maths value based question bank for class xii 2015-16

Prepared by: M. S. KumarSwamy, TGT(Maths) Page - A -

MATHEMATICS

VALUE BASED QUESTIONS

QUESTION BANK

for

CLASS – XII

2015 – 16

CHAPTER WISE COVERGAE OF EXPECTED

VALUE BASED QUESTIONS

Prepared by

M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed.

Kendriya Vidyalaya GaCHIBOWLI

Prepared by: M. S. KumarSwamy, TGT(Maths) Page - B -

PREFACE

It gives me great pleasure in presenting the Question Bank for Value Based

Questions for Class XII Mathematics for March 2016 Board Examination. It is in strictly

according to the latest guidelines issued by CBSE.

The CBSE has introduced the concept of value based questions in question

papers of all main subject from the session 2012-13. Total 3-5 marks questions will be

value based from this year onward in CBSE question papers. These questions are based

on values & key messages brought out on the basis of prescribed text. Students have the

liberty to go beyond the set text.

I avail this opportunity to convey my sincere thanks to respected sir, Shri S. Vijay

Kumar, Director, KVS ZIET Gwalior, respected madam Smt. R. Kalavathi, Deputy

Commissioner, KVS RO Hyderabad, respected sir Shri Isampal, Deputy Commissioner, KVS

RO Bhopal, respected sir Shri P. V. Sairanga Rao, Deputy Commissioner, KVS RO

Varanasi, respected sir Shri P. Deva Kumar, Deputy Commissioner, KVS RO Ahmedabad,

respected sir Shri Y. Arun Kumar, Assistant Commissioner(Acad), KVS Headquarter, New

Delhi, respected sir Shri Sirimala Sambanna, Assistant Commissioner, KVS RO Hyderabad,

respected sir Shri. K. L. Nagaraju, Assistant Commissioner, KVS RO Bangalore, respected

sir Shri.Gangadharaiah, Assistant Commissioner, KVS RO Bangalore and respected Shri

M.K. Kulshreshtha, Assistant Commissioner, KVS RO Chandigarh for their blessings,

motivation and encouragement in bringing out this notes in such an excellent form.

I also extend my special thanks to respected sir Shri. P. S. Raju, Principal, KV

Gachibowli, respected Smt. Nirmala Kumari M., Principal, KV Mysore & respected Shri. M.

Vishwanatham, Principal, KV Raichur for their kind suggestions and motivation while

preparing this Question Bank.

I would like to place on record my thanks to respected sir Shri. P. K. Chandran,

Principal, presently working in KV Bambolim. I have started my career in KVS under his

guidance, suggestions and motivation.

Inspite of my best efforts to make this notes error free, some errors might have

gone unnoticed. I shall be grateful to the students and teacher if the same are brought to

my notice. You may send your valuable suggestions, feedback or queries through email

to [email protected] that would be verified by me and the corrections would be

incorporated in the next year Question Bank.

M. S. KUMARSWAMY

mailto:[email protected]

Prepared by: M. S. KumarSwamy, TGT(Maths) Page - C -

DEDICATED

TO

MY FATHER

LATE SHRI. M. S. MALLAYYA

Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1 -

RELATIONS AND FUNCTIONS 1. Let L be the set of all the people in a locality. Let relation R in set L is defined by

(x, y) R x and y are like minded. Do you think R is an equivalence relation ? There are equivalence classes under the head Honest, Punctual, Care for environment, Late comers. For which equivalence class you would like to be its member ?

2. Let A be the set of all human beings in a town at a particular time. The relation R

is given by (x, y) R x and y are associated with ‘SAVE THE RARE SPECIES’ campaign. Find whether the relation R is reflexive, symmetric and transitive ? Do you want to campaign for the above mentioned cause ?

3. Let A be the set of 50 students (numbered 1 to 50) who like riding a bicycle and

make environment pollution free. Let f : A N, N the set of natural number defined by f (x) = number of student x. Show that f is one-one but not onto. If we restrict N from 1 to 60, in what way we can make f onto ? Do you want to be a part in making function onto ?

4. Let set A be the set of people of different age groups associated with

‘CLEANLINESS’ drive, Let * be a binary operation of set A defined by a * b = older of (a, b), a, b A. Is * commutative, associative in A ? Are you also a part of set A ?

5. Let L be the set of all the lines in a plane and R be the relation in L defined as R =

{(l1, l2) : l1 is parallel to l2}. Show that R is an equivalence relation. Find all the lines related to y = 3x – 5. Let us relate parallel lines to parallel thinking. Let L be the set of people with same thinking ‘BE HONEST’. Would you like to be member of this equivalence relation ? Give your views in two lines.

6. Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :

P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5 ? Do you think persons having the same ideology form an equivalence relation ?

7. Let T be the set of all triangles in a plane and relation R on T be defined by x R y

x is similar to y, i.e., R = {(x, y) : x is similar to y}. Show that R is an equivalence relation on T. Consider three right triangles x with sides 3, 4, 5; y with sides 5, 12, 13 and z with sides 6, 8, 10. Which triangles among x, y, z are related ? Are the campaign for ‘SAVE ENVIRONMENT’, ‘SAVE NATURAL RESOURCES’, ‘SAVE WILD LIFE’ inter related ?

8. 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?

9. Consider the functions f and g, f: {1, 2, 3} {a, b, c} and g: {a, b, c}

{punctuality, honesty, sincerity} defined as f(1) = a, f(2) = b, f(3) = c, g(a) =


punctuality, g(b) = honesty, g(c) = sincerity. Show that f, g, gof are invertible. If a, b, c are three students who are awarded prizes for the three values given in the function g, which value would you prefer to be rewarded and why?

10. Show that the function f : R+ R+ defined by f(x) = 3x + 4 is an invertible

function. If x represents the number of systematic hours of study that a student puts in and f(x) represents the marks scored by him, from the graph of the above function, which value will be rewarded?

11. Let L be then set of all the lines in a plane and R be the relation in L

defined by R = {(L1, L2) : L1 || L2 }. Show that R is an equivalence relation. L1 represents the ideologies of Gandhiji and L2 represents the ideologies of Netaji Subhash 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?

12. Let Xbe a non-empty set. Let be * a binary operation on the power set

P(X) defined by A * B = A U B. Prove that * is commutative and associative. What is the identity element for the operation ? If X is a set of people in a locality, A is a set of children and B is a set of citizens aged above 75 years in the same locality, is a binary operation for these sets as defined above? What qualities would you suggest that elements of A should have towards elements of B?

13. Consider a relation R in the set A of people in a colony. Defined as aRb iff

a and b are members of joint family. Is R is an equivalence relation? Staying with Grand parents in a joint family imbibes the moral values in us. Can you elicit 2 such values.

14. A is the set of all people in our country India and B is the set of all country

in the world. F is a function from A to B defined by x is patriotic to y, where x A and y B . Which type of function is f ? What is the value point in the question?

15. Let A be the set of all students in class XII of your school. A relation R on

A is defined by xRy if x is honest to y. Is R an equivalence relation on A. 16. Let A be the set of all states in India and B be the set of all languages

spoken in India. A function from A to B is defined by y is the mother tongue of the people belongs to x where x A and y B , which type of function is f? According to this can you state why Hindi is selected as our national language? What is the value point?

17. If Alicia rides a bike at an average of 11.5 mph, the distance that she rides

can be represented by d= t2 +2t+ 6.wheret is the time in hours. a. Find d(1) and d(1.5). b. Interpret the meaning of the function values found in part (a). what is the hazard of rash driving in the road?


18. Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),(3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive. How different relations play a major role in our life?

19. Show that the relation R = { (a,b) : 2 divides ( a – b)} in the set Z of

integers is an equivalence relation. Also give two examples of social relations which are equivalence relations.

20. Let A = Set of boys of XII Class in some school. Let a R b, if and only if a

and b use unfair means in the exam, a,b A (i) Is R a transitive relation? (ii) Should a student use such means? If not why 21. Prove that f : R → R is a bijection given by f (x) = x2 +3. Find f –1 (x). Does

the truthfulness and honesty may have any relation? 22. Set A = {a1, a2, a3, a4, a5} and B = {b1, b2, b3, b4} where ai’s and bi’s are

school going students. Define a relation from a set A to set B by xRy iff y is a true friend of x. If R = {(a1, b1), (a2, b1), (a3, b3), (a4, b2), (a5, b2)}. Is R a bijective function? Do you think that true friendship is important in life? How?

23. If h denotes the number of honest people and p denotes the number of

punctual people and a relation between honest people and punctual people is given as h = p + 5. If P denotes the number of people who progress in life and a relation between number of people who progress and honest people is given as P = (h/8) + 5. Find the relation between number of people who progress in life and punctual people. How does the punctuality important in the progress of life?

24. Let A be the set of all students of class XII in a school and R be the

relation having the same sex (i.e., male or female) on set A, then prove that R is an equivalence relation. Do you think, co-education may be helpful in child development and why?

25. Everyone 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) Does this relationship from (0, )x X to (0, )y , form a function?

(iii) For what level of dishonesty one can achieve 14

th

level of perfection?

(iv) What will be the change in level of perfection when the level of dishonesty changes from 4 to 2?


MATRICES 1. Two schools A and B decided to award prizes to their students for three

values Honesty (x), Punctuality(y) and Obedience (z). School A decided to award a total of Rs. 15,000 for the three values to 4, 3 and 2 students respectively, while school B decided to award Rs. 19,000 for the three values to 5, 4 and 3 students respectively. If all the three prizes together amount to Rs. 5,000, then

(i) Represent the above situation by a matrix equation and form linear equation using matrix multiplication. (ii) Which value you prefer to be rewarded most and why ?

2. A store in a mall has three dozen shirts with ‘SAVE ENVIRONMENT’

printed, two dozen shirts ‘SAVE TIGER’ printed and five dozen shirts with ‘GROW PLANTS’ printed. The cost of each shirt is Rs. 595, Rs. 610 and Rs. 795 respectively. All these items were sold in a day. Find total collection of the store using matrix method. Which shirt you would like to buy and why ?

3. In section A of class XII of a particular school students were asked to

grade each student under values Empathy, Honest, Respectful. Grades are to be given as A, B, C. The following information was received.

Empathy Honest Respectful A 26 20 30 B 12 10 10 C 7 15 5

The same activity was carried out in section B of class XII of the same school and the following information was received.

Empathy Honest Respectful A 30 16 38 B 10 20 7 C 8 12 3

Represent the above information in the matrix form and use matrices to find how many students got A grade in the three values ? Which value according to you is most important to acquire ?

4. 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.270 & 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?


5. 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, 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?

6. 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 pays 7% 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.1800. What are the values reflected in this question?

