MATHS

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8TH YEAR

OMTEX CLASSES 8 Years of Success

MATHEMATICS & STATISTICS

NAME

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STANDARD: -S.Y.J.C (Second year junior college)

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“You don’t know what you can do until you try” “IF YOU ARE SATISFIED WITH OUR TEACHING TELL TO OTHERS IF NOT TELL TO US”

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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 & STATISTICS perfect and useful in every respect. However, suggestions, if any, for its improvement are most welcome.

Omtex classes [REG. NO: - 760076951]

Note: - No part of this book may be copied, adapted, abridged or translated, stored in any retrieval system, computer system, photographic or other system or transmitted in any form or by any means without a prior written permission of the Omtex classes.

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MATHS – II CH. NO. 1. THEORY OF ATTRIBUTES EX. NO. 1 NINE SQUARE TABLE OF TOTAL FREQUENCIES

1. Find the missing frequencies in the following data of two attributes A and B. 2. For a data for 2 attributes, it is given that find the 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 was conducted in which 120 passed. If 10 girls failed, find the number of (i) boys who failed, (ii) girls who passed.

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5. If for 3 attributes A, B and C, it is given that (ABC) = 210, (A), (B), (C), (AB), (AC) and (BC). 6. 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. 2. 3. 4.

EX. NO. 3 IMPORTANT POINTS TO REMEMBER If

then, A and B are independent attributes OR A and B have no association.

If

then, A and B have positive association.

If

then, A and B have negative association.

YULE’S COEFFICIENT OF ASSOCIATION.

PROPERTIES OF Q Q always lies between – 1 to 1. If Q = 1 then A and B have perfect positive association. If Q = -1 the A and B have perfect negative association. If Q > 0 then A and B have positive association. If Q < 0 then A and B have negative association. If Q = 0 then A and B have no association OR A and B are independent attributes. 1. Discuss the association of A and B if i. N = 100, (A) = 50, (B) = 40, (AB) = 20. ii. (AB) = 25, 2. Discuss the association between attributes A and B if i. N = 100, (A) = 40, (B) = 60, (AB) = 30. 4

HSC MATHS 3.

4.

5.

6.

7.

8.

9. 10.

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ii. N = 500, . 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 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. Find Yule’s 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 88 persons are classified according to their smoking and tea drinking habits. Find Yule’s coefficient and draw your conclusion. Smokers Non – smokers Tea Drinkers 40 33 Non Tea Drinkers 3 12 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 10 300 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. Out of 700 literates in town, 5 were criminals. Out of 9,300 literates in the same town, 150 were criminals. Find Q. Examine the consistency of the following data and if so, find Q. N = 200, (AB) = 24,

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CH. NO. 2. NUMERICAL METHOD EX. NO. 1. NEWTON’S FORWARD INTERPOLATION FORMULA.

1. Using Newton’s Interpolation formula, find f(5) from the following table. (8.75) 2 4 6 8 4 7 11 18 2. Given the following table find f(24)using an appropriate interpolation formula. (485.84) 20 30 40 50 512 439 346 243 3. The population of a town for 4 year was as given below. Find f(1985) (60.4375) Year 1980 1982 1984 1986 Population (in Thousand) 52 54 58 63 4. For a function f(x), f(0) = 1, f(1) = 3, f(2) = 11, f(3) = 31. Estimate f(1.5), using Newton’s Interpolation formula. (5.875) 5. For a function f(x), f(1) = 0, f(3) = 25, f(5) = 86, f(7) = 201. Find f(2.5) using Forward Difference interpolation formula. (16.078125) 6. Construct a table of values of the function for x = 0,1,2,3,4,5. Find (2.5) and f(2.5)2 using Newton’s Forward Interpolation Formula. (6.25, 39.0625) 7. Estimated values of logarithms upto 1 decimal are given below find log(25). (1.35625) 10 20 30 40 1 1.3 1.4 1.6 8. Estimated values of sinx up to 1 decimal are given below find sin(450). (0.708125) 00 300 600 900 0 0.5 0.87 1 9. Find f(x), if f(0) = 8, f(1) = 12, f(2) = 18. (x2+3x+8) 10. f(x) is a polynomial in x. Given the following data, find f(x). Also find f(1.1). (3x2+2x+2, 7.83) 1 2 3 4 7 18 35 58 11. 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. (196 students) Marks 0-19 20-39 40-59 60-79 80-99 No. Of Candidates 41 62 65 50 17

