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OMTEX CLASSESTHE HOME OF TEXT

SYJC MATHS NOTES FOR COMMERCE STUDENTS Page 1

Preface

It gives us great pleasure to present this thoroughly revised edition of

OMTEX MATHEMATICS & STATISTICS for Standard XII , prepared

according to the pattern prescribed by the board.

A thorough study and practice of this edition with the help of Omtex

guidance (teaching + coaching) will enable the students to pass the HSC

Examination with flying colours.

Meticulous care has been taken to make this edition of OMTEX

MATHEMATICS & STATISTICSperfect and useful in every respect. However,

suggestions, if any, for its improvement are most welcome.

Omtex

Note: - No part of this book may be copied, adapted, abridged or translated, stored in any

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or by any means without a prior written permission of the Omtex classes.


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MATHS II


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CH. NO. 1. THEORY OF ATTRIBUTES

EX. NO. 1

1. Find the missing frequencies in the following data of two attributes A and B.

= 800,

= 120,

= 500,

= 300.

2. For a data for 2 attributes, it is given that = 500, = 150, = 100, = 60,findthe other class frequencies.3. In a population of 10,000 adults, 1290 are literate, 1390 are unemployed and 820 are literate

unemployed. Find the number of (i) literate employed. (ii) literates, (iii) employed.

4. In a co educational school of 200 students contained 150 boys. An examination wasconducted in which 120 passed. If 10 girls failed, find the number of (i) boys who failed, (ii)

girls who passed.

5. In a sample of 240 persons, 40 were graduates and 5 were graduates employed. If 40 non graduates were employed, find the number of unemployed non graduates and the number of

unemployed persons.

6. If for 3 attributes A, B and C, it is given that (ABC) = 210, = 280, = 180 = 240, = 250, = 160, = 360, = 32, (A), (B), (C),(AB), (AC) and (BC).

7. If for 3 attributes A, B, C, it is given that (ABC) = 370, = 1140, = 230, = 960, = 260, = 870, = 140, = 1030,, , , .8. If N = 800, (A)=224, (B) = 301, (C) = 150, (AB) = 125, (AC) = 72, (NC) = 60 and (ABC) = 32, find.

EX. NO. 2

Check the consistency of the following data.

1.

= 100, = 150, = 60, = 500.2. = 100, = 150, = 140, = 500.3. = 300, = 400, = 200, = 1000.4. 150, = 45, = 125, = 200.5. = 40, = 70, = 160, = 200.6. = 75, = 50, = 55, = 300.7. = 50, = 79, = 89, = 782.8. = 200, = 300, = 300, = 1000.

EX. NO. 3

1. Discuss the association of A and B ifi. N = 100, (A) = 50, (B) = 40, (AB) = 20.

ii. (AB) = 25, = 30, = 25, = 20.2. Discuss the association between attributes A and B ifi. N = 100, (A) = 40, (B) = 60, (AB) = 30.

ii. N = 1000, (A) = 470, (B) = 620, (AB) = 320.iii. N = 500, = 300, = 350, = 60.iv. N = 1500, = 1117, = 360, = 35.


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3. Find the association between literacy and unemployment in the following data.Total No. Of adults 1000

No. Of literate 130

No. Of unemployed 140

No. Of literate unemployed 80

4.

Find the association between literacy and employment from the following data.Total Adults 10000 Unemployed 1390

Literates 1290 Literate unemployed 820

Comment on the result.

5. Show that there is very little association between the eye colour of husband s and wives from thefollowing data.

Husband with light eyes and wives with light eyes = 309

Husband with light eyes and wives with dark eyes = 214

Husband with dark eyes and wives with light eyes = 132

Husband with dark eyes and wives with dark eyes = 119

6. 88 persons are classified according to their smoking and tea drinking habits. Find Yules coefficientand draw your conclusion.

Smokers Non smokers

Tea Drinkers 40 33

Non Tea Drinkers 3 12

7. Show that there is no association between sex and success in examination from the following data.Boys Girls

Passed examination 120 40

Failed examination 30 10

8. Find Yules coefficient to determine if there is association between the heights ofspousesTall Husbands Short Husbands

Tall Wives 60 10

Short Wives 10 509. 300 students appeared for an examination and of these, 200 passed. 130 had attended a coaching

class and 75 of these passed. Find the number of unsuccessful students who did not attend the

coaching class. Also find Q.

10.Calculate Yules coefficient of association between smokers and coffee drinkers, from the followingdata.

Coffee Drinkers Non coffee Drinkers

Smokers 90 65

Non smokers 260 110

11.Out of 700 literates in town, 5 were criminals. Out of 9,300 literates in the same town, 150 werecriminals. Find Q.

12.Examine the consistency of the following data and if so, find Q.N = 200, (AB) = 24, = 160, = 70.

13.Find Yules coefficient of association for the following data.Intelligent husbands with intelligent wives 40

Intelligent husbands with dull wives 100

Dull husbands with intelligent wives 160

Dull husbands with dull wives 190


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CH. NO. 2. NUMERICAL METHODEX. NO. 1. NEWTONS FORWARD INTERPOLATION FORMULA.

1. Using Newtons Interpolation formula, find f(5) from the following table. 2 4 6 8

(

) 4 7 11 18

2. Given the following table find f(24)using an appropriate interpolation formula. 20 30 40 50() 512 439 346 2433. In an examination the number of candidates who scored marks between certain limits were as

follows. Estimate the number of candidates getting marks less than 70.

Marks 0-19 20-39 40-59 60-79 80-99

No. Of Candidates 41 62 65 50 17

4. The population of a town for 4 year was as given below.Year 1980 1982 1984 1986

Population (in Thousand) 52 54 58 63

5. For a function f(x), f(0) = 1, f(1) = 3, f(2) = 11, f(3) = 31. Estimate f(1.5), using NewtonsInterpolation formula.

6. For a function f(x), f(1) = 0, f(3) = 25, f(5) = 86, f(7) = 201. Find f(2.5) using Forward Differenceinterpolation formula.

7. Construct a table of values of the function = 2 for x = 0,1,2,3,4,5. Find (2.5) and f(2.5)2 usingNewtons Forward Interpolation Formula.

8. Estimated values of logarithms upto 1 decimal are given below find log(25) 10 20 30 40 1 1.3 1.4 1.69. Estimated values of sin upto 1 decimal are given below find sin(450)

00 300 600 900

0 0.5 0.87 110.Find f(x), if f(0) = 8, f(1) = 12, f(2) = 18.11.f(x) is a polynomial in x. Given the following data, find f(x) 1 2 3 4() 7 18 35 58

Also find f(1.1)

EX. NO. 2. LAGRANGES INTERPOLATION FORMULA.

1. By using suitable interpolation formula estimate f(2) from the following table. -1 0 3() 3 1 192. By suing Lagranges Interpolation formula, estimate f(x) when x = 3 from the following table.

0 1 2 5() 2 3 10 1473. A company started selling a new product x in the market. The profit of the company per year due

to this product is as follows:

Year 1st 2nd 7th 8th

Profit (Rs. In lakh) 4 5 5 5

Find the profit of the company in the 6thyear by using Lagranges Interpolation formula.


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4. Using the Lagranges Interpolation formula, determine the percentage number of criminals under35 years.

Age % number of criminals

Under 25 years

Under 30 years

Under 49 yearsUnder 50 years

52

67.3

84.194.4

5. The function y = f(x) is given by the points (7,3), (8,1), (9,1), (10, 9). Find the value of y at x = 9.5using Lagranges formula.

6. Given 1010 = 1, 1012 = 1.1, 1015 = 1.2 1020 = 1.3. find1013 = ? [Valuesare approximate and rounded off to 1 decimal place].

EX. NO. 3. FORWARD DIFFERENCE TABLE

1. Form the difference table for f(x) = x2 +5 taking values for x = 0, 1 , 2 , 3.2. Write down the forward difference table of the following polynomials f(x) for x = 0(1)5

a. f(x) = 4x-3b. f(x) = x2 4x 4.

3. Obtain the difference table for the data. Also what can you say about f(x). From the table? x 0 1 2 3 4 5

f(x) 0 3 8 15 24 35

4. By constructing a difference table, obtain the 6 th term of the series 7, 11, 18, 28, 41.5. Estimate f(5) from the following table. 0 1 2 3 4() 3 2 7 24 596. By constructing a difference table, find 6th and 7th term of the sequence 6, 11, 18, 27, 38.7. By constructing a difference table, find 7th and 8th term of the sequence 8, 14, 22, 32, 44, 58.8. Given u4 = 0, u5= 3, u6 = 9 and the second difference are constant. Find u2.9. Find u9, if u3 = 5, u4 = 12, u5= 21, u6 = 32, u7= 45.

EX. NO. 4

1. Estimate the missing term by using " " "" from the following table.a. x 0 1 2 3 4

y 1 3 9 - 81

b. x 1 2 3 4 5 6 7

y 2 4 8 - 32 64 128

c.

1 2 3 4 5

() 2 5 7 - 322. Findeach of the following case, assuming the interval of difference to be 1.

i. = 1 2 3.ii. = 2 + . iii. =

2 2 + 4.iv. = 2 + 3.

3. Given = 2 + 3 + 5 taking the interval of differentiating equal to 1. Findand2.4. Given = 2 8 + 2, taking the interval of differentiating equal to 1. Findand2.


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5. Find2 if = + 1 + 2.6. Evaluate

i. 22 + 5 ii. sin + iii. cos ( + )7. Evaluate

i.

3

5

ii. 4( )iii. 2

3

iv. 2 33 v.

2

1

vi. 8. Show that2 . 2 = 9. Show that = log 1 + 10. If = . Show that, , 2, . are in geometric progression.11.Given: u0 = 3, u1 = 12, u2 = 81, u4 =100, u5= 8, find50.12.Given: u2 = 13, u3 = 28, u4 = 49, find22.13.Given: u2 = 13, u3 = 28, u4 = 49, u5= 76. Compute 32 + 23.14.Prove the following:

i. 4 = + 3 3 + 2 + 3 + ii. 4 = + 4 4 + 3 + 6 + 2 4 + +.iii. + 3 = + 3 + 32 + 3.iv. + 5 = + 5 + 102 + 103 + 54 + 5.