7. Using matrix method, solve the following system of equations: x + 2y + z =

7, x – y + z = 4, x + 3y +2z = 10. Suppose x represents the number of persons who take food at home, y represents the number of persons who take junk food in market and z represent the number of persons who take food at hotel. Which way of taking food you prefer and why?

8. A school has to reward the students participating in co-curricular activities

(Category I), with 100% attendance (Category II) and 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 students of category III by 2 and add to the number of students of category I to the result, we get 7. By adding II and III category students to three times the I category students, we get 12. Form the matrix equation and, hence solve it.

9. For keeping fit, X people believe 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 diseases who believe in morning walk, yoga and gym respectively. Total no. of such people is 21. If morning walk costs Rs. 0, yoga costs Rs.500/month and gym costs Rs. 400/month and total expenditure is Rs. 23000. (i) Formulate a matrix problem. (ii) Calculate the no. of each type of people. (iii) Why exercise is important for health?

10. 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 schemes 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?


11. 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?

12. Two schools A and B decided to award prizes to their students for three

values honesty (x), punctuality (y) and obedience (z). School A decided to award a total of Rs.11000 for the three values to 5, 4 and 3 students respectively while school B decided to award Rs.10700 for the three values to 4, 3 and 5 students respectively. If all the three prizes together amount to Rs.2700, then: i. Represent the above situation by a matrix equation and form linear equations 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?

13. Three friends A, B and C visited a Super Market for purchasing fresh fruits.

A purchased 1kg apples, 3kg grapes and 4kg oranges and paid Rs.800. B purchased 2kg apples, 1kg grapes and 2kg oranges and paid Rs. 500. While C paid Rs.700 for 5kg apples, 1kg grapes and 1kg oranges. Find the cost of each fruit per kg by using matrix method. Why are the fruits good for health?

14. Mr. Manoj has invested a part of his income in 10% (bond A) and another

part of his income in 15% (bond B). His interest during a certain period is Rs.4000. Had he invested 20% more in bond A and 10% more in bond B, his interest would have been increased by Rs.500 for the same period. Then: (i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication. (ii) Is it possible to solve the system of equations so obtained by matrices? If yes, solve it too.

15. In XII class examination, 25 students from school A and 35 students from

school B appeared. Only 20 students from each school could get through the examination. Out of them, 15 students from school A and 10 students from school B secured full marks. Write down this information in matrix from.

16. A 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 used for the financial benefits of the trust. Using matrix multiplication, determine how to divide Rs.30,000 among two types of bonds if the trust fund obtains an Rs. annual total interest of Rs.1800. (i) What are the values reflected in the question? (ii) Why is it required to help orphan children?


17. In a survey of 20 richest person of three cities A, B and C it is found that in city A, 5 believe in honesty, 10 in hardwork, 5 in unfair means while in city B, 5 believe in honesty, 8 in hardwork, 7 in unfair means and in city C, 6 believe in honesty, 8 in hardwork, 6 in unfair means. If the per day income of these richest persons of cities A, B and C are Rs. 32.5K, Rs.30.5K and Rs.31K respectively, then find the per day income of each type of person by matrix method.

(i) Which type of persons have more per day income? (ii) According to you, which type of person is better for country?

18. The cost of 4 chocolates, 3 samosas and 2 apples is Rs.60 and that of 2

chocolates, 4 samosas, and 6 apples is Rs.90. The cost of 6 chocolates, 2 samosas, 3 apples is Rs.70. Find the cost of each item by matrix method. What do you think is the healthiest diet? Suggest an item that could replace chocolates and samosas to make the diet healthier?

19. There are 3 families. Family A consists of 2 men, 3 women and 1 child.

Family B has 2 men 1 woman and 3 children. Family C has 4 men, 2 women and 6 Children. Daily expenses of men, women and children are 200,155 and 200 respectively. Only men and women earn and children do not. Using matrix multiplication calculate the daily expenses of each family. What impact does having more children in the family create on the society? Comment.

20. Gaurav donates 3 pens, 2 bags and 1 instrument box worth Rs.41,

Dheeraj donates 2 pens, 1 bag and 2 instrument boxes worth Rs29, while Ankur donates 2 pens, 2 bags and instrument boxes to a rural school. Translate the problem into a system of linear equations. Solve & find the cost of each item. By the act of these three people what values do you learn?

21. There are 2 families A & B. There are 4 men, 6 women and 2 children in

family A, 2 men, 2 women and 4 children in family B. The recommended daily allowance for calories is 2400 for men, 1900 for women, 1800 for children and 45gms of proteins for men, 55gms for women and 33gms for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the 2 families. What awareness can you create among people about the balanced diet?

22. Last year 1 packet of tea and 3 packets of sugar together cost Rs.96.This

year, the rate of tea increased by 15% and that of sugar by 10%. So, the same amounts of tea and sugar now cost Rs.108.60. Find the rates of sugar and tea per packet last year and this year using matrix method. What do you think is the impact of inflation on family expenses?

23. A part of the monthly expenses of a family is constant while the remaining

varies with the price of rice, fuel etc., When the price of rice is Rs 25/Kg the monthly expenses of the family is Rs.1000. when it is Rs 24/Kg the monthly expenses is Rs 980. Find the total monthly expenses of the family


when the cost of rice is Rs 35/Kg. Is this family below poverty line? Give some suggestions to improve their standard of living.

24. A dietician to mix two types of food,Food1 and Food2 in such a way that

the vitamin content of the mixture contains 8 units of vitamin A and 10 units of vitamin C. Food 1 contains 2units/kg of vitamin A and 1 units/kg of vitamin C. While Food2 contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. How many kg of Food1 and Food2 to mixed for getting her requirement? Formulate this problem as matrix form and solve by matrix method. What is the importance of vitamin content food in our diet than junk food

25. Three shopkeepers A, B, C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils.C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A note book costs 40 paise, a pen costs Rs. 1.25 and a pencil cost 35 paise. Use matrix multiplication to calculate each individual’s bill. What values should not come across the shopkeeper's mind to compete?

26. Consider the following information regarding the number of men and

women workers in three factories I, II and III

Factories Men workers Women workers I 30 25 II 32 20 III 29 23

Represent the above information in the form 3x2 matrix. What does the entry in the third row and second column represent? Is it necessary to equalize the number of men and women workers? What is your opinion? Explain the importance of gender equality in employment.

27. In his saving of Rs 200000, father divides the amount between his two

children and a servant. He gives 45% each to his son and daughter and the remaining to his servant. Write this in matrix form and calculate how much does the servant get and what is the value point in this question.

28. In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters. The cost per contact (in Paisa ) is given in matrix A as

40 Telephone100 House call50 Letter

A

. The number of contacts of each type made in two

cities X and Y is given in matrix B as Telephone House call Letter

1000 500 50003000 1000 10000

XB

Y

Find the total amount spent by the group in the two cities. What is your opinion to reduce the cost of election?


29. There are two families P and Q . There are 3 men, 3 women and 12 children in family P and 2men, 2 women and 4 children in family Q. The recommended daily allowance for calories is : Man : 2400, woman : 2000, child : 1400 and man : 60g for proteins is : woman : 40g and child : 35g. (i) Represent the above information by matrices. Using matrix multiplication, calculate the total requirement of calories and proteins of each of the two families.

(ii) Which family is an ideal family and why ?

30. Represent the following problem by a system of linear equations : Rs 154500 is the Cost of 3 cycles, 2 motorbikes and one car, the cost of one cycle, one motorbike and 2 cars is Rs 226500. The cost of 4 cycles and 3 motorbikes is Rs 81000. Use matrix to Find the cost of each . Which mode of conveyance preferred the most and why?

31. Two firms P and Q decided to award prizes to their employees for three

values Punctuality (x), Honesty (y) and obedience (z). Firm P decided to award a total of Rs 22000 for the three values to 10,8 and 6 employees respectively, while firm Q decided to award Rs 21400 for the three values to 8,6 and 10 employees respectively. If all the three prizes together amount to Rs 2700, then (i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication. (ii) Is it possible to solve the system of linear equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?

32. Two schools A and B decided to award prizes to their students for the

three values Truth (x), patriotism (y) and Non - violence (z). School A decided to award a total of Rs 11000. For the three values to 5,4 and 3 students respectively. While school B decided to award Rs 10700 for the three values to 4,3 and 5 students respectively . If all the three prizes together amount to Rs 2700, then

(i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication. (ii) Is it possible to solve the system of equations so obtained using matrices? (iii) Which value you prefer to be awarded most and why?

33. A sum of saving amount of Rs 10000 is put into three investments at the

rate of 10%,12% and 15% p.a. The combined income of first and second investment is Rs 190 less of the income of third investment. If the combined income is Rs 1310, then find the Investment in each using matrices. Keeping nation’s growth in mind, justify the value of saving in individual life.


34. (a) A trust fund has Rs. 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year & the second bond pays 7% interest per year. Using matrix multiplication , determine how to divide Rs.30000 among the two types of bonds if the trust fund must obtain an annual total interest of Rs.2000.

(b)If the same interest is obtained by investing in social welfare measures, which type of investment you would prefer & why.

35. Two institutions P and Q decided to award prizes for academic, sports and

all-round achievements’ awarded rupees 12000 to 3,1,1 students while Q awarded rupees 7000 1o 1 , 0, 2 students in the above category. All the three prizes together amount to rupees 6000.Find the matrix representation of the above representation. Solve the system of equations using matrices. Do you agree that prizes should be given for honesty and good character also? Give reasons.

36. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method. What do you think is the impact of inflation on family expenses?


APPLICATIONS OF DERIVATIVES 1. The total cost C(x) of planting x plants in a garden is given by C(x) =

0.005x3 – 0.02x2 + 30x + 6000. Find the marginal cost when 200 trees are planted. Do you think plantation helps in saving the environment ?

2. The total revenue received from the sale of x souvenirs in connection with

‘PEACE DAY’ is given by R(x) = 3x2 + 40x + 10. Find the marginal revenue when 100 souvenirs were sold. What is the importance of celebrating Peace Day in our life ?

3. A rectangular plot is to be developed for a meeting to be organised on

‘SAVE WILD LIFE’. It is given that the plot must have a fixed perimeter. Find the area of the plot so that maximum persons can attend the meet. Would you also like to attend the meet ? What are your views on ‘SAVE WILD LIFE’?

4. Two sign boards, one circular and one square are to be made using a wire

of length 40 m and cutting it into two pieces. The sign boards are to depict “BE HONEST” and “BE PUNCTUAL’ and these are to be displayed near the main gate of the school. What should be the lengths of the two pieces, so that the combined area of the square and the circle is minimum ? Do you think these values are important in life ?