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EX. NO. 2. LAGRANGE’S INTERPOLATION FORMULA. BY LAGRANGE’S INTERPOLATION FORMULA

1. By using suitable interpolation formula estimate f(2) from the following table. (9) -1 0 3 3 1 19 2. By suing Lagrange’s Interpolation formula, estimate f(x) when x = 3 from the following table. (35) 0 1 2 5 2 3 10 147 3. 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. (5.1904) Year 1st 2nd 7th 8th Profit (Rs. In lakh) 4 5 5 5 th Find the profit of the company in the 6 year by using Lagrange’s Interpolation formula. 4. Using the Lagrange’s Interpolation formula, determine the percentage number of criminals under 35 years. (77.405%) Age % number of criminals Under 25 years 52 Under 30 years 67.3 Under 49 years 84.1 Under 50 years 94.4 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 = 9.5 using Lagrange’s formula. (3.625) 6. Given are approximate and rounded off to 1 decimal place]. (1.139)

find

[Values

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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 – 3 b. 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 6th 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 59 6. 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

SHIFT OPERATOR IS REPRESENTED BY ‘E’

1. Find i. ii. iii. iv. 2. Given 3. Given 4. Find

each of the following case, assuming the interval of difference to be 1. . . . taking the interval of differentiating equal to 1. Find , taking the interval of differentiating equal to 1. Find

and and

. .

if

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HSC MATHS 5. a. x y b. x y c.

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Estimate the missing term by using

8TH YEAR

from the following table.

0 1 2 3 4 1 3 9 - 81 1 2 3 4 5 6 7 2 4 8 - 32 64 128 1 2 3 4 5 2 5 7 - 32

6. Evaluate i. ii. iii. 7. Evaluate i. ii. iii. iv. v. vi. 8. Show that 9. Show that 10. If Show that are in geometric progression. 11. Given: u0 = 3, u1 = 12, u2 = 81, u4 =100, u5 = 8, find . 12. Given: u2 = 13, u3 = 28, u4 = 49, find 13. Given: u2 = 13, u3 = 28, u4 = 49, u5 = 76. Compute 14. Prove the following: i. ii. iii. iv. 15. Assuming that the difference interval h = 1, prove the following. i. ii. iii.

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BY TRAPEZOIDAL RULE.

8TH YEAR

EX. NO. 5

WHERE :BY SIMPSON’S

RULE.

WHERE :NOTE: This rule is applicable only when ‘n’ is even. BY SIMPSON’S

RULE.

WHERE :NOTE: This rule is applicable only when ‘n’ is a multiple of 3. 1. Find the approximate value of

by using trapezoidal rule by dividing the interval [0,1] in to 5

equal parts. [0.34]

2. Find the approximate value of

by using trapezoidal rule by dividing the interval [0,10] in 10

equal parts. [0.335]

3. Using trapezoidal rule calculate the approximate value of

by taking 7 equidistant

ordinates. [115] 4. Find the approximate value of

5. Solve 6. Evaluate

using trapezoidal rule where

. [0.9125]

number of division is 6. [2.02] by using trapezoidal rule by taking 7 equidistant ordinates. [1.4095]

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with 7 ordinates use trapezoidal rule. [261]

7. Calculate

8. Using trapezoidal rule estimate approximate area of cross section of river 80 m wide and the depth (d) in meters at a distance (x) of the river from one bank to another is given . The cross section of the river 80 m wide is given by

x

0 10 20 30 40 50 60 70 80

d 0 4

7

9

12 15 14 8

3

dx. Where y=d = depth of the river in meter from one bank at a

distance x. [705 sq. m.] 9. A curve is drawn to pass through the points given by the following tables. [8.075 sq unit.] x

1 1.5 2

2.5 3

3.5 4

y 2 2.4 2.7 2.8 3.6 2.6 2.1 Using trapezoidal rule estimate the area bounded by the curve, on the x – axis and the lines