15.Assuming that the difference interval h = 1, prove the following.i. 4 = 3 + 2 + 21 + 31.

ii. 7 = 6 + 5 + 24 + 34.iii. 5 = 4 + 3 + 22 + 31 + 41.iv. 2 = 1 + 0 + 21 + 31.


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CH. NO. 3. BINOMIAL AND POISSON DISTRIBUTIONEX. 1

1. An unbiased coin is tossed 6 times. Findthe probability of getting 3 heads. (5/16)

2. Find the probability of getting atleast 4heads, in 6 trials of a coin. (11/32)

3. An ordinary coin is tossed 4 times. Findthe probability of getting

a. No heads(1/16)b. Exactly 1 head(1/4)c. Exactly 3 tails(1/4)d. Two or more heads(11/16)

4. On an average A can solve 40% of theproblems. What is the probability of A

solving

a. No problems out of 6.(729/15625)

b. Exactly four problems out of 6.(432/3125)

5. The probability that a student is not aswimmer is 1/5. Out of five students

considered, find the probability that

a. 4 are swimmers. (256/625)b. Atleast 4 are swimmers/

(2304/3125)

6. In a certain tournament, the probabilityof As winning is 2/3. Find the probabilityof As winning atleast 4 games out of 5.

(112/243)

7. A has won 20 out of 30 games of chesswith B. In a new series of 6 games, what is

the probability that A would win.

a. 4 or more games. (496/729)b. Only 4 games. (80/243)

8. If the chances that any of the 5 telephonelines are busy at any instant are 0.1, findthe probability that all the lines are busy.

Also find the probability that not more

than three lines are busy. (1/100000)

(99954/100000)

9. It is noted that out of 5 T.V. programs,only one is popular. If 3 new programs are

introduced, find the probability that

a. None is popular. (64/125)b. At least one is popular. (61/125)

10.A marks mans chance of hitting a targetis 4/5. If he fires 5 shots, what is the

probability of hitting the targeta. Exactly twice (31/625)b. Atleast once. (3124/3125)

11. It is observed that on an average, 1 personout of 5 is a smoker. Find the probability

that no person out of 3 is a smoker. Also

find that atleast 1 person out of 3 is

smoker. (64/125) (61/125).

12.A bag contains 7 white and 3 black balls.A ball drawn is always replaced in the

bag. If a ball is drawn 5 times in this way,

find the probability of we get 2 white and

3 black balls. (1323/100000)

EX. 2. BINOMIAL DISTRIBUTION

NOTE: - For a binomial variate parameter means n, p and q.

1. A biased coin in which P(H) = 1/3 and P(T) = 2/3 is tossed 4 times. If getting a head is successthen find the probability distribution.

2. An urn contains 2 white and 3 black balls. A ball is drawn, its colour noted and is replaced inthe urn. If four balls are drawn in this manner, find the probability distribution if success

denotes finding a white ball.

3. Find Mean and Variance of Binomial Distribution. Ifa. n = 12; p = 1/3 b. n = 10; p = 2/5 c. n = 100; p = 0.1

4. Find n and p for a binomial distribution, ifa. Mean = 6; S.D. = 2.b. = 6, variance = 5c. = 12, = 10.2d. = 10, = 3.


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EX. 3. POISSON DISTRIBUTION

Note: For a random variable x with a Poisson distribution with the parameter, theprobability of success is given by.

=

!

Note: - For a Poisson distribution Mean = Variance = .For a Poisson variate parameter is known as and = . If & .1. For a Poisson distribution with = 0.7, find p(2).

2. For a Poisson distribution with = 0.7, find( 2).3. If a random variable x follows Poisson distribution such that p(1) = p(2), find its mean and

variance.

4. The probability that an individual will have a reaction after a particular drug is injected is 0.0001.If 20000 individuals are given the injection find the probability that more than 2 having reaction.

5. The average number of incoming telephone calls at a switch board per minute is 2. Find theprobability that during a given period 2 or more telephone calls are received.

6. In the following situations of a Binomial variate x, can they be approximated to a Poisson Variate?a. n = 150 p = 0.05b. n = 400 p = 0.25

7. For a Poisson distribution with = 3,find p(2) , 3.8. The average customers, who appear at the counter of a bank in 1 minute is 2. Find the probability

that in a given minute

a. No customer appears.b. At most 2 customers appear.

9. The probability that a person will react to a drug is 0.001 out of 2000 individuals checked, find theprobability that

a. Exactly 3b. More than 2 individuals get a reaction.

10.A machine producing bolts is known to produce 2% defective bolts. What is the probability that aconsignment of 400 bolts will have exactly 5 defective bolts?

11.The probability that a car passing through a particular junction will make an accident is 0.00005.Among 10000 can that pass the junction on a given day, find the probability that two car meet

with an accident.

12.The number of complaints received in a super market per day is a random variable, having aPoisson distribution with = 3.3. Find the probability of exactly 2 complaints received on a givenday.

13.For a Poisson distribution if p(1) = p(2), find p(3).14. In a manufacturing process 0.5% of the goods produced are defective. In a sample of 400 goods.

Find the probability that at most 2 items are defective.

15. In a Poisson distribution, if p(2) = p(3), find mean.16. In a Poisson distribution the probability of 0 successes is 10%. Find its mean.


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CH. NO. 4. ASSIGNMENT PROBLEMS AND SEQUENCINGEx. No. 1

1. Solve the following minimal assignmentproblem.

A B C D

12

3

4

1625

10

15

110

25

7

60

2

14

1110

14

10

2. A Departmental Store has 4 wormers to packtheir items. The timing in minutes required

for each workers to complete the packing per

item sold is given below. How should the

manager of the store assign the job to the

workers, so as to minimize the total time of

packing?

Books Toys Crockery Cattery

AB

C

D

212

3

4

102

4

15

912

6

4

72

1

9


A B C D

1

2

3

4

3

5

1

4

4

6

2

10

6

10

3

6

5

9

2

4

4.

For an examination, the answer papers of thedivisions I, II, III and IV are to be distributed

amongst 4 teachers A, B, C & D. It is a policy

decision of the department that every

teacher corrects the papers of exactly one

division. Also, since Mr. As son is in Division I,he cannot be assigned the corrections of that

division. If the time required in days, for

every teacher to asses the papers of the

various divisions is listed below find the

allocation of the work so as to minimize the

time required to complete the assessment.

A B C DI

II

III

IV

-

4

6

1

5

5

6

6

2

3

2

3

6

8

5

4


A B C D

I 12 1 11 5

II

III

IV

3

3

2

11

4

13

10

6

11

8

1

7

6. A Departmental head has four subordinatesand four task to be performed. The time each

man would take to perform each task is

given below.

A B C D

I

II

III

IV

12

1

28

10

20

16

9

17

11

2

8

15

5

14

5

1

7. Minimise the following assignment problem. A B C D

I

II

III

IV

2

9

10

7

13

12

2

6

3

6

4

1

4

13

15

9

8. A team of 4 horses and 4 riders has enteredthe jumping show contest. The number of

penalty points to be expected when each

rider rides each horse is shown below. How

should the horses be assigned to the riders so

as to minimise the expected loss? Also find

the minimum expected loss.

HORSESRIDERS H1 H2 H3 H4

R1

R2

R3

R4

12

1

11

5

3

11

10

8

3

4

6

1

2

13

11

7

9. The owner of a small machine shop has fourmachinists available to assign jobs for the

day. Five jobs are offered to be done on the

day. The expected profits for each job done

by each machinist are given below. Find the

assignment of jobs to the machinists that will

results in maximum profit. Also find themaximum profit. [One machinist can be

assigned only one job]

JOBS

MACHINISTS

A B C D E

M1

M2

M3

M4

62

71

87

48

78

84

92

61

50

61

111

87

101

73

71

77

82

59

81

80


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10.A Chartered Accountants firm has acceptedfive new cases. The estimated number of

days required by each of their fiveemployees for each case are given below,

where -means that the particular employeecannot be assigned the particular case.

Determine he optimal assignment of cases tothe employees so that the total number of

days required completing these five caseswill be minimum. Also find the minimum

number of days.

CASES

EMPLOYEES

I II III IV V

E1

E2

E3

E4

5

3

6

4

2

4

3

2

4

-

4

2

2

5

1

3

6

7

2

5

E5 3 6 4 7 3

11.The cost (in hundreds of Rs.) of sendingmaterial to five terminals by four trucks,

incurred by a company is given below. Find

the assignment of trucks to terminals which

will minimize the cost. [One truck is

assigned to only oneterminal] Which

terminal will not receive material from thetruck company? What is the minimum cost?

TRUCKS

TERMINALS

A B C D

T1

T2

T3

T4

T5

3

7

3

5

5

6

1

8

2

7

2

4

5

6

6

6

4

8

3

2

EX. NO. 2

1. Find the sequence that minimises the totalelapsed time, required to complete the

following jobs on two machineries.

Job A B C D E F G

M1 7 2 3 2 7 4 5

M2 4 6 5 4 3 1 4

2. Solve the following for minimum elapsed timeand idling time for each machine.

Job A B C D E

M1 5 1 9 3 10

M2 2 6 7 8 4

3. Solve the following problems for minimumelapsed time. Also state the idling time for the

machine.

Job 1 2 3 4 5 6 7 8 9

M1 2 5 4 9 6 8 7 5 4

M2 6 8 7 4 3 9 3 8 11

4. Solve the following problem for minimumelapsed time. Also state the idling time for

each machine.

Job 1 2 3 4 5

Machine A 8 10 6 7 11

Machine B 5 6 2 3 4

Machine C 4 9 8 6 5


each machine.