5. A cylindrical box is to be made, which is open at the top and has a given

surface area. Souvenirs of different life values are to be stored in the box, so we would like to have maximum volume of the box. What should be the dimensions of the cylindrical box ? Name some of the values which are important to each person.

6. An organisation wants to inculcate awareness among people about ‘SAVE

ENERGY’. For this it wants to display a big balloon in the air. A balloon which always remains spherical is being inflated by pumping in gas at the rate of 900 cm3/sec. Find the rate at which the radius of the balloon is increasing when the radius of the balloon is 15 cm. Do you think we should be a part of ‘SAVE ENERGY’ campaign ?

7. The performance of a student in x months in governed by the relation f (x)

= 2x3 – 15x2 + 36x + 17 in [0, 4]. Find during which month his performance is increasing or decreasing. What steps a student should take to improve the performance.

8. Rectangular boards of different areas are to be made for use as display

boards. On these boards life skills are to be displayed. The area of expanding rectangle is increasing at the rate of 48 cm2/sec. The length of rectangle is always equal to square of breadth. At what rate the length is increasing when breadth is 4.5 cm ? Name two life skills which you would like to acquire.


9. Verify Lagrange’s Mean Value theorem for the function f(x) = (x – 3) (x – 6)

(x – 9) in the interval [3, 5]. On the graphical representation for the above function, two moral values are engraved, one on the line segment joining (3, 0) and (5, 8) and the other at ‘C’. Suggest two moral values to be engraved.

10. A rectangular piece of tin of sides 45 cm and 24 cm is to be made into a

box without top by cutting a square piece from each corner and folding the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ? On four sides of this box write four life skills.

11. Define absolute value function | x |. Let ‘f’ be a function defined as :

5 | | , 0( )

, 06

x x when xf x x

when x

Show that ‘f’ is discontinuous at x = 0. What are your views about ‘LEARNING’. Is learning a continuous process.

12. Find the equation of the tangent to the curve y = x – sinxcosx at 2

x .

Along the tangent write two like skills / moral values which one should acquire.

13. A cone is 10 cm in diameter and 10 cm deep. Water is poured into it at the

rate of 4 cubic cm/minute. At what rate is the water level rising at the instance when depth is 6 cm ? Water is a natural resource. What is the importance of water in our daily life?

14. A window is in the shape of a rectangle surmounted by a semicircle. If its

perimeter is 30 m, then find the dimensions of the window so that it may admit maximum light. Do you think that by getting maximum light we can save electricity ? Do you agree that we should save electricity ?

15. Show that semi-vertical angle of a right circular cone, of given total surface

area and maximum volume, is 1 1sin3

. On the surface is written “Respect

for beliefs and cultures of various religions.” Do you believe in mutual respect ?

16. Show that the semi-vertical angle of a cone of maximum volume and given

slant height is 1tan 2 . This statement will help us in making a conical tent for a gathering on ‘CLEANLINESS’. What are your views about the CLEANLINESS ?

17. The volume of a spherical balloon is increasing at the rate of 20 cm3/sec.

Find the rate of change of its surface area at the instant when its radius is 8 cm. On the balloon “Make India Polio Free” is printed. Do you think that we should work seriously towards Polio eradication ?


18. Water is leaking from a conical funnel at the rate of 5cm3/sec. If the radius of the base of the funnel is 5 cm and its altitude is 10 cm, find the rate at which the water level is dropping, when it is 2.5 cm from the top. Is leaking of water leads to wastage of water ? Do you think we can afford to waste our natural resources ?

19. Show that the curves 2x = y2 and 2xy = k, cut at right angles if k2 = 8.

Suggest one moral value to be written at the point of intersection of curves.

20. Show that the altitude of the right circular cone of maximum volume that

can be inscribed in a sphere of radius r is 43r . This statement will help us

in making a conical cup for serving ‘WATER’ to the people. What are your views about the serving water to the people?

21. Show that height of the cylinder of greatest volume which can be inscribed

in a right circular cone of height h and semi vertical angle a is one-third

that of the cone and the greatest volume of cylinder is 3 24 tan27

h . This

statement will help us in making a cylindrical glass for serving ‘WATER’ to the people during summer season. What are your views about the serving water to the people?

22. Show that the height of the cylinder of maximum volume that can be

inscribed in a sphere of radius R is 23R . Also find the maximum volume.

This statement will help us in making a cylindrical glass for serving ‘WATER’ to the people during summer season. What are your views about the serving water to the people?

23. A tank with rectangular base and rectangular sides, open at the top is to

be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank? On the top of tank is written “Respect for beliefs and cultures of various religions.” Do you believe in mutual respect ?

24. An open topped box is to be constructed by removing equal squares from

each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box. On the top of the box “Make India Polio Free” is printed. Do you think that we should work seriously towards Polio eradication ?

25. A water tank has the shape of an inverted right circular cone with its axis

vertical and vertex lowermost. Its semi-vertical angle is 1tan 0.5 . Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m. On the top of tank is written “GO GREEN INDIA.” Give some of values about green India?


26. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 time the radius of the base. This statement will help us in making a cylindrical glass for serving ‘WATER’ to the people during summer season. What are your views about the serving water to the people?

27. The radius of a spherical diamond is measured as 7 cm with an error of

0.04 cm. Find the approximate error in calculating its volume .if the cost of 1 cm³ diamond is rupees 1000,what is the loss to the buyer of the diamond? What lesson do you get?

28. Let x be the number of years a person is engaged in the share market. His

income is given by f(x) = x4 - 8 x3 + 22 x2 + 10 , 0< x ≤ 10. Find the intervals in which his income is increasing and decreasing. Do you think that dealing in ‘share market’ is a risky affair? Give reasons

29. A window is in the form of rectangle surmounted by a semicircular

opening. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum sunlight through the whole opening. Explain the importance of sunlight.

30. An Apache helicopter of enemy is flying along the curve given by y = x2 +

7. A soldier, placed at (3,7) wants to shoot down the helicopter when it is nearest to him. Find the nearest distance. What is the main quality needed for the soldier? What steps can be taken to ensure peace between countries?

31. A window is in the form of rectangle surmounted by a semicircular

opening. The total perimeter of the window is 20m. Find the dimensions of the window to admit maximum light through the whole opening. What is the benefit of using maximum area for the window?

32. Show that the percentage error in calculating the volume of a cubical box if

an error of 1% is made in measuring the length of edges of the cube is 3%. What is the value you learn from that?

33. The total area of a page is 150cm2. The combined width of the margin at

the top and bottom is 3cm and the side 2cm. What must be the dimensions of the page in order to get the maximum area of printed matter?

34. A stone is dropped into a quiet lake and waves move in circles at a speed

of 4cm per second. At the instant, when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing? What steps can you take to maintain a clean lake?

35. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then

find the approximate error in calculating its volume. What will you do if you get an incorrect value in your practical exam?


36. Find local maximum and local minimum values of the function f given by f (x) = 3x4 + 4x3 – 12x2 + 12. What can you do to maximize your efforts to get maximum marks in exam?

37. A car starts from a point P at time t = 0 seconds and stops at point Q. The

distance x, in metres, covered by it, in t seconds is given by 2 23tx t

.

Find the time taken by it to reach Q and also find distance between P and Q. What steps should you take for safe driving?

38. The edge of a cubical gold is measured as 8 cm with an error of 0.03 cm. Find the approximate error in its volume. What is the loss to the buyer of the gold if the cost of 1 cubic cm of gold is Rs. 3000. What lesson do you get?

39. The bottom of a rectangular fish tank is 30 cm X 15 cm. Water is pumped

into the tank either by man-made pump at the rate of 50 cubic cm/minute or by motor pump at the rate of 75 cubic cm/minute. Find the rate at which the level of the water in the tank is rising when water is pumped by motor pump. Which pump would you prefer and why?

40. A jet of an enemy is flying along the curve 2 7y x . A solider is placed at

the point (3,7). Find the nearest distance between the solider and the jet. The soldier is refused to fire the jet when it is nearest to him? Is this act of the solider justifiable. Discuss.

41. Find the volume of the largest cylinder that can be scooped out from a

given wooden solid cone. Give a valued method of using the left out wood and justify?

42. A car driver is driving a car on the dangerous path given by

1 , 1( ) ,1 , 1

1

mx xf x m Nx xm

Find the dangerous point (point of discontinuity) on the path. Whether the driver should pass that point or not? Justify your answers.

43. 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?

44. 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?


45. 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.

46. 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.

47. 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?

48. 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.

49. Check whether the function 100( ) sin 1f x x x is strictly increasing or

strictly decreasing or none of both on (–1, 1) . Should the nature of a man be like this function? Justify your answers.

50. If 3

4

3xy x , when x denotes the number of hours worked and y

denotes the amount (in Rupees) earned. Then find the value of x (in interval) for which the income remains increasing? Explain the importance of earning in life?


APPLICATIONS OF THE INTEGRALS 1. A sign board in the shape of a parabola y2 = 16x is to be made such that

area is bounded by the parabola y2 = 16x and the line x = 4. On the sign board life values such as “OBEDIENT” “OBSERVANT” “EMPATHY”, “SINCERE” etc., are to be written. What is the area of the sign board? What do you think about acquiring value “EMPATHY” ?

2. Three hoardings are displayed at the points A, B and C displaying A (DO NOT

LITTER), B (KEEP YOUR PLACE CLEAN) and C (GO GREEN). If these three points form a triangle ABC. Find the area of ABC, using method of integration, if points A, B and C are (0, 2), (4, 8) and (8, 4) respectively. Give your views in two lines about “GO GREEN”.

3. A field is in the form of parabola x2 = 4y. A farmer has planted trees in the

exterior to the region bounded by the parabola and y = |x| and the remaining part for playing games for children. Find the area of the ground where the students are playing games. What is the importance of games in student’s life

4. Evaluate 2

2 1x dx

x . State any one reason to have integrity in real life

5. The area between x = y2 and x=4 is divided into equal parts by the line x =

a, find the value of a. A man constructed a house in one of the area and planted trees in the other area. Find the area where he constructed house? What is the importance of plantation of trees?

6. Evaluate / 4

0log(1 tan )x dx

.What can be the possible limits of integrity

in real life according to you? 7. A farmer has a plot in the shape of a circle x2+ y2 = 4. He divides his

property among his son and his daughter in such a way that for son he gives the area interior to the parabola y2= x and for daughter the area interior to the parabola y2 = -x. How much area the son got? Have both of them got equal share? What is the value for life behind this?

