provided that the interest rate in financial benefits (IInd bond) is more than the Ist bond (money given to orphanage) trust decides to invest fund equally. It reflects that the motive of the trust is not to only to earn the interest but also to help the needy orphan children. This charity should be a concern of every one. (ii) The children living in orphanage are also talented and possess potential. If they are given the proper brought up and opportunity, they can contribute to the development of the society and country and will become good citizens. 3. f the students in a school; it is known that 30% has 00% attendance and 70% students are irregular. Previous year results report that 70% of all students who has 00% attendance attain grade and 0% irregular students attain grade in their annual examination. t the end of the year, one student is chosen at random from the school and he has grade. What is the probability that the student has 00% attendance? (i) Write any two values reflected in this question. (ii) Is regularity required only in school? Justify your answer Let : tudent has 00% attendance : tudent is irregular : tudent attains grade : tudent attains grade given that she has 00% attendance : tudent attains grade given that she is irregular : tudent has 00% attendance given that she attains grade sing ayes theorem

19 Value ased Questions ol. P P P ( ). P P ( ). P( ).P (i) Regularity and intelligence (ii) Regularity is the value which is required at every stage of our life. In our childhood, during school education we can inculcate this value in our personality. Regularity increases our capabilities and makes us able to put the best of our potential. We are able to achieve certain targets due to regular efforts.. In a survey of 0 richest person of three residential society,, C it is found that in society, 5 believes in honesty, 0 in hard work, 5 in unfair means while in, 5 in honesty, 8 in hard in work, 7 in unfair means and in C, 6 in honesty, 8 in hard work, 6 in unfair means. If the per day income of 0 richest persons of society,, C are `3,500, 30,500, 3,000 respectively, then find the per day income of each type of people by matrix method. (i) Which type of people has more per day income. (ii) ccording to you, which type of person is better for country. Let x, y, z be the per day income of person believing in honesty, hard work and unfair means, respectively. The given situation can be written in matrix form as X =, Where x X y, z 3000 X X Now for = lso, C ( ) ( 8 56) (i) = 5(8 56) 0(30 ) + 5(0 8) = 0 0 (xix) V L D Q T I N

20 (xx) Xam idea Mathematics XII V L D Q T I N 5 7 C ( ) ( 30 ) C 3 ( ) ( 0 8) C ( ) ( 60 0) C ( ) ( 30 30) C 3 ( ) ( 0 60) C 3 ( ) ( 70 0) C 3 ( ) ( 35 5) C 33 ( ) ( 0 50) T dj () = adj( ) 8 0 = = Putting the value of X,, in (i) we get 3 x y z x 500 y 000 z x 000 y 000 z 3000

21 Value ased Questions x 500, y 000, z 000 Hence, per day income of person who believe in honesty = `,500 Per day income of person who believe in hard work = `,000 Per day income of person who believe in unfair means = `,000 (i) person, who believe in hard work has more per day income. (ii) person, who believe in hard work and honesty, are better for country. 5. The male-female ratio of a village increases continuously at the rate proportional to the ratio at any time. If the ratio of male-female of the villages was 000 : 980 in 999 and 000 : 950 in 009, what will be the ratio in 09? (i) Why gender equality is value for society? (ii) What should society do to reduce the male-female ratio to? ol. Let male-female ratio at any time be r. dr dr Given r kr where k is the constant of proportionality. dt dt We have dr k dt r Integrating both sides we get log r kt log c where log c is the constant of integration. log r log c kt log r c r c e kt kt...(i) Let us start reckoning time from the year 999 for this problem o in 999, t 0 and r ubstituting in (i) we get 50 0 c. e c (i) becomes r 50 9 e kt lso in the year 009, t 0 and r ubstituting in (ii) we get k e 9 9 ubstituting in (ii) we get k 98 e 95...(ii) (xxi) V L D Q T I N

22 (xxii) V L D t t 50 0k r e ( )...(iii) In the year 09, t 0 r ~. 085 ~ 085 : 000 Thus in the year 09, the male-female ratio will be 085 : 000 (i) Gender equity promotes economic growth, reduce fertility, child mortality and under nutrition. (ii) (a) top female-foeticide. (b) mpower women to realise their rights. (c) Provide special opportunities to women to come at par with men in all walks of life. 6. window is in the form of rectangle surmounted by a semi-circular opening. Total perimeter of the window is 0 m. What will be the dimensions of the whole opening to admit maximum light and air? (i) How having large windows help us in saving electricity and conserving environment? (ii) Why optimum use of energy is required in the Indian context? Xam idea Mathematics XII Q T I N ol. Let CD be required window having length x and width y. If is the area of window. Then xy x x( 0 x x) x 0x x x x Given, Perimeter = 0 x y y x 0 y 0 x x 0x x x 0x x bviously, window will admit maximum light and air if its area is maximum. d Now, x 0 dx D y x C

23 Value ased Questions (xxiii) ol. For maxima or minima of d dx 0 0 x 0 0 x( ) 0 0 d x ( ) < 0 dx 0 For maximum value of, x and thus y 0 Therefore, for maximum area i.e., for admitting maximum light and air, 0 Length of rectangular part of window = x 0 Width = (i) Large windows allow more light during daytime and hence will reduce the use of electricity. aving energy (electricity) helps in conserving environment as electricity is produced by using natural resources which we should conserve for the sake of future generation. (ii) India is the nd most populated country in the world so have more consumers of energy but less sources of its production. Therefore, in Indian context energy saving is like energy production. 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 centimeter of gas per second. Find the rate at which the radius of the balloon is increasing when its radius is 5 cm. (i) Which values have been reflected in this question? (ii) Why child labour is not good for society? Let r be the radius and V be the volume of the balloon. Then dv dt 900 cm3 /sec dr?, when r = 5 dt V 3 r 3 Differentiating both sides w.r.t t dv dr 3r dt 3 dt 900 ( 5) dr dr dt dt cm/sec V L D Q T I N