Job 1 2 3 4 5 6

Machine A 8 3 7 2 5 1

Machine B 3 4 5 2 1 6

Machine C 8 7 6 9 10 9


each machine.

Job A B C D E F G

Machine A 2 7 6 3 8 7 9

Machine B 3 2 1 4 0 3 2

Machine C 5 6 4 10 4 5 11

7. Five jobs have to go through the machines A,B, C in order ABC. Following table shows the

processing times in hours for the five jobs.

Job J1 J2 J3 J4 J5

Machine A 5 7 6 9 5Machine B 2 2 4 5 3

Machine C 3 6 5 6 7

Determine the sequence of jobs, which will

minimise the total elapsed time.

8. Determine the eptimum sequence so as tominimize the total elapsed time.

Type of

Chairs

Number

To be

processed/day

Processing

time on

12

3

4

5

6

46

5

2

4

3

Machine

A B4

12

14

20

8

10

8

6

16

22

10

2


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CH. NO. 5. VITAL STATISTICS, MORTALITY RATES AND LIFE TABLECRUDE DEATH RATE (C.D.R.)

1. For the following data, find the crude death rate.Age group 0-25 25-50 50-75 Above 75

Population 5000 7000 6000 2000

No. of deaths 800 600 500 100

2. Compare the crude death rate of the two given population.Age group 0-30 30-60 60 & above

Population A

Deaths in A

4000

180

8000

120

3000

200

Population B

Deaths in B

7000

250

9000

320

4000

230

3. Compare the crude death rate of the two given population.Age group 0-25 25-50 50-75 Above 75

Population A in thousands

Deaths in A

60

250

70

120

40

180

30

200

Population B in thousandsDeaths in B

20120

40100

30160

10170

4. For the following data. Find if the C.D.R. = 31.25 per thousand.Age group Population Deaths

0-3535-70

Above 70

40003000

1000

80120

5. For the following data. Find if the C.D.R. = 3.75Age group 0-20 20-40 40-60 Above 60

Population in thousands 58 71 41 30

Deaths 195 130

245

6. For the following data. Find if the C.D.R. = 50Age group 0-25 25-40 40-70 Above 70Population in thousands 25 28 15Deaths 1250 1000 1570 1680

SPECIFIC DEATHS RATES (S.D.R.)

1. Find the Age Specific deaths rates (S.D.R.) for the following data.Age group Population No. of deaths

0-15

15-4040-60

Above 60

6000

200001000

4000

150

180120

1602. Find the age Specific deaths rates (S.D.R.) for population A and B of the following.Age group 0-30 30-60 60 and abovePopulation A in thousands

Deaths in A

50

150

90

180

30

200


60120

100160

20250


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3. Find the Age specific deaths rates (S.D.R.) for population A and B for the following.Age group 0-30 30-60 60-80 Above 80Population A in thousands

Deaths in A

30

150

60

120

50

200

20

400

Population B in thousands

Deaths in B

50

200

100

140

90

270

70

350

STANDARD DEATHS RATES (S.T.D.R.)1. Find the Standard Deaths Rates for the following data:

Age group 0-30 30-60 Above 60Population A in thousands

Deaths in A

60

240

90

270

50

250

Standard Population in thousands 20 30 20

2. Find the Standard Deaths Rates for the following data.Age group 0-25 25-50 50-75 Over 75Population A in thousands

Deaths in A

66

132

54

108

55

88

25

100


34102

58116

5278

1680

Standard Population in thousands 40 60 80 20

3. Taking A, as the standard population. Compare the standardized death rates for thepopulation A and B for the given data.

Age group 0-30 30-60 Above 60Population A in thousands

Deaths in A

5

150

7

210

3

120


Deaths in B

6

240

8

160

2.5

7.5

4. Taking A, as the standard population. Compare the standardized death rates for thepopulation A and B for the given data.

Age group 0-20 20-40 40-75 Above 75Population A in thousandsDeaths in A

7

140

15

150

10

110

8

240


Deaths in B

9

270

13

260

12

300

6

150

LIFE TABLES

1. Construct the life tables for the rabbits from the following data.x 0 1 2 3 4 5 6

lx 10 9 7 5 2 1 0

2. Construct the life tables for the following data.x 0 1 2 3 4 5 6lx 50 36 21 12 6 2 0

3. Construct the life tables for the following data.x 0 1 2 3 4 5

lx 30 26 18 10 4 0


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4. Fill in the blanks in the following tabled marked by ? sign.Age lx dx qx px Lx Tx e0x50

51

60

50

?

-

?

-

?

-

?

-

240

?

?

?

5.Fill in the blanks in the following table marked by ? sign.Age lx dx qx px Lx Tx e0x56

5758

400

250120

?

?-

?

?-

?

?-

?

?-

3200

??

?

??

CH. NO. 6. INDEX NUMBEREX. NO. 1.SIMPLE AGGREGATIVE METHODI. FIND THE INDEX NUMBER.1. Find Index number. [Ans. 137.73]

Commodities Prices in

2002 (P0)

Prices in

2003 (P1)

III

III

IVV

21.355.9

100.2

60.570.6

30.788.4

130

90.185.72. Find Index number. [Ans. 180]

Commodities Prices in

1990 (P0)

Prices in

2002 (P1)

A

B

C

D

E

12

28

10

16

24

38

42

24

30

463. Find Index number. [Ans. 107.1, 109.375]Commodities Prices

in

2000

Prices

in

2003

Prices

in

2006

TrucksCars

Three wheelersTwo wheelers

800176

10044

830200

12743

850215

115434. Find Index number. [Ans. 64.06, 39.06]

Commodities 1998 2000 2005

P0 P1 P1

Stereo 10 6 5

T.V.Computer

Mobile

3080

8

2050

6

1525

55. Find the index number for the year 2003 and2006 by taking the base year 2000. [Ans. 48,

75.4359]

Security at

Stock market

2000 2003 2006

P0 P1 P1A

B

CD

E

160

2400

8003500

150

180

35

5502000

600

210

8

8504000

2206. Calculate Index Number. [Ans. 69.078,238.15]

Real Estate

Area wise

1990 1998 2006

A

B

CD

100

35

512

65

22

711

250

75

12257. Calculate Index Number. [Ans. 113.0952]

Items 2000 2005

Wheat

Rice

DalMilk

Clothing

500

400

70020

60

600

430

77032

68


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8. Calculate the Index Number. [Ans. 412.19,92.68]

Security at

Stock market

1988 1991 1994

P0 P1 P1

ABC

D

E

6501200530

270

1450

350013504700

5050

2300

7001300200

100

1500

9. Compute the Index Number. [Ans. 110.526,126.31579]

Food

Items

Units 2004 2005 2006

P0 P1 P1

PotatoOnionTomato

Eggs

Banana

KgKgKg

Dozen

Dozen

101212

24

18

122525

2

20

141616

26

24

II. THE INDEX NUMBER BY THE METHOD OF AGGREGATES IS GIVEN IN EACH OF THEFOLLOWING EXAMPLE. FIND THE VALUE OF X IN EACH CASE.

1. Index Number = 180Commodity Base year Current Year

P0 P1

AB

C

D

E

122826

24

3841

25

36

40

[Ans. = 10]2. Index Number = 112.5

Commodity Base Year Current Year

P0 P1I

IIIII

3

1640

5

2535

IV

V

7

14

10[Ans. = 15]3.

Index Number = 120

Commodity Base Year Current Year

P0 P1I

IIIII

IV

V

40

805030

60

9070

110

30

[Ans. = 100]

EX. NO. 2.WEIGHTED AGGREGATIVE INDEX NUMBERS.1. For the following data find Laspeyres, Paasches, Dorbish Bowleys and Marshall Edgeworth

Index Numbers. [Ans. 134.2, 130, 132.1, 132.05]

Commodities Base Year Current Year

Price Quantity Price Quantity

A

B

C

D

20

30

50

70

3

5

2

1

25

45

60

90

4

2

1

3

2. For the following data find Laspeyres, Paasches, Dorbish - Bowleys and Marshall Edgeworth Index Numbers. [Ans. 144.11, 149.2, 146.66, 147.422]Commodities Base Year Current Year


1

2

34

10

40

3060

3

4

12

20

60

5070

3

9

42


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OMTEX CLASSESTHE HOME OF SUCCESS

17

3. Find Fishers Price Index Number. [Ans. 132.1] {using log table} Commodities Base Year Current Year


A

B

C

D

20

30

50

70

3

5

2

1

25

45

60

90

4

2

1

3

4. Find Walschs Price Index Number.[Ans. 116.21]Commodities Base Year Current Year


I

II

IIIIV

10

40

3050

4

5

10.5

20

3

5060

9

5

42

5. Calculate Price Index Number by using Walschs Method. [Ans. 126.83]Commodities Base Year Current Year


AB

C

52

10

46

9

73

12

16

4

6. The ratio of Laspeyres and Paasches Index number is 28:27. Find x. [Ans. x = 4]Commodities 1960 1965


A

B

1

1

10

5

2

X

5

2

7. For the following the Laspeyres and Paasches index number are equal, find .Commodity P0 Q0 P1 Q1A

B

4

4

6

6

4

5

4

EX. NO. 3.COST OF LIVING INDEX NUMBERTHERE ARE TWO METHODS TO CONSTRUCT COST OF LIVING INDEX NUMBER.