8. Find the integration of cos2xcos4xcos6x. What is the importance of

integration of different religions for our country peace? 9. Using the method of integration find the area bounded by the curve |x| + |y|

= 1 . What is the value of truthfulness in our life? 10. Find the integration of ex sin x dx. Write any famous proverb for unity. 11. Integrate sin (ax + b) cos (ax + b). Write any two characteristics of

Mahatma Gandhi which you adore the most?


12. Find the area lying above x-axis and included between the circle x2 + y2 =

8x and the parabola y2 = 4x. What is the importance of non-violence in our life?

13. 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 y2 = 4x and x = 4 and to divide the area equally he draws a linewhat is the value of a? What is the importance of equality among the people?

14. 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.

15. A poor deceased farmer has agriculture land bounded by the curve y=

cosx, between x = 0 and x=2 π. He has two sons. Now they want to distribute this land in three parts (As already partitioned). Find the area of each part. Which parts should be given to the farmer & why? Justify your answer.

16. 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?

17. A parking lot in an IT company as an area bounded by the curve y = 4 – x2

and the lines y = 0 and y = 3. The line y = 3 divides the area into two parts out of which the greater area is allotted for car owners who practice car pooling. Find this area using integration. Write any two benefits of carpooling.

18. Integrity is an integral part of a student’s life. Elucidate. Similarly find the

value of the integral 2 2 7

dxx x

19. A construction company is constructing a gated community in a plot of 10

grounds. When he decides to allocate sufficient space for a park and gym

in an area given by

3 4

3a

a

xd x

. Find ‘a’ using / 2

3

0 0

2 sina

xdx a xdx

.

Highlight the values reflected by the gesture of the company.

20. Evaluate using limit as a sum : 3 2

1(2 5 1)x x dx

Mention two suitable situations in your everyday life


21. Using integration find the area of the bounded region y = 1 + | x + 1 | ; The required area is divided into two regions and the students are asked to put a posters on “Child Labour” and “Education for all” in those two. Which theme according to you deserves a bigger space? Justify.

22. Integrate exsin2x with respect to x. What values are integrated and instilled

in children with the singing of the National Anthem? 23. Find the area of the region enclosed by the curve y = x2 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 a wooden fence to be erected. Who would you favour? Mention two values demonstrated by the son.

24. Draw the graph and evaluate the integral 3

1| 3 |x dx . What geometric

property does the shaded region exhibit about the line x = 3. Write any two values you in real life that you can correlate with this property.

25. Evaluate as the limit of a sum : 3 2

1(2 3 1)x x dx . Mr. X. crossed the

limit while speaking at a public function by using abusive language. What values should one possess while speaking in a public forum. Mention any two.


VECTOR, THREE-DIMENSIONAL GEOMETRY 1. Two groups of students representing ‘SAVE MOTHER EARTH’ and ‘GO

GREEN’ are standing on two planes represented by the equation .( 2 ) 5r i j k

and .(2 ) 8r i j k . What is the angle between the two

planes ? Name few activities which should be taken up to Save Mother Earth.

2. Find the coordinates of the foot of the perpendicular drawn from the point

A(1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,1). Represent foot of the perpendicular with one life skill.

3. Find the image of the point (1,2,3) in the plane x+2y+4z=38. Represent the

image of the point with one life skill. 4. Three hoardings are displayed at the points A,B and C displaying A(Do not

litter), B(Keep your place clean) and C(Go green). If these 3 points form a triangle ABC, find the area of the triangle ABC using method of vectors if points A,B and C are (0,2,1),(4,8,2) and (8,4,3) respectively. Give your views in two lines about “GO GREEN”.

5. If , ,a b c

are position vectors of vertices A,B,C of a triangle ABC, show that

the area of the triangle is 12

a b b c c a . A student takes honesty,

truthfulness and complacency as the three sides of the triangle. Which side of the triangle do we prefer to take? Give your suggestion.

6. Let A and B be two points whose position vectors are 3 4 8i j k and

4 7 4i j k respectively. Find AB

and BA

. If point A represents a person who is regular and systematic and B represents a lazy person, which vector AB

or BA

would you choose for your success?

7. In metropolitan city a bypass road to be constructed passing through the

point (1,1,1) and parallel to the road whose equation given by .( 2 3 ) 5r i j k

. Find the equation of the bypass road. State one benefit that people get from this?

8. Find the reflection of the point (1,2,-1) in the plane 3x – 5y + 4z = 5. What

is the value of fixing mirrors on T roads? 9. Find the foot of the perpendicular from the point (2,3,4) to the line

4 12 6 3

x y z . Also find the perpendicular distance from a given point to

a line. If you want to construct a road joining point and the line which way you prefer to make and why?


10. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane. What should we have in mind while crossing a road?

11. Find the equation of the plane through the line of intersection of the planes

x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Which value will you choose if you have to choose between good and truth?

12. Find | a×b|, if a = 2 3i j k and b = 3 5 2i j k . We should be firm and

determined about __________________ in life.(Hint: some value as you perceive is a possible answer). Why?

13. Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2,

3, 1) as its vertices. What is the difference between finding its area and a triangle in which all points are in a single plane?

14. Find the coordinates of the point where the line through the points A (3, 4,

1) and B(5, 1, 6) crosses the XY-plane. Name any two values which we get from our friends.

15. Considering the earth as a plane having equation 5x + 9y – 10z + 138 = 0,

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?

16. Let the point P(5, 9, 3) lies on the top of Qutub Minar, Delhi. Find the

image of the point on the line: 1 2 32 3 4

x y z . Do you think that the

conservation of monuments is important? Why? 17. Two bikers are running at the speed more than allowed speed on the road

along the lines (3 )r i j k i j and 4 (2 3 )r i k i k

. Using Shortest distance formula check whether they meet to an accident or not? ‘While driving, the driver should maintain the speed limit as allowed.’ Comment.

18. Two airships are moving in space along the following lines

3 5 71 2 1

x y z and 1 1 1

7 6 1x y z

. An astronaut wants to move

from one ship to another ship when two airships are closest. What is the least distance between the ships that he has to travel?

19. Find the area of a hoarding board with logo “GO GREEN” in the form of

triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices. Give your views in two lines about “GO GREEN”.


20. Find the vector equation of the line passing through (1, 2, 3) and parallel to the roads .( 2 ) 5r i j k

and .(3 ) 6r i j k . What should we have in

mind while crossing a road? 21. A bird is located at the point A(3, 2, 8) in space. It wants to move to the

plane whose equation is given by 3x + 2y + 6z + 16 = 0 in the shortest time. Find the distance she covered.

22. From the point A(2, 3, 8) in space, a shooter aims to hit the target at P(6,

5, 11). If the line of fire is 2 3 84 2 3

x y z , what do you think about the

success of the shooter? 23. A bird at A(7, 14, 5) in space wants to reach a point P on the plane 2x + 4y

– z = 2 when AP is least. Find the position of P and also the distance AP travelled by the bird.

24. Show that a powerful bomb shot along the line of fire 1 2 32 3 4

x y z

will never hit a helicopter flying in the plane 2x + 4y – 4z + 11 = 0. 25. A gunner who is hiding himself from the enemy is at the point G(2, 1, 3)

and observes an enemy bomber flying along the plane 3x + 6y + 2z +10 = 0. What is the least distance of G from the plane?

26. Find the area of a hoarding board with logo “GO GREEN” in the form of

parallelogram whose adjacent sides are given by the vectors 3 4a i j k

and b i j k . Give your views in two lines about “GO GREEN”.

27. Find the coordinates of the point where the line through (5, 1, 6) and (3,

4,1) crosses the ZX-plane. What should we have in mind while crossing a road?

28. If the lines 1 2 3 1 1 63 2 2 3 1 5

x y z x y zandk k

are perpendicular,

find the value of k. If you want to construct a road joining point and the line which way you prefer to make and why?

29. Find the distance of the point (– 1, – 5, – 10) from the point of intersection

of the two roads 2 2 (3 4 2 )r i j k i j k and .( ) 5.r i j k

What should we have in mind while crossing a road?

30. If the coordinates of the hoarding showing display “SAVE WATER” which

is in the quadrilateral with A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD. Give your views in two lines about “SAVE WATER”.


DIFFERENTIAL EQUATIONS 1. Form the differential equation of the family of circles touching y axis at

origin. What are the values conveyed by the family of circles formed? 2. The temperature T of a cooling object drops at a rate which is proportional

to the difference T – S, where S is the constant temperature of the

surrounding medium. Thus ( )dT k T Sdt

, where k > o is a constant and t

is the time. Solve the differential equation if it is given that T(0) =150. What are the demerits of global warming?

3. Form the differential equation of the family of parabolas having vertex at

the origin and axis along positive y-axis. Mosquito bite is the origin for `Dengue’ . Suggest two methods to curb the breeding of mosquitoes.

4. Show that the given differential equation is homogeneous and solve it.

x2 dy + y (x+y) dx = 0. Which is advantageous to life homogeneity or hetrogenity ? Justify your answer.

5. Solve the differential equation: x(x2 – 1 ) dydx

=1 ; y =0 when x = 2. If `y’ is

distance and x is the time. dydx

is velocity. Ravi rides a vehicle beyond the

limit in a highway. What suggestions would you give him to help him understand the risks of overspeeding?

6. In a college hostel accommodating 1000 students, one of the hostellers

came in carrying a flu virus, and the hostel was isolated. If the rate at which the virus spreads is assumed to be proportional to the product of the number N of infected students and the number of non-infected students, and if infected students are 50 after 4 days then show that more than 95% of the hostellers will be infected after 10 days. If Shyam was the first student to be infected what precautions he should have initiated to avoid this situation.

7. The population of a city increases at a rate proportional to the number of

inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010? What is the main awareness to be given among the people to control population?

8. If the interest is compounded continuously at 6% per annum, how much

worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000? (Given e0.6 = 1.822 ). What is the importance of savings among the students?


9. Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis. Suggest one method to enhance the gender equity in our country?

10. Show that the differential equation (x – y) dydx

= x + 2y is homogeneous

and solve it. State one step to eliminate the racial discrimination? 11. Find the equation of a curve passing through the point (0, 1). If the slope of

the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point. In what way the coordination among people will bring harmony in society?

12. Solve the differential equation

(xdy – y dx) y sin(yx

) = (y dx + x dy) x cos(yx

)

In what way human conscience overcomes the human mind? 13. The population of a village increases continuously at the rate proportional

to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009? What are the values coming in your mind considering village life?

14. Solve the differential equation 2( 2 ) dyx y ydx

. Given that when x=2,

y=1.If x denotes the people who are Polite and y denotes the % of people who are intelligent. Find when. A polite child is always liked by all in society. Do you agree? Justify.

15. 0dy ydx x

, where x denotes the percentage population living in a city & y

denotes the area for living a healthy life of population. Find the particular Solution when Is higher density of population is harmful? Justify yours answer.