10. Using Simpson’s

rule calculate the approximate value of

and

by taking 7 equidistant

ordinates. [98] 11. Evaluate

rule. Take 6 equidistant. [1.365]

by using Simpson’s

12. A curve is drawn to pass through the points given by the following tables. [7.9833] x

1 1.5 2

2.5 3

3.5 4

y 2 2.4 2.7 2.8 3.6 2.6 2.1 Using Simpson’s

rule estimate the area bounded by the curve, on the x – axis and the lines

and 13. Using Simpson’s

rule calculate the approximate value of

by taking 7 equidistant

ordinates. [99] 14. A curve is drawn to pass through the points given by the following tables. [8.175] Using Simpson’s

rule estimate the area bounded by the

curve, on the x – axis and the lines

x

1 1.5 2

2.5 3

3.5 4

y 2 2.4 2.7 2.8 3.6 2.6 2.1

and

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CH. NO. 3. BINOMIAL AND POISSON DISTRIBUTION EX. 1 1. An unbiased coin is tossed 6 times. Find the probability of getting 3 heads. (5/16) 2. Find the probability of getting atleast 4 heads, in 6 trials of a coin. (11/32) 3. An ordinary coin is tossed 4 times. Find the probability of getting a. Exactly 1 head(1/4) b. Exactly 3 tails(1/4) 4. On an average ‘A’ can solve 40% of the problems. 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 a swimmer 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 probability of A’s winning is 2/3. Find the probability of A’s winning atleast 4 games out of 5. (112/243) 7. A has won 20 out of 30 games of chess with 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 telephone lines are busy at any instant are 0.1, find the 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 man’s chance of hitting a target is 4/5. If he fires 5 shots, what is the probability of hitting the target a. Exactly twice (31/625) b. Atleast once. (3124/3125) 11. It is observed that on an average, 1 person out 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) = and P(T) = is tossed 4 times. If getting a head is success then find the probability distribution. 2. An urn contains 2 white and 3 black balls. A ball is drawn, its colour noted and is replaced in the 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. If a. n = 12; p = 1/3 b. n = 10; p = 2/5 c. n = 100; p = 0.1 12

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4. Find n and p for a binomial distribution, if a. Mean = 6; S.D. = 2. b. c. variance = 5 d.

EX. 3. POISSON DISTRIBUTION Note: For a random variable x with a Poisson distribution with the parameter , the probability 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 , find p(2) & . 2. If a random variable x follows Poisson distribution such that p(1) = p(2), find its mean and variance and also find p(3). 3. In a Poisson distribution, if p(2) = p(3), find mean. 4. In a Poisson distribution the probability of 0 successes is 10%. Find its mean. 5. For a Poisson distribution with find p(2) & 6. 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. 7. The probability that a person will react to a drug is 0.001 out of 2000 individuals checked, find the probability that a. Exactly 3 b. More than 2 individuals get a reaction. 8. A machine producing bolts is known to produce 2% defective bolts. What is the probability that a consignment of 400 bolts will have exactly 5 defective bolts? 9. 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. 10. 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. 11. The average number of incoming telephone calls at a switch board per minute is 2. Find the probability that during a given period 2 or more telephone calls are received. 12. 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. 13. The number of complaints received in a super market per day is a random variable, having a Poisson distribution with = 3.3. Find the probability of exactly 2 complaints received on a given day. 14. In the following situations of a Binomial variate x, can they be approximated to a Poisson Variate? a. n = 150 p = 0.05 b. n = 400 p = 0.25

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CH. NO. 4. ASSIGNMENT PROBLEMS AND SEQUENCING Ex. No. 1 1. Solve the following minimal assignment problem. [21 units] A B C D 1 16 1 6 11 2 25 10 0 10 3 10 25 2 14 4 15 7 14 10 2. A Departmental Store has 4 workers to pack their 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?[9 minutes] Books Toys Crockery Cattery A 2 10 9 7 B 12 2 12 2 C 3 4 6 1 D 4 15 4 9 3. Solve the following minimal assignment problem. [16 units] A B C D 1 3 4 6 5 2 5 6 10 9 3 1 2 3 2 4 4 10 6 4 4. For an examination, the answer papers of the divisions 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. A’s 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. [13 days] A B C D I - 5 2 6 II 4 5 3 8 III 6 6 2 5 IV 1 6 3 4 5. Solve the following minimal assignment problem. [14 units] A B C D I 12 1 11 5 II 3 11 10 8 III 3 4 6 1 IV 2 13 11 7 6. A Departmental head has four subordinates and four task to be performed. The time each man would take to perform each task is given below. [22 units] A B C D I 12 20 11 5 II 1 16 2 14 III 28 9 8 5 IV 10 17 15 1