1. AGGREGATIVE EXPENDITURE METHOD.2. FAMILY BUDGET METHOD.

1. Taking the base year as 1995, construct the cost of living index number for the year 2000 fromthe following data. [Ans. 137.5]

Group 1995 2000

Price Quantity Price

Food

Clothes

Fuel and Lighting

House Rent

Miscellaneous

23

15

5

12

8

4

5

9

5

6

25

20

8

18

13


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18

2. The price relatives I, for the current year and weights (W), for the base year are given belowfind the cost of living Index number. [Ans. 221.3]

Group Food Clothes Fuel & Lighting House Rent Miscellaneous

I

W

320

20

140

15

270

18

160

22

210

25

3. Find the cost of living Index number. [Ans. 150]Group Food Clothes Fuel & Lighting House Rent Miscellaneous

I

W

200

6

150

4

140

3

100

3

120

4

4. Find the cost of living index number. [Ans. 208]Group 1995 2000


FoodClothes

Fuel and Lighting

House Rent

Miscellaneous

9025

40

30

50

54

3

1

6

20080

50

70

90

5. Find the cost of living index number. [Ans. 86.06]Group 1995 2000


Food

Clothes

Fuel and Lighting

House RentMiscellaneous

30

45

25

1236

15

10

12

820

25

30

20

1535

6. Find if the cost of living index number is 150Commodity Food Clothes Fuel & Lighting House Rent Miscellaneous

I

W

200

6

150

4

140 1003 1204


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19

CH. NO. 7. REGRESSION ANALYSIS1. For a bivariate data the mean of series is 35 and the mean ofseries is 29. The regression co

efficient ofis 0.56. Find the regression equation of. Estimate the value ofwhen = 25.2. For a bivariate data the means of

series is 40 and mean of

series is 35. The Regression co

efficient ofis 1.2. Find the line of Regression of y on x. Estimate the value ofwhen = 28.3. For the following data, find the regression line of. 1 2 3 2 1 6

Hence find the most likely value of when = 4.4. , =

10. 1 2 3 4 5 6 2 4 7 6 5 65.

.

125 .Production 120 115 120 124 126 121Price Rs/unit 13 15 14 13 12 14

6. Compute the appropriate regression equation for the data.[] 2 4 5 6 8 11[] 18 12 10 8 7 57.

Mean

S.D.

13

3

17

2

0.6.

= 15,

= 10.

8. .Mean

S.D.

Adv. Exp (Rs. Lakhs)

10

3

. 90

12 = 0.8. .. 120 .9. . .

MeanS.D.

.

4010

Adv. Exp (Rs. In crores)

61.5 = 0.9. . . . 10 ?. . . . 60 ?

10. 2 4 = 0 = 1,, .11. 2 = + 15


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20

4 = 3 + 25, . . . . . 12. 10 + 3 6 2 = 0

6 + 57 50 = 0. . , . = 2.13. 8 10 + 66 = 0 & 40 18 = 214.

= 9.

,

.

.

.

14. 30 3 4 + 60 = 0. = 40, 2 2 = 925 ,.15. 50 3 5 + 18 = 0. 44 9

16..

16. 50. . = 8500, = 9600, = 60, = 20, = 0.617.

= 5,

,

. = 30, = 40, 2

= 220, 2

= 340, = 214.18.. = 8, = 40, = 32, 2 = 32, 2 = 16 = 6. = 5.19. = 50, = 10000, = 500, 2 = 20000, = 1000,2 = 9800. = 12.

CH. NO. 8. LINEAR PROGRAMMING


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21

MATHS I


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22

CH. NO. 1. LOGIC1.Express the following in the symbolic form

i.Hari is either intelligent or hard working.ii. + 2 = 2 + 2 = 0.

2.Given p x is an irrational number.qx is the square of an integer.

Write the verbal statement for the following.

i. ~ii.~

3.:P: Kiran passed the examination.

S: Kiran is sad.

And assuming that not sad is happy, represent the

following statement in symbolic form.

Kiranfailed or Kiran passed as well as he is

happy

4.Write the following statements in symbolicform.

i.Bangalore is a garden city and Mumbai is ametropolitan city.

ii.Ram is tall or Shyam is intelligent.5.Write the following statements insymbolically.

i.If a man is happy, then he is rich.ii.If a man is not rich, then he is not happy.

6.Write the following statements in symbolicform.

i.Akhila likes mathematics but not chemistry.ii.IF the question paper is not easy then we shall not

pass.

7.Let p : Riyaz passes B.M.S.q : Riyaz gets a job.

r : Riyaz is happy.

Write a verbal sentence to describe the

following.. . ~8.Using appropriate symbols, translate thefollowing statements into symbolic form.

A person is successful only ifhe is a politician or

he has good connections.

9.Express the following statements in verbalform: . . ~: . : .10.Let p: Rohit is tall. q: Rohit ishandsome.

Write the following statements in verbal form

using p & q.

a.~(~)b. (~ )11.a. ~b.~ ~ .

.12.Construct the truth table and determinewhether the statement is tautology,

contradiction or neither.

i.( pq) (q ~q)ii.[ p (~ q p)]p

iii.~( p q)iv.p (qp)v.p (~q p).

vi.~ ( p q).vii.[ p (~ q ~p)]p

viii.( p~q) (q ~q)ix.[q ( p q)]px.~( ~p ~q )

xi.[~(p q) p]13.Do as directed.

i.Prove that the following statements are logicallyequivalent: p q ~q ~p

ii.Show that the statements p q and ~( p ~q) areequivalent.

iii.Write the truth table for Disjunction. Write thedisjunction of the statements: India is a

democratic country. France is in India.

iv.Using the truth table, Prove that p (~p q) p q.

v.Show that p q ( p q ) ( q p ).vi.Using truth table show that, p q (~p q)

vii.Using truth table prove that, p q (~q)(~p)viii.Prove that the statement pattern ( p q)

(~p~q) is a contradiction.

ix.Show that the following pairs of statements areequivalent: p q and ~ (p ~q).

14.Represent the following statements byVenn Diagrams:

i.No politician is honest.ii.Some students are hard working.

iii.No poet is intelligent.iv.Some poets are intelligent.v.Some mathematicians are wealthy. Some poets are

mathematicians. Can you conclude that some

poets are wealthy?

vi.Some parallelograms are rectangles.vii.If a quadrilateral is a rhombus, then it is a

parallelogram.

viii.No quadrilateral is a triangle.ix.Sunday implies a holiday.x.If U = set of all animals.

D = Set of dogs.

W = Set of all wild animals;

Observe the diagram and state


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23

whether the following statements are true or false

a. All wild animals are dogs.b. Some dogs are wild.

xi.Some students are obedient.xii.No artist is cruel.

xiii.All students are lazy.xiv.Some students are lazy.xv.All students are intelligent.

xvi.Some students are intelligent.xvii.

All triangles are polygons.xviii.Some right-angled triangles are isosceles.

xix.All doctors are honest.xx.Some doctors are honest.


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24

CH. NO. 2. LIMITEx. No. 1. [Algebraic Limits]1. lim4 37 2+ 32 53262.

lim2

2+

+1

2+3 1

33. lim2 3+ 212 322 167 4. lim3 324 3+ 236 26335. lim1 33+4 261231 115 6. lim1 43+2342 1117. lim2 32 24+8 35 2+84 418. lim

3

324

3

62

+9[

]

9. lim4 38 2+16 316 010.lim1 32+1 3+ 25+3 1211.lim1

2

8 3143 (3)

12.lim2 2+24 232+4 (3)13.lim3 2+3633 32

14.lim3 3+62+9 3+52+39 3415.lim

1

2+2

2

4+3 3

2

16.lim3 52432 29 1354 17.lim3 24+32 239 2918.lim1 2 211 7219.lim4 364 3154 161120.lim3 2936 2+116 (3)21.lim

1

2+23

1

(5)

22.lim3 2+3312 49 51223.lim2 41625+6 8524.lim2 416 25+6 (32)25. lim1 7+42 32+1 (11)

Ex No 2. [Algebraic Limits]

1. lim2 1

2 2

22 1

22. lim5 15 525 153. lim3 13 9327 (0)4. lim2 12 223+2 325. lim2 1 25+6 12 27+6 36. lim4 1 234 1213+36 ( 225)7. lim

3

1

2

+4+3 1

2

+8+15 1

2

8. lim 123 +22 12 23 +2 ( 32)

9. lim1 1

1 1

2 (1)10.lim1 32+2 42+23 111211.lim3 25+6 29 3272+12 16942 12.lim2 12 4 32 2 (1)13.lim2 12 1 23+2 (1)14.lim3 13 27 43 3 (1)


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25

Ex No 3

=

1. lim

33

10

10

(3

10

7)

2. lim 2525 1515 (5103 )3. lim 55 99 ( 594)4. lim 15 15 20 20 ( 34 5)5. lim2 7128 664 736. lim2 664 10 1024 3807. lim

3

838 12 312

2

243

8. lim5 757 10 510 712509. lim 66 88 (3

2

4)

10.lim 55 77 527 11.lim0 +66 65

12.lim

0

+

8

8

1

8

7

13.lim2 12212 13213 325614.lim 3 3 231615.lim2 2422 423 16.lim2 222 32232 423 17.lim

1

+ 2+ 3++

1

Note: = [ + ]Ans.