16. The velocity v and mass m of a rocket at time t are given by the equation:

0dv dmm Vdt dt

, where V is the constant velocity of emission. If the rocket

starts from rest when t = 0 with mass m0 , then prove that:

0log mv Vm

. Should we encourage the rocket technology? Comment.

17. The male-female ratio of a village increases continuously at the rate

proportional to the rate at any time. If the ratio of male-female of the village was 1000:980 in 1999 and 1000:950 in 2009, what will be the ratio in 2019? (i) Why is the gender equality value-able for the society?


(ii) What should society do to reduce the male-female ratio to 1? 18. Find the equation of a curve passing through the point (0, 2) given that the

sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5. In what way the coordination among people will bring harmony in society?

19. A manufacturer’s marginal revenue function is MR = 275 – x – 0.3x2. Find

the increase in the manufacturer’s total revenue if production is increased from 10 units to 20 units.

20. The marginal cost of a manufacturer is given by 2

50040

MCx

, where x is

the number of units of a product in thousands. If x increases from 3000 to 9000 units, find the total increase in cost.

21. The marginal cost is defined as the rate of change of total cost with

respect to the number of units of the product. The marginal cost of producing x units of a product is given by Marginal Cost = 2 5x x . If the cost of producing 4 units of the product is Rs. 314.40, find the cost function.

22. An electric manufacturing company makes small house-hold switches. The

company estimates the marginal revenue function for these switches to be 2

2( 2)xxMR e

x

, where x represents the number of units (in thousand)

and MR is the rate of change of revenue R w.r.t. x. What is the total revenue function? Use integrals.

23. Solve the differential equation (tan–1y – x) dy = (1 + y2) dx, where x

denotes the number of fruits and y denotes the cost of fruits. Why fruits is good for health?

24. Find the equation of a curve passing through the origin given that the

slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point. In what way the coordination among people will bring harmony in society?

25. Form the differential equation of the family of circles in the second

quadrant and touching the coordinate axes. Suggest one method to enhance the gender equity in our country?

26. Find the particular solution of the differential equation log 3 4dy x ydx

given that y = 0 when x = 0. Which is advantageous to life homogeneity or hetrogenity ? Justify your answer.


27. Find the equation of a curve passing through the point (1, 0). If the slope of

the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point. In what way the coordination among people will bring harmony in society?

28. Find the general solution of the differential equation ydx – (x + 2y2) dy = 0.

If x is a brilliant student and y is a slow learner and x helps y in his studies, what quality does x possess?

29. Find the general solution of the differential equation 22 ( 0)dyx y x x

dx ,

where x denotes the number of students who are punctual and y denotes the number of students who are not punctual. How does the punctuality important in the progress of life?

30. Find the general solution of the differential equation cosdy y x

dx .

Suppose x represents the number of persons who take food at home and y represents the number of persons who take junk food in market. Which way of taking food you prefer and why?


LINEAR PROGRAMMING 1. 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 Rs. 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 a daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost, assuming that both wheat and rice are to be taken in a diet ? What is your opinion about healthy diet name few ingredients necessary for a healthy diet ?

2. A retired person has Rs. 70,000 to invest in two types of bonds. First type

of bond yields an annual income of 8% on the amount invested and the second type of bond yields 10% per annum. As per norms he has to invest minimum of 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 ? Do you think that a person should start saving at an early age for his retirement ? Can you name some avenues ?

3. A company manufactures two types of stickers A : “SAVE

ENVIRONMENT” and B : “BE COURTEOUS”. Type A requires 5 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 are 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 many stickers of each type should the company manufacture in a day to maximize profit ? Give your views about “SAVE ENVIRONMENT” and “BE COURTEOUS”.

4. A farmer decides to plant up to 10 hectares with cabbages and potatoes.

He decides to grow at least 2, but not more than 8 hectares of cabbage and at least 1 but not more than 6 hectares of potatoes. If he can make a profit of Rs. 1,500 per hectare on cabbage and Rs. 2,000 per hectare on potatoes, how should he plan his farming so as to get the maximum profit, assuming the total yield that he gets is sold. What is your opinion about farmer getting maximum profit ? Do you think we should help the farmers in every way to maximise the production ?

5. Asha wants to invest atmost Rs. 20,000 in VIII series National Saving

Bonds and Kisan Vikas Patra. According to rules she has to invest at least Rs. 5,000 in Saving Bonds and atmost Rs. 8,000 in Kisan Vikas Patra. If the rate of interest on Saving Bonds is 10% p.a. and rate on Kisan Vikas Patra is 15% p.a., how much money should she invest in each to earn maximum yearly income. Also, find the maximum yearly income. Do you think our investment in National Saving Bonds and Kissan Vikas Patra help in the well being of the nation ?


6. A factory owner purchases two types of machines A and B for his factory to produce ‘Eco-friendly vehicles. The requirements and limitations for the machines are as follows :

Area occupied

by machine (in sq. m)

Labour force for each machine

(man)

Daily Output in units

Machine A 1000 12 60 Machine B 1200 8 40

He has an area of 9,000 sq. m available and 72 skilled men who can operate the machines. How many machines of each type should he buy to maximise the daily output ? Do you agree we should use Eco-friendly vehicle for transportation ?

7. Two godowns, A and B have a grain storage capacity of 100 quintals and

50 quintals respectively. They supply to three 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 as

Transportation cost per quintal (in Rs.) From To A B

D 6 4 E 3 2 F 2 3

How should the supplies be transported so that the transportation cost is minimum ? What are other positive effects of minimum cost ?

8. If a person drives his car at 25 km/hr, he has to spend Rs. 2/km on petrol.

If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5/km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as a L.P.P. and solve it. Which is a better option as a fuel, petrol or CNG ?

9. A producer has 30 units and 17 units of labour and capital respectively,

which he can use to produce two types of goods X and Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly 3 units of labour and 1 unit of capital are required to produce one unit of Y. If X and Y are priced at Rs. 100 and Rs. 120 per unit respectively, how should the producer use his resources to maximise the total revenue. Solve the problem graphically. Which life skills produces is supposed to develop.

10. A company produces two types of hoardings A and B representing life

skills. Profits on each types of hoarding are Rs. 2 and Rs. 1.5 respectively. A hoarding of type A requires twice as much time as of hoarding B. The company can produce at the most 1000 hoardings of type B per day. Material for only 800 hoardings per day is available. At the most 400 buckles for hoardings of type A and 700 of type B are available per day.


How many hoardings of each type should the company produce so as to maximise the profit ? Which two life skills would you like to be written on A and B ?

11. A company produces two types of goods A and B that require gold and

silver. Each unit of type A requires 1 gram of silver and 2 grams of gold. Each unit of type B requires 2 grams of silver and 1 gram of gold. The company has only 100 grams of silver and 80 grams of gold. Each unit of type A brings a profit of Rs. 500 and each unit of type B brings a profit of Rs 400. Find the number of units of each type the company should produce to maximise the profit. These goods A and B are to be used as souviners during debate on ‘SAVE WILD LIFE’. What do you mean by SAVE WILD LIFE ?

12. A small manufacturing firm produces two types of gadgets A and B which

are first processed in the foundary shop, then sent to another machine shop for finishing. The number of man hours of labour required in each shop for the production of each unit of A and B and the number of man hours for the firm available per week are as follows :

Foundary

shop Machine

shop Gadget A 10 5 Gadget B 6 4 Firm capacity per week (in hours)

1000 600

The profit on sale of A is Rs. 30 per unit and on B is Rs. 50 per unit. Formulate the linear programming model for determining how many units of each gadgets should be produced per week to maximise the firm’s profit. The firm keeps its workers happy and cares for them. Do you think company is in good shape and progress well ? Which life skill company acquires ?

13. A carpenter has 20 and 15 square metres of plywood and sunmica

respectively. He produces products A and B. Product A requires 2 and 1 square metres and product B requires 1 and 3 square metres of plywood and sunmica respectively. If the profit on one piece of product A is Rs. 30 and on one piece of product B is Rs. 20, how many pieces of product A and B should he make to maximise his profit ? These products are to be used as sign boards for a procession regarding awareness among people towards their environment. Suggest two slogans for product A and product B, one each.

14. A hospital dietician wishes to find the cheapest combination of two foods A

and B, that contains at least 0.5 mg of thiamine and at least 600 calories. Each unit of food A contains 0.12 mg of thiamine and 100 calories, while each unit of food B contains 0.01 mg of thiamine and 150 calories. If each food costs Rs. 10 per unit, how many units of each should be combined for minimum cost ? Is balanced food necessary for us to remain active ? What are your views about Junk food ?


15. A dealer wishes to purchase a number of fans and sewing machines. He

has only Rs. 57,600 to invest and has space for atmost 20 items. A fan costs him Rs. 3,600 and a sewing machine Rs. 2,400. Profit on selling a fan and a sewing machine are Rs. 220 and Rs. 180 respectively. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit ? Formulate this problem mathematically and solve it graphically. Do you think sewing machine is environmental friendly ?

16. A housewife wishes to mix together two kinds of foods X and Y, in such a

way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg of food is given below :

Vitamin A Vitamin B Vitamin C

Food X 1 2 3 Food Y 2 2 1

One kg of food X costs Rs. 6 and one kg of food Y costs Rs. 10. Formulate the above problem as a linear programming problem to find the least cost of mixture which will produce the diet. Write two lines about the importance of balanced diet.

17. A company combines two items A and B from gift packs during a festival

season. Each pack must weigh 5 kg and should contain at least 2 kg of A and not more than 4 kg of B. The net contribution of the company is ` 10 per kg of A and ` 12 per kg of B. The company wants to determine the optimum mix. Formulate the L.P.P. to maximise the net contribution per pack. What is the importance of exchanging gifts with friends and relatives?

18. 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 maximum number of units of A that can be sold is 70 and that of B is 125. Profit on each unit of A is Rs. 20 and that on B is Rs. 15. Has many units of A and B should be produced to maximise the profit ? Solve it graphically. Consider products A and B as awareness projects for environment. Name two such environmental needs.

19. Kelloggs is a new cereal formed of a mixture of bran and rice that contain

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 of 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 kilogram. Do you think new cereal formed is a healthy diet ?


20. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs.300 is made on each economy class ticket. The airline reserves at least 20 tickets for first class seats. However, at least 4 times as many passengers prefer to travel by economy class as compared to the first class. Determine how many tickets of each type must be sold in order to maximise the total profit for the airline. Solve the problem by graphical method. Is an aeroplane Eco-friendly mode for travelling ?