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7. Minimise the following assignment problem. [16 units] I II III IV

A 2 9 10 7

B 13 12 2 6

C 3 6 4 1

D 4 13 15 9

8. A team of 4 horses and 4 riders has entered the 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. [14 penalty points] HORSES RIDERS R1 R2 R3 R4

H1

H2

H3

H4

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 ‘four’ machinists 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 the maximum profit. [One machinist can be assigned only ‘one’ job][Rs. 376] JOBS MACHINISTS M1 M2 M3 M4

A

B

C

D

E

62 71 87 48

78 84 92 61

50 61 111 87

101 73 71 77

82 59 81 80

10. A Chartered Accountants’ firm has accepted ‘five’ new cases. The estimated number of days required by each of their ‘five’ employees for each case are given below, where ‘-‘means that the particular employee cannot be assigned the particular case. Determine he optimal assignment of cases to the employees so that the total number of days required completing these ‘five’ cases will be minimum. Also find the minimum number of days. [11 days] CASES EMPLOYEES E1 E2 E3 E4 E5

I

II

III

IV

V

5 3 6 4 3

2 4 3 2 6

4 4 2 4

2 5 1 3 7

6 7 2 5 3

11. The cost (in hundreds of Rs.) of sending material 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 ‘one’ terminal] Which terminal will ‘not’ receive material from the truck company? What is the minimum cost?[Rs. 800] TRUCKS TERMINALS T1 T2 T3 T4 T5

A

B

C

D

3 7 3 5 5

6 1 8 2 7

2 4 5 6 6

6 4 8 3 2

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EX. NO. 2 1. Find the sequence that minimises the total elapsed time, required to complete the following jobs on two machineries. Job M1 M2

A 7 4

B 2 6

C 3 5

D 2 4

E 7 3

F 4 1

G 5 4

2. Solve the following for minimum elapsed time and idling time for each machine. Job M1 M2

A 5 2

B 1 6

C 9 7

D 3 8

E 10 4

3. Solve the following problems for minimum elapsed time. Also state the idling time for the machine. Job M1 M2

1 2 6

2 5 8

3 4 7

4 9 4

5 6 3

6 8 9

7 7 3

8 5 8

9 4 11

4. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job Machine A Machine B Machine C

1 8 5 4

2 10 6 9

3 6 2 8

4 7 3 6

5 11 4 5

5. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job Machine A Machine B Machine C

1 8 3 8

2 3 4 7

3 7 5 6

4 2 2 9

5 5 1 10

6 1 6 9

6. Solve the following problem for minimum elapsed time. Also state the idling time for each machine. Job Machine A Machine B Machine C

A 2 3 5

B 7 2 6

C 6 1 4

D 3 4 10

E 8 0 4

F 7 3 5

G 9 2 11

IMPORTANT POINTS TO REMEMBER

U

R

T

C

Un ticked row. Ticked column.

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CH. NO. 5. VITAL STATISTICS, MORTALITY RATES AND LIFE TABLE CRUDE 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 4000 8000 3000 Deaths in A 180 120 200 Population B 7000 9000 4000 Deaths in B 250 320 230 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 60 70 40 30 Deaths in A 250 120 180 200 Population B in thousands 20 40 30 10 Deaths in B 120 100 160 170 4. For the following data. Find if the C.D.R. = 31.25 per thousand. Age group Population Deaths 0-35 4000 80 35-70 3000 120 Above 70 1000 5. For the following data. Find if the C.D.R. = 3.75 Age group 0-20 20-40 40-60 Above 60 Population in thousands 58 71 41 30 Deaths 195 130 245

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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 6000 150 15-40 20000 180 40-60 1000 120 Above 60 4000 160 2. 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 above Population A in thousands 50 90 30 Deaths in A 150 180 200 Population B in thousands 60 100 20 Deaths in B 120 160 250 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 80 Population A in thousands 30 60 50 20 Deaths in A 150 120 200 400 Population B in thousands 50 100 90 70 Deaths in B 200 140 270 350 STANDARD DEATHS RATES (S.T.D.R.)