[ + ]

18.lim2 103 22 112

Ex No 4 [Rationalizing]

1. lim3 +63 29 1362. lim4 2+13 212 1213. lim4 2+203+44 (24)4. lim5 26+51422 (8)5. lim8 2+1792+72 81536. lim

3

3512

2

2

9

3

(11)

7. lim4 464 2+95 2408. lim4 2168 839. lim3 2+ 2+792 12510.lim2 2422 423

11.lim1 +1

( 2+4+52+1) 1

212. lim2 3422024 613.lim3 2+ +6 12 29 373614.lim1 +32211 1415.lim0 +4 1416.lim

+23

2

2

1

433

17.lim0 + 1218.lim0 +3 3 32 19.lim2 6+1024 182


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26

20.lim0 ++ 1221.lim4 3+44544 35

22.lim1 8+352 2323.lim2 3++210 24 4916

Ex. No. 5 Trigonometric Limits

= & = 1. lim0 25 252. lim0 4 ()3. lim0 44. lim0 54 545. lim0 32 326.

lim0 17. lim0 1

8. lim0 09. lim0 32 3210.lim0 2 3 23

11.lim0 sin 2 5 2 (25)12.lim0 sin 2+5 (5)13.lim0 32 3 (8)14.lim0 sin 2

2

2 14

15.lim

0 4 6

5

2

24

5

16.lim0 3 57 2 757 17.lim0 +2+ 218.lim0 3 2 +23+2 3 8919.lim0 7 +332+ 1020.lim0 83 + 2 73

21.lim0 2 +33 +5 5822.lim0 2+ sin 12+2 2323.lim0 1 (2)24.lim0 1 3 1225.lim0 1 3 2 9226.lim0

1

2 2

2 27.lim0 11 ( 22 )28.lim0 1 (2)

Ex. 6. [Logarithmic Limits]

= 1. 0 + 2. lim0 3. lim0 32 23 4. lim0 5 34 1 5. lim

0

3 2 21

2

6. lim0 7 +8 +9 3+1 7. lim0 3 2 +1 8. lim0 9. lim0 6 4 2 +1

10.lim0 10 2 5 +1 11.lim0 6 3 2 +1 2 12.lim0 4 +

1

4 2 2 13.lim0 5 +5 2 2 14.lim

0

32 1

15.lim0 5 +1

5 2 16.lim0 32 1 17.lim0 +3 +4 3 18.lim0 + 2+1

19.lim0 4 11 3 20.lim0 11 3 21.lim0 3 22.lim0 2 12 +123.lim

0

4 33

+

4

24.lim1 1 25.lim0 8 5 3 +1 410


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27

Ex. 7. Exponential Limits

+ =

1.

lim01 + 25

= 10

2. lim01 + 43 = 123. lim0 1 + 53

4= 203

4. lim0 1 47 5

2=

1

107 5. lim0 1+314

1= 7

6.

lim0 4

+1

141

= 8

7. lim0 2+21

= 8. lim0 484+5

1= 1 134

9. lim0 log 1+ =

10.lim0

log

1+3

= 3

11.lim0 log 5+log 5 = 2512.lim0 10+log (+0.1) = 1013.lim0 1 [10 + log +110 ] = 114.lim0 log 7+log 7 = 2715.lim3 33 = 1316.lim

2

22

=1

2

17.lim 1 = 118.lim1 11 = 19.lim2 1 12 = 20.lim0 3 14 =

Ex. 8. Trigonometric Limits

1. lim0 2 23 = (1)2. lim0 3 = 123. lim0 35 = 24. lim0 32 = (4)5. lim0 2 410 = 1426. lim

0

48

= (24)

7. lim0 2 1410 = 1488. lim0 82 124 = 15329. lim0 = 2222 10.lim0 1 2 +13 = 116

11.lim0 sin + = ()12.lim0 tan + = (sec2 )13.lim = ()14.lim 15.lim 16.lim

17.lim 18.lim

2

2

= 1219.lim1 sin 1 =()20.lim

2

1

22 = 12


28/56


28

21.lim1 1+cos 12 = 22 22.lim

4

14 = 12

23.lim

4

4

=

1

22

24.lim2

3 + 323 =25.lim 5+22 = 18

26.lim3

33 = 4327.lim

6

36 28.lim

41

12

29.lim4

211 12

Ex. 9. Using first principle find or Find +() 1.

=

2. = 2

3. = 34. = 45. = 16. = 7. = 8. = 1

9.

=1

10. = 2 + 111. = 12. = 1 + 213. = 1+3214. = 15. = 16. = 5

17.

=

2

18. = sin2

19. = cos2 20. = 21. = 222. = 2 23. = log3 + 224. = log (2 1)

Ex. 10.

1. = 3 + 2 lim0 2+2 2. lim0 3+3 , = 22 3 + 5 3. lim1 1 21 = 2 + 3 4. lim0 1+1 = 7 2 5. lim0 3+3 = 7 26. lim0 3+3 = 15 7.

lim

0

2+(2)

=

1

2+2

8. lim0 1+(1) = +5+1 ()


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29

ContinuityEx. No. 1.I. Discuss the continuity for the following functions and if the function discontinues,

determine whether the discontinuity is removable.

1. = 3 1 2 0;= 4 = 0, = 0.

2. = + 3 2 1;= 2 = 1, = 1.

3. = 3 12 (1+) 0;= 23 = 0, = 0.

4. = +3 2+ 0;= 4

= 0,

= 0.

5. = +63 29 3;=

1

2 = 3, = 3.

6. = 5 2 3 0;= 1 = 0, = 0.

7. = 33 3 ;=

4

3

=

3

,

=

3

.

8. = 5 12 2 ;= 25 =

2, =

2.

9. = 5 32 1 0;

=log 5

3

2 = 0, = 0.10. = 2164 4;

= 9 = 4, = 4.11. = 2 1 0 < 2;= 4 + 1 2 4 , = 2.

12. = 2 + 3 0 < 2;= 4 2 5, = 2.

13. = 2 + 5 0 < 3;= 2 + 5 3 6, = 3.


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30

Ex. No. 2.1.

=

3 1

,

< 0;

= 1, = 0;=

log (1 + )4 > 0 = 0, & .

2. = 3 12 (1+) , 0; = 0,0.3. = 0

(

) =

,

0;

= = 0.4. = 7 12 (1+5) 0; = 0,0.5. = 3 2 0; = 0,0.6. = 1 0; = 0,, 0 = 3.7. = 15 3 5 +1 , 0; = 0,0.8.

= 0

= 2 + , > 0;= 22 + 1 + < 0&0 = 2.

9. = 0 = 2 + , 0;= 22 + 1 + < 0&2 = 4.

10..

=3

2

2

1

22 15 [0,5]11..

= 3 2212+7+12 [2, 7]


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31

CH. NO. 3. DIFFERENTIATION (DERIVATIVES)EX. NO. 1. .

1.

2 +

+

2. 43 52 + 8 13. + + + 4. 73 + 4 + 25. + 1 3

6. 5

+1

7. 5 + 34 + 28. 1 23 + sin1 + cos1 9. tan1 + sec1 10. + 12

11.

1 +3

12. 1 + 15

EX. NO. 2.1. 5

2. 33. 4 44. 33 5. 6. 17. 18. 9. . 10.11.

2

.

12.(1)(x-2)13.2 + 1.14.2 + 1(2x+1)15. + 1(2)(x+3)16. . 17.. 5 18. +1

19.1+1

20.1+1

21. 2+1 2122.35

2+323. 35+2

2+1 24. 2+1 2+325. 1226.

27.1+

28. 29. 2+1

1+ 30.++131.

1+

32. 1+

33. 1+

34.1+1

35.1+1

36. 1 +137. 3+

1+3 38.1+ 39. +

40.1 21+ 241.1 2

1+ 242. + 43.2 + 3

44.If = . = sec2 EX. NO. 3.

1. 2 + 152. 2 + 343. 23 5 + 1324. 2 25. 1 2

6. 1 2+17. 12+18. 2 + 1 + 1 2+3


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32

9. 11+

10.2 + 111.2 212. 1

2 25+13

13. + 14.sin315.cos8 + 516.sin2 17.sin3 18.19.020.0 + 021.sin122.tan

1

2

23.sin1 224.tan1 225.sin1 26.sin1

2

27.sin1 2

28.sin1 129.log

30.log31.log32.233.sin23 34.sin135.sin( sin1 )36.cos37. sin 1 38.2 39.sin2

+ cos2

40.32 + log 341.log + 42. + 43. 1 cos + sin 44.1 + 2 + 45. 1+ 1+2

46. 1 2+ 2+2+ 247. 2+3 48.2. 49.3 cos 250.

sin + 51.

52. = 7+1 53. + 54.log (tan1 )55.log + 56.(log )357.log ( + 2 258.log + 2 + 259.log

[]

60. 222

+ 22

sin1 61. 22

2 2

2log + 2 2

62. 2+22

+ 22

log + 2 + 263.log 2+3

3

564.log 3 +15

65.log 53

1+2

1 23

66.log 3 3

2+1367.log 1 2

1+ 268.log cos

1sin 69.log

1+ 70.log 2+2+2+271.log 1 72.log 2 2+4

sin 3+273.If = + . . = 1 2.74.If = cot , . . = 2


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33

EX. NO. 4. INVERSE1. sin1 1 22. cos

1

1 23. sin121 24. sin1 11+25. sin1 2

1+26. sin13 437. sin1 4

1+428. cos11 229. sec1 1

12 210.sec

1

1

22111.1 1212

12.sin1 121+2

13.sec1 1+212

14.sec1 2+2 15.cos1 1+

2

16.tan

1

1

2

17.tan1 1218.tan1 1 2

1+ 219.tan1 1

1+ 20.tan1 1

1+21.tan11 + 2 + 22.tan

1

1+21

23.tan1 sin 1+

24.tan1 + 25.tan1 22+ 2226.tan1 2 2227.cos1 2 +313 28.cos1 +3

2

29.cos1 +122 30.sin

1

+

2

31.sin1 +2+2 32.tan1 + 33.tan1 + 34.tan1 5

16 235.tan1 + 36.cot1 32

2+3 37.tan

1

1+562

38.tan1 13142239.tan1

1+12240.tan1 2

1+15241.tan1 4

1442.cot1 1+ 3243.tan1 + 244.

tan

1

1+

45.tan1

1+20

46.cot1 1+3+42+7 47.tan1 +

1348.tan1 8

116249.cot1 5+ 5 5 550.cot1 4 4 4+ 451.cot1 + 52.cot1 1+

1

53.cot1 1333 354.sec1 1+

255.sin1 2 2

1+ 456.sin1 8

1+16257.1 1+25 2

10 58.cos1 136 2

1+36 259.sin1

125 21+25

2

60.cos1 11+61.cos1 12

1+262.sin1 1++1

2

63.sin1 + cos1 1 64.sin1 +

12

65. = 1 + 2

EX. NO. 5. LOGARITHMIC FUNCTIONS1. + 342 + 1523 1132. + 2542 3723 113 3.