21. A machine company has factories at two places A and B. From these

places, supply is made to each of its three agencies situated at P, Q and R. The monthly requirement of the agencies are respectively 40, 40 and 50 packets of machines, while the production capacities of the factories at A and B are 60 and 70 respectively. The transportation cost (in `) per packet from the factories to agencies are given below :

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 is the significance of minimum travel cost ? Is this related to ‘SAVE ENVIRONMENT / NATURAL RESOURCES’ ?

22. A firm manufactures jute bags and cloth bags. The total number of items it

can manufacture is at most 24. A Jute bag requires 1 hour to be made while a cloth bag requires only half an hour. The maximum number of hours available per day is 6 hours. If the profit on a jute bag is Rs.30 and on a cloth bag is Rs.20, how many bags of each type must be made for maximum profit? Solve it graphically. The manufacturer wants to replace cloth bags by plastic bags to increase his profit margin. Is it a good idea? If not, give reasons.

23. A manufacturer makes cycles and scooters. Processing of these products

is done on two machines A and B. The cycle parts need 2 hours on machine A and 6 hours on machine B. Parts of a scooter needs 4 hours on machine A and 2 hours on machine B. Machine A is available for 16 hours per day and Machine B is available for 30 hours per day. Profit gained by the manufacturers from a cycle and a scooter is Rs.1000 and Rs.3000 respectively. Find with the help of a graph what should be the daily production of each of the two products to maximize the profit? Which of the above modes of transport is a better option and why?

24. There are two types of fertilizers that a farmer uses in his farm namely A and B. A consists of 10 % of nitrogen and 6% phosphoric acid while B contains 5% nitrogen and 10% phosphoric acid. The farmer needs at least 14 kg of nitrogen and 14 kg of phosphoric acid. If A costs Rs. 8/kg and B costs Rs. 6/kg, determine how much of each type of fertilizer should be used by the farmer so that the required nutrient levels are met at minimum


cost. What is the minimum cost? The farmer is considering the option of using pesticides to increase his yield. What is your opinion? Can he convert it into an organic farm?

25. A farmer decides to plant upto 10 hectares with cabbages and potatoes.

He decides to grow at least 2 but not more than 8 hectares of cabbages and at least 1 but not more than 6 hectares of potatoes. He can make a profit of Rs. 1500 per hectare on cabbages and Rs.2000 per hectare on potatoes. How should he plan his farming so as to maximize his profit. Keeping in mind the nutritional value of both vegetables, do you think the farmer’s production will help the community? Mention one value point the farmer display?.

26. A manufacturer produces two types of disposable plates, one using plastic

and the other using bamboo. Three machines are required to produce the plates and the time in minutes is given below.

Machines Type of Plate I II III

Plastic 12 18 6 Bamboo 6 0 9

Each machine is available for a maximum of 6 hours per day. If the profit on each plate of plastic is 75 paise and profit on bamboo plate is 50 paise, how many plates of each type should the factory produce to maximize the profit. Which type of plates do you think should be promoted more and mention one value point as the reason.

27. A principal decides to buy colour boxes and books as prizes for children. A

colour box costs Rs.5 and a book costs Rs. 10.He wants to buy at least 4 of each of them. How many of each should he buy so that the expenditure does not exceed Rs.100 and at the same time can give maximum number of prizes? Which of the two , do you think will benefit the children more and mention one value point as the reason.

28. A diet for a sick person must contain atleast 4000 units of vitamins, 50

units of minerals and 1400 calories. Two foods X and Y are available at a cost of Rs 4 and Rs 3 per unit respectively. One unit of food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements? Explain the importance of balanced diet.

29. A farmer has a supply of chemical fertilizer of type A which contains 10%

nitrogen and 5% phosphoric acid, and type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, it was found that at least 14kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type A costs Rs 5 per kg and the type B costs Rs 3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum possible cost ? Using L.P.P. Solve the above problem graphically. Which manure you prefer to be promoted and why ?


30. If a Youngman rides his motorcycle at 25 km/ hour, he had to spend Rs 2

per km on Petrol . If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs 5 per Km. He has Rs 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour . Express this as L.P.P. and solve it graphically. Which mode of transport you suggest to a young man and why.

31. (a)There are two types of fertilizers F1& F2. F1consists of 10% nitrogen &

6% phosphoric acid & F2 consists of 5% nitrogen & 10% phosphoric acid. After testing the soil conditions, a farmer finds that he needs atleast 14 kg of nitrogen & 14 kg of phosphoric acid for his crop. If F1 costs Rs. 6/kg & F2 costs Rs.5/kg , determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost. (b)Joseph found that his uncle ,who did not have experience of growing plants, was planting the seeds. He explained his uncle the proper procedure for planting the seeds. His uncle , following the same procedure , planted the seeds & the same came out successfully. What values& qualities you find with Joseph & with his uncle.

32. A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it.

He has a sum of Rs. 1400 to purchase young trees. He has the choice of two type of trees. Type A requires 10 sq.m of ground per tree and costs Rs 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs 25 per tree. When fully grown, type A produces an average of 20 Kg of fruit which can be sold at a profit of Rs. 2.00 per Kg and type B produces an average of 40 Kg of fruit which can be sold at a profit of Rs. 1.50 per Kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit? What are the advantages that the man received by the plantation of trees.

33. 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 requirements of the schools are respectively40, 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 schools is given below:

Transportation cost per packet (in Rs)

From To A B P 5 3 Q 4 2 R 3 5

How many packets from each kitchen should be transported to school so that the cost of transportation is minimum? Also find the minimum cost. What is the value of giving mid day meals in the schools?


34. Anil’s mathematics teacher has given him three very long list of problems with the instruction to submit not more than 100 of them(correctly solved) for credit. The problem in the first set are worth 5 points each, those in the second set are of worth 4 points each, and those are in the third set are of worth 6 points each. Anil knows from experience that he requires on the average 3 minutes to solve a 5 point problem, 2 minutes to solve a 4 point problem and 4 minutes to solve a 6 point problem. Because he has other subjects to worry about, he cannot afford to devote more than 3 ½ hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than 2 ½ hours work on this type of problem. Under these circumstances, how many problems of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a LPP. What is the importance of time management? According to you why Anil can’t do the entire question in time?

35. A fruit grower can use two types of fertilizer in his garden, brand P and

brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the maximum amount of nitrogen added in the garden?

kg per bag Brand P Brand Q Nitrogen 3 3.5 Phosphoric acid 1 2 Potash 3 1.5 Chlorine 1.5 2

In which way the fruit grower is helping the environment?

36. A dietician wishes to mix two types of foods in such away that vitamin

contents of the mixture contain atleast 8 units of vitamin A and 10units of vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as a linear programming problem to minimise the cost of such a mixture. In what way a balanced and healthy diet is helpful in performing your day-to-day activities?

37. A dealer in rural area wishes to purchase a number of sewing machines.

He has only’ 5,760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him `360.00 and a manually operated sewing machine `240.00. He can sell an Electronic Sewing Machine at a profit of `22.00 and a manually operated sewing machine at a profit of `18.00. 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 solve it graphically. Keeping the rural background in mind justify the ‘values’ to be promoted for the selection of the manually operated machine.


38. 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 of B is 125 . Profit on each unit of A is Rs. 20 and on B is Rs. 15. How many units of A and B should be produced to maximize the profit. Form an L.P.P. and solve it graphically. What safety measures should be taken while working in a factory?

39. A cooperative society of farmers has 50 hectare of land to grow two crops

X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further,no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximize the total profit of the society? How can you protect wildlife from destruction?

40. An oil company has two depots A and B with capacities of 7000 L and

4000 L respectively. The company is to supply oil to three petrol pumps, D, E and whose requirements are 4500L, 3000L and 3500L respectively. The distances(in km) between the depots and the petrol pumps is given in the following table:

Distance in (km.)From / To

A B

D 7 3 E 6 4 F 3 2

Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost? How can you save fuel in your daily routine? any two ways

41. A dealer in rural area wishes to purchase a number of sewing machines.

He only has rupees 5760 to invest and has space for at most 20 items. An electronic sewing machine costs him rupees 360 each while a manually operated sewing machine costs him rupees 240. He can sell an electronic machine at a profit of rupees 22 and that a manually operated machine at a profit of rupees 18. 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 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.


42. 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. (a) Express this as an L .P.P and solve graphically. (b) What is benefit of driving at an economical speed? (c) Should a child below 18years be allowed to drive a motorcycle? Give reasons.

43. 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?

44. 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 20000 bottles of medicine A and 40000 bottles of medicine B but there are 45000 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?

45. A toy company manufactures two types of dolls, A and B. Market tests and

available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit? How is toys useful for the children? Which type of toys is good for the children?

46. A fruit grower can use two types of fertilizer in his garden, brand P and

brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden? Why are the fruits good for health?


47. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 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 maximise the profit for the airline. What is the maximum profit? What precautions have to take while traveling in aerolane? (write any three)

48. A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs

250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag? In what way a farmer is helping the society?

49. A dietician has to develop a special diet using two foods P and Q. Each

packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A? What is the importance of vitamin content food in our diet than junk food

50. A diet is to contain at least 80 units of vitamin A and 100 units of minerals.

Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.


PROBABILITY 1. In a school 12 N.C.C cadets of which 3 are reserve for parade selected for

the Republic day parade. Find the probability distribution of the number of cadets reserves for parade. Explain the importance of being a citizen of republic nation.

2. In a girls hostel 60% of the students read Hindi newspaper, 40% read

English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English newspapers. (b) If she reads Hindi newspaper, find the probability that she reads Hindi newspaper. (b) If she reads English newspaper, find the probability that she reads English newspaper. (c) Explain the role of newspapers in national integration.

3. A man is known to speak the truth 3 out of 5 times. He throws a die and

reports that it is a number greater than 4. (i) Find the probability that it is actually a number greater than 4. (ii) Write about ‘ truth’ as essential human value.

4. To test the quality of electric bulbs produced in a factory , two bulbs are

randomly selected from a large sample without replacement. If either bulb is defective, the entire lot is rejected. Suppose a sample of 200 bulbs contains 5 defective bulbs. Find the probability that the sample will be rejected. Is a compromise on quality at the cost of quantity justifiable ? explain.

5. Paurush throw a die again and again until three sixes are obtained. Find

the probability of obtaining the third six in the sixth throw of die by paurush. There is no substitute of hard work. Comment.

6. There are 40 workers in a factory who are participating in strike for

increase in wages. Out of them 5 workers are sincere to management and believe that management know each thing and do the needful. Management selects 2 workers at random out of them. Write the probability distribution for the selected worker who are sincere. Also find the mean of the distribution. Write the importance of sincerity in service.