SP = Standard Population 1. Find the Standard Deaths Rates for the following data: Age – group 0-30 30-60 Above 60 Population A in thousands 60 90 50 Deaths in A 240 270 250 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 75 Population A in thousands 66 54 55 25 Deaths in A 132 108 88 100 Population B in thousands 34 58 52 16 Deaths in B 102 116 78 80 Standard Population in thousands 40 60 80 20

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3. Taking A, as the standard population. Compare the standardized death rates for the population A and B for the given data. Age – group 0-30 30-60 Above 60 Population A in thousands 5 7 3 Deaths in A 150 210 120 Population B in thousands 6 8 2.5 Deaths in B 240 160 7.5 4. Taking A, as the standard population. Compare the standardized death rates for the population A and B for the given data. Age – group 0-20 20-40 40-75 Above 75 Population A in thousands 7 15 10 8 Deaths in A 140 150 110 240 Population B in thousands 9 13 12 6 Deaths in B 270 260 300 150 LIFE TABLES Age

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 6 lx 50 36 21 12 6 2 0 3. Fill in the blanks in the following tabled marked by ‘?’ sign. Age lx dx qx px Lx Tx e0x 50 60 ? ? ? ? 240 ? 51 50 - ? ? 4. Fill in the blanks in the following table marked by ‘?’ sign. Age lx dx qx px Lx Tx e0x 56 400 ? ? ? ? 3200 ? 57 250 ? ? ? ? ? ? 58 120 - ? ?

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CH. NO. 6. INDEX NUMBER EX. NO. 1. SIMPLE AGGREGATIVE METHOD FORMULA INDEX NUMBER I. FIND THE INDEX NUMBER. 1. Find Index number. [Ans. 137.73] Commodities I II III IV V 2. Find Index number. [Ans. 200] Commodities

Prices in Prices in 2002 (P0) 2003 (P1) 21.3 55.9 100.2 60.5 70.6

30.7 88.4 130 90.1 85.7

Prices in Prices in 1990 (P0) 2002 (P1)

A 12 38 B 28 42 C 10 24 D 16 30 E 24 46 3. Find Index number. [Ans. 107.1, 109.375] Commodities Prices Prices Prices in in in 2000 2003 2006 Trucks 800 830 850 Cars 176 200 215 Three wheelers 100 127 115 Two wheelers 44 43 45 4. Find the index number for the year 2003 and 2006 by taking the base year 2000. [Ans. 48, 75.435] Security at 2000 2003 2006 Stock market P0 P1 P1 A 160 180 210 B 2400 35 8 C 800 550 850 D 3500 2000 4000 E 150 600 220

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5. Calculate Index Number. [Ans. 69.078, 238.15] Real Estate 1990 1998 2006 Area wise A 100 65 B 35 22 C 5 7 D 12 11 6. Compute the Index Number. [Ans. 110.526, 126.315] Food Units 2004 2005 Items P0 P1 Potato Kg 10 12 Onion Kg 12 25 Tomato Kg 12 25 Eggs Dozen 24 2 Banana Dozen 18 20

250 75 12 25 2006 P1 14 16 16 26 24

II.

THE INDEX NUMBER BY THE METHOD OF AGGREGATES IS GIVEN IN EACH OF THE FOLLOWING EXAMPLE. FIND THE VALUE OF X IN EACH CASE. 1. Index Number = 180. [Ans. X = 10] Commodity Base year Current Year P0 P1 A 12 38 B 28 41 C 25 D 26 36 E 24 40 2. Index Number = 112.5. [Ans. = 15] Commodity Base Year Current Year P0 P1 I 3 5 II 16 25 III 40 35 IV 7 10 V 14 3. Index Number = 120. [Ans. = 100] Commodity Base Year Current Year P0 P1 I 40 60 II 80 90 III 50 70 IV 110 V 30 30

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EX. NO. 2. WEIGHTED AGGREGATIVE INDEX NUMBERS. FORMULA 1.

LASPEYRE’S PRICE INDEX NUMBER

2.

PAASCHE’S PRICE INDEX NUMBER

3.

DORBISH - BOWLEY’S PRICE INDEX NUMBER

4.

MARSHALL – EDGEWORTH PRICE INDEX NUMBER

5.

FISHER’S PRICE INDEX NUMBER

6.