12232334


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34

4. 2132+15332135. 31321341

6.

+

2

+ 57. .12 8. 9. 10. 11. 12. 13.(tan1 ) 14.

sin1

cos

15.sin 1 16.cos1 17. 18. 19.

20.log 21. + sin 1 22. + (sin) 23. + sin1 24.

+

25.(sin ) 26. + log +127.cos 28.

1+2 29.

1+ 2 30.sin 1 12 31.

1+232.

(tan

1

)

1+2 33. + + 34. + 2 + 2 + 2235. + 36.1 + 3

37.If = 2 2+1

then show that = 2+1

+2+1 + log 2+138.If = then show that = 1+ 2 2 21+ 2

EX. NO. 6. IMPLICIT FUNCTION1. 2 + 2 + 2 + 2 + = 02. 3 + 3 = 33. 3 + 3 = 324. 2 + 2 + 2 + 2 + 2 + = 0.5. 2 + 2 = + 6. 22 = 2 27. + = + 8.

+ = + 9. =

10.If + = + show that =

11. = cos + 12. = + tan1 13. + = sin 14. + = 15. = 2 16. = log17. = 18. = . 19.23 + 23 = 23

20.If + 1 = 1, Show that = 2 21 21 + 121.If sec + = , =


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35

22.If sin1 2 2 2+ 2 = log , show that = 23.If cos1 2 2 2+ 2 = tan1 , show that = 24.If log 33 3+ 3 = , show that = 25.If cos1

2

2

2+ 2 = 2, show that = tan2 26.If tan1 2 2 2+ 2 = , show that = 11+ 27.If45 = + 9, show that = 28.If3 = + 3+ , show that = 29.If = + + , show that = 30.If = + + , show that = 31.If sin = .sin + , show that = sin 2(+) 32.If = . , show that = 1 33.If = , show that = 1+ 234.If = . , show that = 1+ 1 35.If = + , show that = 1 36.If = , show that = 1+1212 37. + = 38. + = 139.

If = 1 + , show that =

2

40.If = , show that = 2 41.If = , Prove that = 1log 242.If = , Prove that = 2143.If log 2 + 2 = tan1 show that = + 44.If = 5 , show that = 5 5

EX. NO. 7. PARAMETER FUNCTIONSIn the following problems , , are parameters

1. If = 2, = 2, = 1 2. If acos , = sin , + cot = 03. Differentiate 2 + 1 w.r.to 34. Differentiate w.r.to sin


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36

5. If = sin2 and = tan , = 46. If = sec , = tan , show that = 7. If = sin3 , = cos3 , + cot = 08. If

=

+

,

=

1

,

= = tan2

9. If = , = 1 cos , = cot210.If = 3 2sin3 , = 3 2cos3 , = tan 11.If = cos , = cos + , = 12.If =

1+3 , = 1+ 3 , = 2 313 2 13.If = 12

1+2 , = 21+2 , = 212 14.If = 2 , = 2 , + 2 2 2 = 0.15.If

=

2

,

=

2

,

=

16.If = sin1 1+2 = cos1 11+2 , = 117.If = 2

1+2 , = 121+2 , = 2 218.Differentiate . with respect to 19.Differentiate tan1 12 with respect to sec1 12 2120.Differentiate log 1+2+1+2 with respect to log21.If = 22 , = 22, = tan 32 22.Differentiate

with respect to

.

23.Differentiate log10 with respect to cos 24.Differentiate tan1 12 with respect to cos121 225.Differentiate tan1

1+ with respect to sec1 26.Differentiate cos122 1with respect to 1 227.Differentiate cos12 1 with respect to 1 228.If = , =

2 + , = cot2.


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37

CH. 4. APPLICATION OF DERIVATIVESEx. No. 1 Approx.1. Find approximately, the value of100.1, 64.1

2. Find approximately, the value of283 tothree decimal place.

3. Find approximately, the value of26.963 tofour decimal places.

4. Find approximately, the value of9973 , 633 5. Find approximately, the value of4.14&3.0746. Find approximately, the value of

(45030 ) given 10 = 0.0175 7. Find approximately, the value of310 , 10 = 0.0175 , 300 =0.0866, 300 = 0.5

8. Find approximately, the value of cos(89030),given 10 = 0.0175

9. Find approximately, the value of cos(30030),given

10 = 0.0175 300 = 0.0866, 300 =0.5

10.Find approximately, the value of11.001, 10.99911.Find approximately, the value of2.12 = 7.38912.Find approximately, the value of1.002 , = 2.71828.13.Find approximately, the value of10101610 = 2.3026.14.Find approximately the value of

101

10 = 203026

15.Find approximately, the value of9.01given 3 = 1.0986.16.Find approximately, the value of

51113 , 80.71417.Find approximately, the value of =

23 + 7 + 1 = 2.00118.Find approximately, the value of52 +

80

= 5.08319.Find approximately, the value of

32.01,

3 = 1.0986

31.5

5

Ex. No. 2 Error1. Radius of the sphere is measured as 12 cm

with an error of 0.06cm. Find

a. Approximate errorb. Relative errorc. Percentage error in calculating

the volume.

2. Radius of a sphere is measures as 25 cmwith an error of 0.01cm. Find

a. Approximate errorb. Relative errorc. Percentage error in calculating

the volume.

3. Radius of a sphere is found to be 24cmwith the possible error of 0.01cm. Find

approximately

a. Consequent errorb. Relative errorc. Percentage error in the surface

area of the sphere.

4. The side of a square is 5 meter isincorrectly measured as 5.11 meters. Find

up to one decimal place the resulting

error in the calculation of the area ofsphere.

5. If an edge of a cube is measured as 2mwith an possible error of 0.5 cm. Find the

corresponding error in calculating the

volume of the cube.

6. Find the approx error in the surface areaof the cube having an edge of 3m. If an


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38

error of 2cm is made in measuring the

edge. Also find the percentage error.

7. The volume of a cone is found bymeasuring its height and diameter of base

as 7 cm and 5 cm respectively. It is found

that the diameter is measured incorrectlyto the extent of 0.06 cm. Find the

consequent error in the volume.

8. The diameter of a spherical ball is foundto be 2cm with a possible error of

0.082mm. Find approximately the possible

error in the calculated value of the volume

of the ball.

9. Side of an equilateral triangle is measuredas 6cm with a possible error of 0.4mm.

Find approximate error in the calculatedvalue of its area.

10.Find the approximate % error incalculating the volume of a sphere, if an

error of 2% is made in measuring its

radius.

11. If an error of 0.3% in the measurement ofthe radius of spherical balloon, find the

%error in its volume.

12. If the radius of a spherical balloonincreases 0.1%. Find the approximate %

increase in its volume.

13.Under ideal conditions a perfect gassatisfies the equation PV = K; where P =

Pressure, V = Volume and K = Constant. IfK = 60 and Pressure is found by

measurement to be 1.5 unit with error of

0.05 per unit. Find approximately the

error in calculating the volume.

14. In ABC, B is measured using theformula = 2+22

2 . Find the errorin calculation ofB if an error of 2% is

made in the measurement of side b.

15.Area of the triangle is calculated by theformula 1

2 . IfA is measured as 300with 1% error. Find the % error in the

area.

16.Time (T) for completing certain length (L)is given by the equation = 2 where

g is a constant. Find the % error in the

measure of period, if the error in the

measurement of length (L) is 1.2%.

Ex. No. 3. MAXIMA AND MINIMA1. Examine each of the function for Maximum and Minimum.

i. 3 92 + 24ii. 23 152 + 36 + 10

iii. 83 752 + 1502. Output, is given by = 10 + 60 + 7 2

2 3

3. Where x is the input. Find Input for which

output Q is maximum.

3. Find the position of the point P on seg AB of length 12cm, so that2 + 2 is minimum.4. Find two Natural Number whose sum is

i. 30 and product is maximum.ii. 18 and the sum of the square is minimum.

iii. 16 and the sum of the cube is minimum.5. Find two Natural numbers x and y such that

i. + = 6 2.ii. + = 60 3.

6. Product of two natural numbers is 36. Find them when their sum is minimum.


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39

7. Product of two Natural Number is 144. Find them when their sum is minimum.8. Divide 70 in two part, such that

i. Their product is maximumii. The sum of their square is minimum.

9. Divide 100 in two part, such that the sum of their squares is minimum.10.Divide 12 in two parts, so that the product of their square of one part and fourth power of the

other is maximum.

11.Divide 10 in two part, such that sum of twice of one part and square of the other is minimum.12.The perimeter of a rectangle is 100 cm. Find the length of sides when its area is maximum.13.Perimeter of a rectangle is 48cm. Find the length of its sides when its area is maximum.14.A metal wire 36cm long is bent to form a rectangle. Find its dimensions when its areas is

maximum.

15.A box with a square base and open top is to be made from a material of area 192 sq. cm. Findits dimensions so as to have the largest volume.

16.An open tank with a square base is to be constructed so as to hold 4000 cu.mt. of water. Findits dimensions so as to use the minimum area of sheet metal.

17.Find the maximum volume of a right circular cylinder if the sum of its radius and height is 6mts.


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40

CH. NO. 5. INDEFINITE INTEGRATIONEx. No. 1 Integrate the following functions1. 4

3

2. 323. 23 4. 1

2 35. 16. 34 7. 2 + 58. 1329. 13234 10. 12334 11.2 + 1 + 13+512. 132313. 173314.7 15.53 16.3 2 17.5 + 35 18.4 3 19. 2 20.4 52

21. 12

22.2 + 2 + 323.+2+3 24.2 3+5 2+4 25. 32 2+57+ 2 26. 2+32+7 27. 1

3

1

3 +

+ 2

28.+1(+2)2 29. 1 1 + 130. + 1231. 1132. 1+33. 13+103734.