7. There are 40 workers in a night shift of a factory out of a which 10 are

drug- addicted. Two persons are selected at random out of them. Write the probability distribution for the selected workers who are drug addicted. Also find the mean of the distribution. Write the drawbacks of drug addiction.

8. There are 80 workers in a publication house. Out of which 20 are

hardworking and honest. Three workers are selected at random out of them. Write the probability distribution for the selected workers who are hard working and honest. Also, find the mean of the distribution. Write the importance of hardworking and honesty in One’s life.


9. A man is known to speak the truth 5 out of 6 times. He throws a die and reports that it is a six. (a) Find the probability that it is actually a six. (b) Find the probability that it is actually not a six. (c) How would telling a lie hamper one’s character development in future?

10. Suppose that the reliability of a HIV test is specified as follows : Of people

having HIV , 90% of the test detect the disease but 10% go undetected. Of People free of HIV, 99% of the test are judges HIV negative but 1% are diagnosed as showing HIV positive. From a large population of which 0.1% have HIV,one Person is selected at random, given the HIV , test and the pathologist reports him /her as HIV positive. (i) What is the probability that the person actually has HIV ? (ii) What is the full form of HIV. Write the disease associated with HIV.

11. In a town of 6000 people 1200 are blood donors and 2000 are female. It is

know that 30% of the females are blood donors. What is the probability that a random chosen individual from the town is either female or blood donor ? (a) Name the organization world wide helping the victims in war also ? (b) Blood donors and eye donors are helping the society and strengthen the love to Mankind. Comment.

12. In a group of 100 patriotic people, 20 believe in non-violence. If two

persons are selected at random out of them, (i)Find the probability distribution for the selected persons who are non- violent. (ii)Also find the mean & variance of the distribution.

13. 5000 cyclists, 4000scooter drivers& 3000 motor bike drivers are insured to an insurance company. Their respective probability of an accident are 0.01, 0.03& 0.25. If one of the insured persons met with an accident, find the probability that (i)He is a bike driver. (ii)Which mode of transport would you suggest to a student & why.

14. (a)Suppose that the reliability of a HIV test is specified as follows:Of people having HIV, 90% of the test detects the disease but 10% go undetected. Of people free of HIV , 99% of the test are judged HIV –ve but 1% are diagnosed as showing HIV +ve.From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV +ve. What is the probability that the person actually has HIV? (b) A & B are working in the same company . ‘A’noticed that ‘B ‘is suffering from a serious disease of which ’B’ is not aware. When asked for check up ‘B’ refused. ‘A’ convinced ‘B’ how important it is to get check up done. What according to you are the values displayed by ‘B’ in helping him.

15. A man is known to speak truth 3 out of 4 times. He throws a die and reports that I is a six. Find the probability that is actually a six. As a student what is the probability to speak truth.


16. Three students Rahul , Rahim & Rakesh appear for an interview and their respective chances of selection is 0.5 , 0.3 & 0.4.Find the probability that (i) none is selected (ii) atleast one is selected. (b).Rahul has to attend to an interview. As Rahul was not keeping well, he vomited on the way on his dress. Rahim, the friend of Rahul who was with him washed his shirt and provided him enough water and sprayed scent in his shirt to get rid of the foul smell. Rahul attended the interview, performed well and finally selected. What values do you find in Rahim& Rakesh .

17. Assume that the chances of a patient having a heart attack are 40%. It is

also assumed that a meditation and yoga course reduce the risk of heart attack by 30 % and prescription of certain drug reduces its chance by 25 %. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient select at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. In a student life state any one point how yoga and meditation influence studies.

18. There is a group of 50 people who are patriotic out of which 20 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. Also find the mean of the distribution. Explain the importance of Non violence in patriotism.

19. In answering a question on a MCQ test with 4 choices per question, a

student knows the answer, guesses or copies the answer. Let ½ be the probability that he knows the answer, ¼ be the probability that he guesses and ¼ that he copies it. Assuming that a student, who copies the answer, will be correct with the probability ¾, what is the probability that the student knows the answer, given that he answered it correctly? Arjun does not know the answer to one of the questions in the test. The evaluation process has negative marking. Which value would Arjun violate if he resorts to unfair means? How would an act like the above hamper his character development in the coming years?

20. An insurance company insured 2000 cyclists, 4000 scooter drivers and

6000 motorbike drivers. The probability of an accident involving a cyclist, scooter driver and a motorbike driver are 0.01, 0.03 and 0.15 respectively. One 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?

21. In a factory which manufactures bolts, machines A, B and C manufacture

respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?Should company get profit from sale of defective products or loss from not selling the non-defective products?

22. 70% men and 30% women eat panmasala.10% of these men and 20% of

these women eat “x” brand. What is the probability that a person eating “x” brand will be a man? Why should you discourage your friends to eat pan masala?


23. A biased die is twice as likely to show an even number as an odd number.

The die is rolled three times. If occurrence of an even number is considered a success, then write the probability distribution of number of successes. Also find the mean number of successes. Which human value is violated in this case?

24. In a backward state, there are 729 families having six children each. If the

probability of survival of a girl is 1/3 and that of a boy is 2/3, find the number of families having 2 girls and 4 boys. Do you believe that a female child is neglected in backward areas? What steps will you take to restore respect of a female child in society?

25. There are 40 hardworking scholars in a class out of which 10 are sports

persons.3 scholars are selected out of them. Write the probability distribution for selected persons who are also sports persons. Find the mean distribution. Explain the importance of sports in education.

26. An anti aircraft gun of Indian army can take maximum number of shots at

an enemy plane moving away from it. The probabilities of hitting the plane at the first, second third and fourth shots are 0.8, 0.6, 0.5 and 0.2 respectively. What is the probability that the plane will be shot down? If 125 enemy planes one after another are fired by the gunner, how many planes are likely to be shot down? Do you think that the success will boost the morale of the Indian army?

27. In all Indian dancing competition, the odds that a dance will be favorably

reviewed by three independent examiners are 5:2 , 4:3 , and 3:4 respectively. What is the probability that none of the examiners will approve the dance? What is the probability that a majority will approve the dance? Do you think that dancing is a part of Indian culture?

28. An examinee either guesses or copies or knows the answer of a multiple

choice question with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6.the probability that the answer is correct, given that he copied the answer is 1/8.find the probability that he knew the answer when he answered it correctly. What steps should be taken to discourage guess and copy in the examination?

29. 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?

30. 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?

31. 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.


32. 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?

33. 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?

34. 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?

35. 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.

36. 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.

37. 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?

38. 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 students 9 respectively, 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.

39. 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?

40. 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?


41. A manufacturer has three machine operators A (skilled) B (Semi- skilled)

and C (non-skilled).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?

42. 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?

43. 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 and violence will ruin the country?

44. 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?

45. 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 it is 1/7. He prepared well for the two posts by getting all the possible informations. What is the probability that he is selected for at least one of the post ? Which values in life he is representing ?

46. A speaks truth in 60% of the cases and B in 90% of the cases. In what

percentage of cases are they likely to contradict each other in stating the same fact ? Which values A is lacking and should improve upon ?

47. There is a group of 100 people who are patriotic out of which 70 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. Also find the mean of the distribution. Explain the importance of non-violence in patriotism.

48. In answering a question on a multiple choice questions test with four

choices per question. A student knows the answer, guesses or copies the answer. If 1/2 be the probability that he knows the answer, 1/4 be the probability that he guesses it and 1/4 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 unfairness ? How would his personality would be hampered ?

49. An insurance company insured 4,000 cyclists, 8,000 scooter drivers and

12,000 car drivers. The probability of an accident involving a cyclist, scooter driver and a car driver are 0.02, 0.06 and 0.30 respectively. One of the insured person 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 ?

50. In an examination an examinee either guesses or copies or knows the answer of

multiple choice questions with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6.The probability that his answer is correct, given that he copied it, is 1/8. Find the probability that he knew the answer to the question, given that he correctly answered it. While copying and giving correct answer, which life value the examinee is lacking?

51. A target is displayed as ‘BE TRUTHFUL’. The probability of A hitting a

target is 4/5 and that of B hitting it is 2/3. They both fire the target. Find the probability that (i) at least one of them will hit the target. (ii) only one of them will hit the target. Write your views in two lines about truthfulness.

52. A student is to be selected out of two students for a competition regarding

debate on ‘SAVE NATURAL RESOURCES’. Both are competent. It was decided to use die to select the students. Two students A and B take turns in throwing two dice. The first to throw a sum of 10, is being selected. Show that, if A has the first throw, their chances of winning are in the ratio 12 : 11. Do you wish to be student A ?

53. A is known to speak truth three times out of five times. He throws a die

and reports that it is 4. Find the probability it is actually 4. Do you think A needs to improve his acquired life value about truth ?

54. The probability that a bulb produced in a factory will fuse after 150 days of

use is 0.05. Find the probability that out of 5 such bulbs (i) none (ii) not more than one (iii) more than one (iv) at least one will fuse after 150 days of use. Do you prefer bulbs or other means which saves energy ?

55. Two students are to be chosen at random from 52 students of a class

having 4 boys and 48 girls, to be a part of ‘CLEANLINESS DRIVE’. Find the probability distribution for the number of boys. Do you want to be a part of a drive ?

56. An urn contains 10 white and 3 black balls. Another urn contains 3 white

and 5 black balls. Two balls are drawn from first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that it is a white ball. If white ball represents ‘ + life skills’ and black ball represents ‘– life skills’. Which ball you want to be ? Mention two ‘+ life skills’.


57. A problem in Mathematics is given to three students whose chances of

solving are 1 1 1, ,3 4 5

. What is the probability that the problem is solved ?

Looking at the probability of three students solving a problem, do you think they need to do more to improve performance ? What life skill they need to acquire ?

58. There are three probable candidates A, B and C for the post of a principal.

The chances of their selection are in proportion 4 : 2 : 3 respectively. The probability that A, if selected, will introduce co-education in college is 0.3. The probability of B and C doing the same are respectively 0.5 and 0.8. What is the probability that there will be co-education in the college ? Find the probability that principal B introduced co-education in college. Give your views about co-education and mention some life skills for the same.

59. By examining the chest X-ray, the probability that T.B is detected when a

person is actually suffering from it is 0.99. The probability of incorrect diagnosis is 0.001. In a certain city 1 in 1000 persons suffers from T.B. A person is selected at random and is diagnosed to have T.B. What is the chance that he actually has T.B. ? Give two life skills which a doctor must acquire.

60. The odds that a book will be received favourably by three independent

critics are 5 : 2, 4 : 3 and 3 : 4 respectively. What is the probability that majority will be in favour of the book ? Do you think we must make books our friends, and when we get time we must read value based books besides our subject books ?