24 (xxiv) Xam idea Mathematics XII V L D Q T I N ol. (i) Three values reflected are bravery, sympathy for child labour and raising voice against child labour. (ii) We know that child labour is illegal and harmful to both society and country. We should spread awareness in society so that child labour should be abolished. In the childhood they should be sent to school for their education so that they can contribute for the development of the society. 8. manufacturing company makes two type of teaching aids and of mathematics of class XII. ach type of requires 9 labour hours for fabricating and labour hour for finishing. ach type of requires labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 80 and 30, respectively. The company makes a profit of `80 on each piece of type and `0 on each piece of type. How many pieces of type and should be manufactured per week to get a maximum profit? What is the maximum profit per week? Is teaching aid necessary for teaching learning process? If yes, justify your answer. Let x and y be the number of pieces of type and manufactured per week respectively. If Z be the profit then, bjective function, Z = 80x 0y We have to maximize Z, subject to the constraints 9x y 80 3x y 60...(i) x 3y 30...(ii) x 0, y 0...(iii) The graph of constraints are drawn and feasible region C is obtained, which is bounded having corner points ( 0, 0), ( 0, 0), (, 6 ) and C ( 0, 0) 3x + y = 60 x + 3y = 30 X Y Y C(0, 0) (, 6) (0, 0) X

25 Value ased Questions (xxv) ol. Now the value of objective function is obtained at corner point as Corner point Z = 80x 0y (0, 0) 0 (0, 0) 600 (, 6) 680 Maximum C (0, 0) 00 Hence, the company will get the maximum profit of `,680 by making pieces of type and 6 pieces of type of teaching aid. Yes, Teaching aid is necessary for teaching learning process as (i) it makes learning very easy. (ii) it provides active learning. (iii) students are able to grasp and understand concept more easily and in active manner. 9. village has 500 hectares of land to grow two types of plants, X and Y. The contribution of total amount of oxygen produced by plant X and plant Y are 60% and 0% per hectare respectively. To control weeds, a liquid herbicide has to be used for X and Y at rates of 0 litres and 0 litres per hectare, respectively. Further no more than 8000 litres of herbicides should be used in order to protect aquatic animals in a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total production of oxygen? (i) How do you think excess use of herbicides affect our environment? (ii) What are the general implications of this question towards planting trees around us? Let plants X and Y be grown in x and y hectares. o, x 0 and y 0 x y (i) Contribution of oxygen by the plants = 60% of x + 0% of y 6x y z 0. 6x 0. y 0 0 lso, mount of liquid herbicides required = ( 0x 0y) litres Given 0x 0y 8000 x y (ii) The LPP for given problem is Maximum, Z 0. 6x 0. y.t. x y (iii) and x y (iv) x, y 0 V L D Q T I N

26 (xxvi) V L D Xam idea Mathematics XII ketching a graph for the above LPP, we get the region shown in the figure Y (0, 800) (0, 500) (0, 0) (00, 0) C (500, 0) X Q T I N olving x y 500 and x y 800, we get ( 300, 00) Corner point x + y = 800 Value of the optimizing function (0, 0) Z = = 0 (00, 0) Z = = 0 (300, 00) Z = = = 60 x + y = 500 (0, 500) Z = = 00 Maximum Production of oxygen will be achieved when plant X is planted in 300 hectare and plant Y is planted in 00 hectare. (i) xcess herbicide will get absorbed in the soil and may contaminate the water source also. Thus it can affect the whole ecosystem. (ii) Care should be taken while planting trees that the variety of the plants is such that they provide more oxygen for our environment.

27 Value ased Questions (xxvii) 0. In shop, 30 tin pure ghee and 0 tin adulterated ghee are kept for sale while in shop, 50 tin pure ghee and 60 tin adulterated ghee are there. ne tin of ghee is purchased from one of the shop randomly and it is found to be adulterated. Find the probability that it is purchased from shop. ol. (i) How adulteration is dangerous for humanity? (ii) What you can do against adulteration? Let the event defined be as = election of shop. = election of shop. = Purchasing of a tin having adulterated ghee. P( ), P( ) P 0, P P required P P P ( ). P P ( ). P( ). P (i) dulteration is dangerous as it is harmful for user s health. (ii) To prevent adulteration, we should spread awareness against it in society.. In a self-assessment survey 60% persons claimed that they never indulged in corruption, 0% persons claimed that they always speak truth and 0% say that they neither indulged in corruption nor tell lies. person is selected at random out of this group. (i) What is the probability that the person is either corrupt or tells lie? (ii) If the person never indulged in corruption, find the probability that she/he tells, truth. (iii) If the person always speaks truth find the probability that she/he claims to have never indulge in corruption. (iv) What values have been discussed in this question? V L D Q T I N

28 (xxviii) Xam idea Mathematics XII V L D Q T I N ol. (v) Why is it must for all to practice values in our life? Let : et of persons never indulged in corruption : et of persons always speak truth Then, P( ) 60 00, P ( 0 ) 00 and P( ) 0 00 (i) P(ither or ) = P( ) P( ) P( ) P( ) (ii) P P( ) 00 P( ) (iii) P P( ) 00 P( ) 0 00 (iv) The following values have been discussed (a) We should never indulge in any type of corruption. (b) We should never tell lies i.e., we should always speak truth. (v) Values contribute to intellectual development, use of abilities, achieve creativity, personal development and development of society.