+1

235. +2+336.+1212

37.2

38. sin 2 39.tan2 3 sin4 + 340. 112 2 241.cot2 sin5 + 3 +

1

.42. 149 243. 1

5

3

2

44. 154 245. 1

9+246. 1

3 2+547. 1

92+2548. 1

3 2+449. 15 2+450. 1

42

+25

51. 13 2+252. 1

259 253. 1

49 2Ex. No. 2. Integrate the following functions1. sin2 2. cos2 3. sin2 24. cos2

2

5. cos2 36. tan2 7. cot2 8. sin3

9. cos3 10.. 11.

sin

2

cos2

12.sin3 313.214. 1

sin 2 2 15. 1

1+ 16. 1

1

17. 11 2

18.1+1

19.1 + 220.1 221.1 + 22.1 23.1 + 224.1 +


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41

25.1 26. 1

1 27.

1+ 28.

1

29. + 30.

1 231. 2

cos 2 2

32.2 33.sec2 3 134. sin 2

1+ 35. sin 2

1+ 2

36.12sin 2

37.341 2

38.3439.53

40.3441.5742.3443.5744.sin

1

45.tan1 1+ 46.tan1

1 47.tan1 2

1+ 2Ex. No. 3. Integrate the following functionsNote: - Whenever the degree (Highest Power of a polynomial equation) of the numerator isgreater than or equal to the degree of the denominator then divide the numerator bydenominator.

1. +12. +333. 2+3+1 4. 2+1+2 5. 5+4+2 6. 2+1

3

2

7. 2+2

8. 2+11 9. 2 2+1 10. 3+5 2+2+3

21 11.5 26+3

2+1 12.

5

2+3

+1

21

13.5 2+11 14. 21 2+115. +1216. 22+312

Ex. No. 4. Integration by Substitution1. 2 2. 2 2 3. 4. 5.

1

6. 1 7. 2 sec2 38. 32 tan2 3 9. 2 410. 2 +1

11. 42 12.

25cos 2 13.

4sin 2 14.log 1+1+1 15.log (tan 2)sinx 16. 17.cos 3 18.2219.sin3

20.[tan 1 )21+2

21. 1+ 3222.2+ 223. 1+

cos 2

24. 12 25.

1+sin 2 26. 1 2227. 1 .


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42

28. 1 .log 29. 3 30. sec2 31.

+1

+

2

2 32.(cos 1 ) 212 33. sin 1 12 34.

sin 4 35.(sin 1 ) 312 36.cos 1 12 37.

tan 1 1+ 2

38.sin (tan 1 )1+x2

39. 12 40. 1+ 23 41. 12+3 242.243. 12 44.

+1

sin 2 45.1 2

1+ 4 46. + 47.1 1 1 248. 2+12++5

49. 4+10 2+5150. 5

2 2+351. 2+6 2+6+1052.

2

6+4 253. 3

1254. +3 55. .

log 56. 13 +1 cos 2 57. 11+ 1 58. 1+ 1

+

59.1+ 2+sin 2 60.

log 61. 1

2+ 62. 1 + 263. 1+164. 33+ 23265.

2

acos 2 +266. 1

1+ 67. 2 1 2 +168. +1 169. 1 +1

70. 2 171. 1+72.1+

1

73.11+ 74. 1

1+ 75. + 76. 2 + 277. 2+178. 2279.

1

2

80. 2+162+581. 2+3 2+3182. +183. 1+ 84. 23sin 2 85. 2286.

+ 87. 22 sin 2 +2 cos 2


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43

Ex. No. 5. Integration of the type + or

+

1. 2 + 1 + 12. 3+433. 2+14. 2 + 15. 2+1+1 6.

2+

+3

1 7. 22 18. (2sin 2 +3)1 9. 2 2 +9 +5) +1 10.tan 1+ 11.cot1+ 12.tan 2+3

13. 14.sec 15. 16. 11

17. 11+ 18. 11 19.

sin +20.

cos 21.

cos 22.sin sin +23.

cos 24.cos +

cos 25.sin

sin 26.cos

cos 27. 1

sin sin 28. 1

cos cos

29. 1sin cos

30. 1cos sin

31.1+tan 2 1tan 2

32. 1 .cos 2

33. 162 34. 21+635. 4+2536. 2 937. sec 2 3tan 2 +238. 25cos 2 39. sec 2

4tan 2 940. 1 2+941.

4sin 2 342.

4+sin 2 43.

4sin 2 +544.

+

45.+11


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44

Ex. No. 6. Integration of the type ++ OR

+

+

1. 12+6+102. 5

4 2+4153. 1

3 24+54. 1

962+6+55. 12++16.

1

15+442

7. 542 2

8. 14+432

9. 1 2+4+310. 1324+211. 1 2+4+512.

1

424+3

13. 13 24314. 13+44215. 19+8 216. 2 +4 +13

Ex. No. 7. Integration of the type +++ OR +++

1. 2+32+3+12. 2

5

25+23. 326+44. 1

3+2 25. 2+12+3+56. 3+7

2 2+32

7. +3 2+48.

2

+1

9429. 3+5 2+4+510. 3 +5 2+12+10 2+4+5 11. 1

3+43212. 21 2+3

13. 2323+414. 2

+1

2+3415.+4

316.2 3+8 236 2+6+10

Ex. No. 8. [Important] Integration of the type

+ OR + Or ++ 1. 1

32 2. 1

5+4 3. 1

53

4. 13+2

5. 11+3

6. 14+9

7. 154

8. 1 + 9. 1


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45

10. 1 + 11. 1

1+ 12. 1

13+3 +

13. 13+2 2

14. 11+ 2

15. 14+5 2

16. 15+3 2

17. 112 2

Ex. No. 9. [Important] Integration of the type + OR + Or ++

1. 132sin 2

2. 12+3sin 2

3. 123 cos 2

4. 13sin 2

5. 11+7cos 2

6. 15cos 2

7. 14+5sin 2

8. 12 sin 2 +3 cos 2

9. 14 cos 2 +3sin 2

10. 12 sin 2 +2 cos 2

Ex. No. 10. [Important] Integration of the type

1. sin2 2. cos2 3. sin3

4. cos3 5. cos4 6. cos5 7. sin6 8. cos7 9. sin7 10.sin3 411.sin2 312.sin5

13.sin3 314.sin3 15.sin 3

cos4

16.sin5 217.cos5 18.tan3 19. 20..cos3 21.sin5 322.cot3 23.sec4

24.tan4 25.426.tan5

27.628.829.5330.3431.332.43233.53

Ex. No. 11. [Important] Integrate the following .1. 1+1+22. 1+2

3. 2+1+1+234. 32 23+2

5. +1 246. 1

1++ 2+ 3


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46

7. 1+128. +112

9. 1+122+110. 3+2 2+6 2+2

Ex. No. 12. [Important] Integrate the following. = .

1. 2. 23. 4. log + 45. log

6. 4 3

7. tan1 8. log2 + 49. 3 10.211.

1+ 212.

13.314. 15.216.

Ex. No. 13. [Important] Integrate the following [ + ] = +

1. + 2.

1

+

3. [ + log]4. [1 ]5. sin

+ cos

6. tan + sec2DEFINITE INTEGRATIONEx. No. 1.1. 1

0

2. 1331

3. 1 394 4. 0 5.

322

1

6. 1+2

11 7. 12

1+ 2 10 8. 120

9. 20

10. 5 2+4 20 11.

cos

4

0

12. 5320 13. sin3 2

0

14. 1 + 23

15. 240

16. (sin 1 )312 12

0

17. sin2 . 20

18.

22

2

2

19. 362 2 63 20. 42221 21. 1+ 3

2

0

22. 20


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47

23. 5+3

2

0

24. 12 1+ 2 11

25. 12 2121

2

0

26. 11+

4

0

27. 10

Ex. No. 2. [Important] PROPERTIES1. = 2. = 3. = + 4.

0=

0

5. = + 6. 2

0=

0+ 2

0

7. = 2 0 = 0 .

1. +3 21 2. +2+2+5 21 3. 54+5 45 4.

+

2

2

0

5. +1 10 6. +44+44 + 94 50 7. +2+2+5 30

8. + 2

0

9. + + 2

0

10. +2

0

11.

1

1+

2

0

12. 11+ 36 13.

1+ 0 14. +16240

15. 1+1+ 20 16. 1+92 30 17. +220 18.

1+cos 2

0

19. log1 + 40

20. + 0 21. 1 1

0

22. 4 40


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48

CH. NO. 6. DIFFERENTIAL EQUATIONEX. NO. 1.A. Form the differential equations by eliminating the arbitrary constant.1. = 2 + 2. =. 3. = 4. = 25. = 2 + 6. 2 + 2 = 27. 2 = 48. 2 = 4 + 9.

2 +

2

2

= 0

10. = 4 211. = 2 + 112. = 13. = 14. = sin15. = cos + 16. = + 17. = 2 + 18. + = 1

19. 2 2 + 2

2 = 120. =3 + 3 21. = 22. = 23. =. 2 + . 5 24. = 2 + 2 25. = + 26. =7727.

+

= 1

28.2 + 2 = 429. = 12 + 230.3 + 2 = 5 (Note: Important sum use

the condition for consistency)31. 2 2 +

2

2 = 132.12 + 22 = 5

EX. NO. 2.1. Solve 2 = 2 + + 22. Solve the differential equation = 03. Solve the differential equation = 2 4. Solve = sin + sin .5. Solve = sin + + cos + + = .6.

Find the particular solution of the differential equation 1 + = 0 when = = 2.

7. Solve the differential equation sin = 2 by substituting = .8. Solve = 4 + 3 12 by using substitution 4 + 3 1 = .9. Solve + 2 + 1 2 + 4 + 3 = 0.10.Find the particular solution of the differential equation 1 1 + = 0, =

4 = 2.


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49

11.Solve = 1+2

1+212.Solve the D.E. = ++12+113.Solve the D.E. 2 + 2 2 = 014.Solve

=

2 +

. Hence find the particular solution if

= 2

= 1.