61. An unbiased dice is thrown again and again until three sixes are obtained.

Find the probability of obtaining third six in the sixth throw of a dice. If with each obtaining ‘six’ we are allowed to choose a life skill tag. Which three life skill tags you will chose ?

62. A speaks truth in 60% of the cases and B in 90% of the cases. In what

percentage of cases they are likely to contradict each other in stating the same fact ? Which out of the two A and B you would like to make your friend ? Why ?

63. If the people having HIV, 90% of the tests detect the disease but 10% go

undetected. Of people free of HIV, 99% of the tests are judged HIV – ve but 1% are diagonosed as showing HIV + ve. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test and is found to be HIV + ve. What is the probability that a person actually has HIV ? Are you aware about HIV ?

64. A factory has three plants A, B and C. Their daily production is 500, 1000

and 2000 units. Out of these 0.5%, 0.8% and 1% units respectively are found to be defective. An item is chosen at random and is found to be defective. What is the probability it came from plant A ? A manufacturer knows that the product is defective, even then sells it in the market. Is he doing the right thing ? Which life skill he is lacking ?


65. 20% of the bulbs produced by a machine are defective. Find the probability distribution of the number of defective bulbs in a sample of 4 bulbs chosen at random. Do you think that company producing 20% defective items is in fair shape ? Which life skill company is lacking ?

66. There are 2,000 scooter drivers, 4,000 car drivers and 6,000 truck drivers

all insured. The probabilities of an accident involving a scooter, a car, a truck are 0.01, 0.03, 0.15 respectively. One of the insured drivers meets with an accident. What is the probability that he is a scooter driver ? Are these modes of transport Eco-friendly ? They should switch to which fuel to be Eco-friendly.

67. Two coins are tossed once. What is the probability of getting 2 heads, if it

is known that at least one head comes up ? If ‘head’ represents acquired skill value ‘EMPATHY’. Would you like to get head each time ?

68. Three students A, B and C are given a question to solve. Their respective

probabilities to solve the question are 1 1,2 3

and 14

. What is the probability

that (i) any one of them will solve the question (ii) the question will be solved ? Which of the three students need to work hard to improve his/her performance? Name the life skills to be acquired.

69. In a hurdle race, a player has to cross 10 hurdles. The probability that he

will clear each hurdle is 56

. What is the probability that he will knock down

fewer than 2 hurdles ? Which life skills a player has acquired ? 70. A man is known to speak the truth 3 out of 4 times. He throws a die and

reports that it is a six. Find the probability that it is actually a six. Besides ‘TRUTHFUL’ name the life skills which we should acquire ?

71. A pair of dice is thrown 7 times. If getting a total of ‘7’ is considered a

success, find the probability of getting. (i) not success (ii) exactly 6 successes (iii) at least 6 successes (iv) atmost 6 successes Name two life skills to be successful in life.

72. A speaks truth in 75% and B in 80% of the cases. In what percentage of

cases are they likely to contradict each other in narrating the same incident ? Give your views about ‘non-agreement’ during the discussion. Name the life skills to be in positive disagreement with other persons.

73. Two cards are drawn simultaneously from a well-shuffled pack of 52 cards.

Find the mean and variance for the number of aces. Do you think during the game, we should always play fair game?

74. Three persons A, B, C throw a die in succession till one gets a ‘six’ and

wins the game. Find their respective probabilities of winning. Here we notice C’s chance of winning is least. Should C take it supporting or suggest a way, so that C does not have to be a third player in each try ?


75. A, B and C shoot to hit a target. If A hits the target 4 times in 5 trials; B hits

it 3 times in 4 trials and C hits it 2 times in 3 trials, what is the probability that the target is hit by at least 2 persons ? Do you think that hitting a target helps us in developing the life skills. Name one of the life skills.

76. A bag contains tickets numbered 1, 2, 3, ..., 50 of which five tickets x1, x2,

..., x5 are drawn at random and arranged in ascending order of magnitude x1 < x2 <x3 < x4 < x5. What is the probability that x3 = 30 ? If x1, x2, ..., x5 represent different life skills. Can you decide about the relation x1 < x2 < x3< x4 < x5. i.e., prefer one life skill in comparison to other ?

77. Three urns A, B and C contain 6 red and 4 white; 2 red and 6 white and 1

red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If a ball drawn is found to be red, find the probability that the ball was drawn from the urn A. An urn is chosen at random. This relates to which life skill ?

78. A and B toss a coin alternately till one of them gets a head and wins the

game. Find the probability of each winning the game, if A starts. Do you recommend that always A should start new game ?

79. A doctor is to visit a patient. From the past experience it is known that the

probability that he will come by train, bus, scooter or by other means of

transport are 3 1 1 2, ,10 5 10 5

and respectively. The probabilities that he will be

late are 1 1 1, ,4 3 12

and if he comes by train, bus and scooter respectively.

But when he comes by other means of transport he will not be late. When he arrives he is late. What is the probability that he comes by train ? What is your opinion about being ‘late’. Which life skills a person is lacking ?

80. By examining the chest X-ray, the probability that T.B. is detected when a

person is actually suffering from it is 0.99. The probability that the doctor diagnosis incorrectly that a person has T.B. on the basis of X-ray is 0.001. In a certain city 1 in 1000 persons suffer from T.B. A person is selected at random and is diagnosed to have T.B. What is the chance that he actually has a T.B. ? Give two precautions measure to remain free from diseases (if possible).

81. Two persons appear for an interview for two vacancies available. The

chance of A’s selection is 17

and B’s selection is 15

. What is the chance

that : (i) both are selected ? (ii) both are rejected ? (iii) only one of them is selected ? (iv) at least one of them is selected ? Name two qualities for a person to possess, while appearing for an interview.

82. Five dice are thrown successively. If the occurrence of an even number in

a single die is considered a success, find probability of atmost 3 successes. How does the word ‘SUCCESS’ strike you ? Name a life skill to be successful in life.


83. Suppose that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. What is the probability that he is a male assuming that males and females are equal in proportion ? Should colour blindness, if any be taken care of immediately ?

84. In shuffling a pack of 52 playing cards, four are accidently dropped. Find

chance that missing cards should be one from each suit. If four cards represent four life skills. Name the life skills which you would always abide with ?

85. A man is known to speak the truth 2 out of 4 times. He throws a die and

reports that it is a six. Find the probability that it is actually a six. Besides ‘TRUTHFUL’ name the life skills which we should acquire ?


SAMPLE PAPER – I SUBJECT : MATHEMATICS

Class – XII

Time Allowed: 3 hrs Maximum Marks:100 General Instructions: i. All questions are compulsory ii. The question paper contains 3 pages and is divided into three sections A, B and C.

Section A comprises 6 questions each 1 mark, Section B comprises 13 questions each 4 marks and Section C comprises 7 questions each 6 marks.

iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question.

iv. There is no overall choice. However internal choice has been provided in 4 questions of 4 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.

v. Use of calculators is not permitted. However you may ask for mathematical tables.

SECTION – A

1. Find the principal value of 1 3tan tan4

.

2. If 1 1tan ( 3) cot2

x , find x.

3. Fins x – y + z, if 957

x y zx zy z

.

4. If cos sinsin cos

x xA

x x

, then for what value of x, A is an identity matrix.

5. If 1 24 2

A

, then find k if |2A| = k .|A|.

6. The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. Find the marginal revenue when x = 5 and write which value does the question indicate.

SECTION – B

7. Let A = N x N and X be a binary operation on A defined by (a, b) x (c, d) = (a + c, b + d). Show that X is commutative and associative. Also find it identity element for X on A, if any.

8. Prove that 2

1 12

1cos tan sin cot2

xxx


OR

Solve for x: 1 1 2tan 2cot3

x x

9. Using the properties of determinants, prove that

2

2 2 2 2

2

4a ab ac

ba b bc a b cca cb c

.

10. If ( sin )x a and (1 cos )y a , find 2

2

d ydx

11. If y x yx e , show that 2

loglog( )

dy xdx xe

12. Find the value of ‘a’ if the function f(x) defined by 2 1, 2

( ) , 21, 2

x xf x a x

x x

is

continuous at x = 2. OR

Find the values of a and b such that the function defined by as follows is

continuous 2 2

( ) 2 53 2 2

x xf x ax b x

x x

13. Evaluate: sin cossin 2x xdx

x

OR

Evaluate: 22

(1 )(1 )dx

x x

14. Evaluate: 5( 3)dx

x x

15. Evaluate: cos

cos cos0

x

x xe dx

e e

16. Find a vector of magnitude 5 units and parallel to the resultant of 2 3a i j k

and 2b i j k .

17. Find the angle between the pair of lines given by 2 5 (3 2 6 )r i j k i j k

and 7 6 6 ( 2 2 )r i j k i j k

18. Find the shortest distance between the lines (1 ) (2 ) ( 1)r i j k and

2 (2 2 )r i j k i j k

19. Past experiences shows that 80% of operation performed by a doctors an

successful. If he performs 4 operations in a day what is the probability that at least three operations will be successful. What values are reflected by the doctors?.

SECTION – C

20. Find the intervals in which the function is 2 3( ) 20 9 6f x x x x (a) strictly increasing (b) strictly decreasing.


OR Show that of all rectangles of given area the square has the smallest perimeter.

21. Find the area of the region enclosed by the parabola x2 = 4y and the line x = 4y – 2.

22. Solve the differential equation: 23 dxy x xdy

23. Write the vector equation of the following line and hence find the distance

between them 1 2 42 3 6

x y z and 3 3 5

4 6 12x y z

.

OR Find the coordinates of the foot of the perpendicular drawn from the point (0, 2, 3)

on the line 3 1 45 2 3

x y z , Also find the length of the perpendicular.

24. In a bolt factory machines A, B and C manufactures 25%, 35% and 40% of the total production respectively. Out of their total outputs 5%, 4% and 2% are defective bolts. A bolt is drawn at random and found to be defective, what is the probability that it is manufactured by machine B.

25. A retired person has Rs. 70,000 to invest in two types of bonds. First type of bond yields an annual income of 8% on the amount invested and the second type of bond yields 10% per annum. As per norms he has to invest minimum of 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 ? Do you think that a person should start saving at an early age for his retirement ? Can you name some avenues ?

26. Two schools A and B decided to award prizes to their students for three values Honesty (x), Punctuality(y) and Obedience (z). School A decided to award a total of ` 15,000 for the three values to 4, 3 and 2 students respectively, while school B decided to award ` 19,000 for the three values to 5, 4 and 3 students respectively. If all the three prizes together amount to ` 5,000, then

(i). Represent the above situation by a matrix equation and form linear equation using matrix multiplication.

(ii). Which value you prefer to be rewarded most and why ?

maths value based question bank for class xii 2015-16

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