15.Solve the equation + 1 + = 016.Verity that = 2 + is a solution of 2 2 = 0.17.Verify that =3 + 3 is the general solution of the differential equation

2 2 + 9 = 0.

18.Find the particular solution of the differential equation: + 1 1 = 2 when = 1 & = 0.19.Solve the differential equation = 4+622+3+3, by taking 2 + 3 = .20.

Verify that

2

+ 2

= 2

is a solution of the D.E. = + 1 + 2

.

21.Find the order and degree of the D.E. = 11+ 23.

22.Determine the order and degree of the differential equation. 2 2 + 31 2 = 0.23.Determine the order and degree of the D.E. 2 2 + 1 2 = .24.Determine the order and degree of the differential equation 5 2 = 10 1 .


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50

CH. NO. 7. MATRICESEx: 1A. 1. Consider the Matrices = [2 1 3 4], =

4

61, =

2 33 1, D =

4

6 3

2 4 16 5 2,E = 3

3 , F = [5 6 7], G = [5 7 6]Answer the following questions.a. State the orders of the matrices A, C, D, G.b. Which of these are row matrixes?c. If G is a triangular matrix. Find a.d. If e11 = e12. Find a.e. For D, state the values of d21, d32, d13.

2. A = [ ]23 such that = +. Write down A in full.3. Find which of the following matrices are non singular. = 3 38 8 , = 5 204 16 , =

1 2 3

3 1 27 0 7

, = 2 1 38 2 612 0 12

, = 6 5 42 1 010 6 3

4. If = 6 34 is a singular matrix, find a.

5. If = 6 5 14 2 114 1

is a singular matrix, find k.B. 1. Consider the matrices.

= 1 21 3 , = 3 54 2 , = 1 12 3 , = 21 + , = 3 2,

= 2 1 31 2 4 , = 2 11 2

3 4

, = 2 sin 2 3 2 4Answer the following questions.i. , , , .ii. = ,.iii.

=

,

.

2. If 4 56 + = 11 56 5 ,.

3. Find , + 2 2 + = 2 31 2


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51

Ex: 21. = 2 1

3 5 , = 3 2 1

6 1 5 , = 6 21 5 , = 4 3

2 22 1 , = 5 2 4 , = 2 63 5

Answer the following.a. Can you find,i. A + B; ii. A + C; iii. B + D, iv. B +D; v. A + A; vi. D + D; vii. C + F.b. If A + F = 0, find b.c. If C E = I, Find a.

2. = 3 1 24 3 5 , = 1 2 48 1 3 , = 8 2 42 3 7

Verify the following.a. A + B = B + Ab. A + (B + C) = (A + B) + Cc. A (B C) = A B + C.d. 3(A + B C) = 3A +3B 3Ce.A + B = A + B.

3. If = 6 3

2 1 , = 0 1

3 2 3 .4. Find 4 5

3 6 + = 10 1

0 55. If = 1 2

3 4 , + = 0.

6. If = 1 23 4 2 + 3 = 0, find the matrix B.7. If = 3 21 5find the matrix X such that 2 = 1 87 6.8. If = 1 2 23 1 0 , = 1 0 12 1 3 Find the matrix C such that A + B + C is a zero matrix.9. If = 2 1

2 4 , = 1 23 0 Find the matrix X such that 2X + 3A 4B = 0.

10.Find the matrix X such that 3 + 4 51 3 = 7 118 9 .

11.Find the values of x and y satisfying the matrix equation.a. 1 0 2 4 + 3 1 24 3 2 = 4 2 26 5 2b. 2 + 1 1 1

3 4 4 + 1 6 43 0 3 = 4 5 56 12 712.Find x, y & z if + 2 = 3 11 1


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52

Ex: 31. Find the following products:. [2 3 1] 41

3

. [3 4] 21

. [2 4] 30

. [6 5 1] 222

. [4 0] 30

. [2 ] 1 . [8 4] 52 . [ ] . [ ] 2. Find x in the following cases.. [3 2] 1 = [1] . [4 1]

32 = [8] . [4 ] = [21]

. [5 1] 24

= [20] . [ 2 3] 21 = [0] . [ ] = [5]

Ex: 4A.1. Find AB and BA whenever they exist in each of the following cases.2. = 2 5

2 5 , = 3 1

1 3

3. = 2 31 2 , = 1 2 30 1 2

4. = 1 0 21 1 0 , = 1 30 1

5. = 2 3 15 1 03 2 1

, = 0 2 01 2 31 1 2

6. = [3 1 2], = 4357. = 1 0 34 3 2

1 2 4

, = 16 6 918 7 105 2 3

2. = 1 24 3

, = 5 67 8

, = [2 08 3

] Then verify the following

. = . = . + = +

. = =, 3. If = 2 13 3

, = 2 5 73 2 1 , = [1 6 43 2 1] verify the following.. + = +. . =.

4. If = 1 2 22 1 22 2 1

show that2 4 is a scalar matrix.


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53

5. If = 0 3 33 0 43 4 0, = ,

= [ ] . . .6. (A) Find the values of a and b from the matrix equation:

3 24 1 15 = 4 53 56. (B) Find the values of x and y

a. 1 23 2

5 32 5 = 5 3 77 7 1 b. 4 1 2 02 1 3 2 1 3 12 3 8

2

01

= 6. (C). Find x, y, z values in each of the following cases.

i.

1 3 4

2 0 6

5 2 3

=

9

8

4

.

3

2

2 [1 2

] 3

1

=

ii. [ ] 1 0 52 3 74 2 1 = [4 4 7] . 5 1 0

0 1

1 1

3 1 22 33 1

21

= 7. Find x, y, z, a, b, c if 1 2

3 2 3

3 1 = 7 0 7 8. If = 1 00 1

1 1

, = 1 22 33 1

, = 21

, = Find the values of x, y, z if5 3 =

9. If = 4 1

5 23 4 , =

1

6 4

2 0 3 Find the Matrix AB and without computing the Matrix BA, showthat AB BA.10. If = 3 5

2 0 , = 1 2

3 4 Verify that AB BA.

11.

i. If = 1 23 21 0 , = 1 3 24 1 3 ,

ii. If = 2 1

0 3 , = 1 2

3 2 = . ||12. If = 2 0 1

1 2 3 , = 0 12 3

1 1show that AB is a Non singular matrix.13. If = 2 41 2, Show that2 is a null matrix.


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14. = 1 11 1 show that2 = 2.15. . = [2 4

1 1] Show that A satisfies the Matrix Equation2 = 3 + 2.

. = 1 23 4 ,

2

5 2.16. If = show that = = +

Ex: 51. If = 3 2

12 8 = 8 412 6show that AB = 0.

2. If = 1 21 2 = 4 35 3 , = 2 17 5 show that BA = CA.3. Show that AB = AC does not imply that B = C.4.

.

=

3 4

4 3,

=

5 6

6 5show that AB = BA.

. = 3 62 4 show that2 =.5. = 3 11 3 , = 2 55 2 show that + = 2 26. = 3 2

12 8 , = 6 189 27 show that. + 2 =2 + + 2 . + =2 + 2.

7. If = 8 410 5

, = 5 410 8show that

.

+

2 =

2 +

+

2

.

+

=

2

2.

8. If = 2 2 41 3 41 2 3 =

1 2 4

1 2 41 2 4 + 2 =2 + 29. If = 1 1

2 1 and = 1 4 such that + 2 =2 + 2 & .10. If = 1 21 2 , = 2 1 and + 2 =2 + 2, find a and bEx: 6I. Write down the following equation in the Matrix Form and hence find values of x, y, z using

Matrix method.

1. + 3 + 3 = 12; + 4 + 4 = 15 ; + 3 + 4 = 13.2. + + = 6; 3 + 3 = 10 ; 5 + 5 4 = 3.3. + + = 3; 3 2 + 3 = 4; 5 + 5 + = 11.4. + + 4 = 0 ; 3 2 + 3 = 4; 5 + 5 + = 11.5. 4 3 + = 1; + 4 2 = 10; 2 2 + 3 = 4.6. + + = 1; 2 + + 2 = 10 ; 3 + 3 + 4 = 21.


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55

II. Solve the following equation by the methods of reduction.1. + + 2 = 7; 3 + 5 = 6; 2 + 2 3 = 7.

2. + + 4 = 4; 2 + 3 + 6 = 5; 3 2 = 4.3. + = 1; 3 + 2 = 1; 2 2 + 3 = 2.4.

+

+

= 3; 7

+

+

= 9; 2

+ 3

= 4

5. 4 + 2 = 3; 2 + = 8; 2 + = 76. 3 + 3 4 = 2; + = 1; 2 = 1.Ex: 7A. Find the inverse of each of the following Matrices by using elementary transformations.

1. 1 32 5

2. 3 12 4

3. 4. 5. 6. 2 0 10 1 2

1 0 1

7. 1 2 21 3 00 2 1 8.

7 3 31 1 01 0 1 9. 0 0

0 0 1

10. 0 00 0 1

11. 0 00 0 1

12. 0 00 0 1

. 1. = 3 11 2 , 2 5 + 7 = 0, 1.2. = 2 4

1 1 , 2 3 = 2, 1.

3.

=

1 3

0 3,

2

4

+ 3

= 0,

1.

4. = 1 2 22 1 22 2 1

, 2 4 = 5, 1.. 1. = 3 11 2 , = 7 30 6 = .

2. = 1 01 1 , = 1 2 34 5 6 , = .3. + 2 2 = 5; + 3 = 0; 2 + = 3, .4. + = 1 3 41 1 32 3 1 , + .5. = , 2 = 2 22 2


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6. = 2 0 02 1 3 , = 0 1

2 31 1 ,

1.7. = 2 1

1 0 , = 3 1

2 1 ,1.

8. = 1 2 2 12 1 , = 2 12 1 1 2 = 0,.

9. 1 = 11, = 1 10 1

, = 2 41 3

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