WALSCH’S PRICE INDEX NUMBER

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1. For the following data find Laspeyre’s, Paasche’s, Dorbish – Bowley’s and Marshall – Edgeworth Index Numbers. [Ans. 134.2, 130, 132.1, 132.05] Commodities Base Year Current Year Price Quantity Price Quantity A 20 3 25 4 B 30 5 45 2 C 50 2 60 1 D 70 1 90 3 2. Calculate Price Index Number by using Walsch’s Method. ,Ans. 126.83Commodities Base Year Current Year Price Quantity Price Quantity A 5 4 7 1 B 2 6 3 6 C 10 9 12 4 3. Find Walsch’s Price Index Number.,Ans. 116.21] Commodities Base Year Current Year Price Quantity Price Quantity I 10 4 20 9 II 40 5 3 5 III 30 1 50 4 IV 50 0.5 60 2 4. The ratio of Laspeyre’s and Paasche’s Index number is 28:27. Find x. ,Ans. x = 4Commodities 1960 1965 Price Quantity Price Quantity A 1 10 2 5 B 1 5 X 2 5. For the following the Laspeyre’s and Paasche’s index number are equal, find . Commodity P0 Q0 P1 Q1 A 4 6 6 5 B 4 4 4 6. Find Fisher’s Price Index Number. ,Ans. 132.1- *using log table+ Commodities Base Year Current Year Price Quantity Price Quantity A 20 3 25 4 B 30 5 45 2 C 50 2 60 1 D 70 1 90 3

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EX. NO. 3. COST OF LIVING INDEX NUMBER THERE ARE TWO METHODS TO CONSTRUCT COST OF LIVING INDEX NUMBER. 1. AGGREGATIVE EXPENDITURE METHOD. COST OF LIVING INDEX NUMBER = 2.

FAMILY BUDGET METHOD. COST OF LIVING INDEX NUMBER = NOTE: W = I=

1. Taking the base year as 1995, construct the cost of living index number for the year 2000 from the following data. [Ans. 137.5] Group 1995 2000 Price Quantity Price Food 23 4 25 Clothes 15 5 20 Fuel and Lighting 5 9 8 House Rent 12 5 18 Miscellaneous 8 6 13 2. Find the cost of living index number. [Ans. 208] Group 1995 2000 Price Quantity Price Food 90 5 200 Clothes 25 4 80 Fuel and Lighting 40 3 50 House Rent 30 1 70 Miscellaneous 50 6 90 3. The price relatives I, for the current year and weights (W), for the base year are given below find the cost of living Index number. [Ans. 221.3] Group Food Clothes Fuel & Lighting House Rent Miscellaneous I W

320 140 270 160 210 20 15 18 22 25 4. Find the cost of living Index number. [Ans. 150] Group Food Clothes Fuel & Lighting House Rent Miscellaneous I 200 150 140 100 120 W 6 4 3 3 4 5. Find if the cost of living index number is 150. [x = 3] Commodity Food Clothes Fuel & Lighting House Rent Miscellaneous I W

200 6

150 4

140

100 3

120 4

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CH. NO. 7. REGRESSION ANALYSIS 1. For a bivariate data the mean of series is 35 and the mean of series is 29. The regression co – efficient of is 0.56. Find the regression equation of Estimate the value of when

2. For a bivariate data the means of series is 40 and mean of series is 35. The Regression co – efficient of is 1.2. Find the line of Regression of y on x. Estimate the value of when

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. 1 2 3 4 5 6 2 4 7 6 5 6 5. Production 120 115 120 124 126 121 Price 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 5 7. Mean S.D. 8. Adv. Exp (Rs. Lakhs) Mean S.D.

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9. Adv. Exp (Rs. In crores) Mean S.D.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20. Find both the regression co – efficient and the measure of the acute angle between the regression lines n=10;

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CH. NO. 8. LINEAR PROGRAMMING 1. A manufacturer makes two types of Radio I and II. To produce radio I require 2 hours in plant A and 3 hours in plant B. To produce radio II requires 3 hours in plant A & 1 hour in plant B. Plant A can be operated at the most 15 hours a day & plant B can be operated for at the most 12 hours a day. If manufacturer makes a profit of Rs. 4 on radio I and Rs. 12 on radio II, in order to produce number of radios of each type, to maximize the profit. Formulate the above problem as L.P.P. 2. A carpenter has 60 units of wood & 50 units of board. From this material, he can prepare chairs and tables. A chair requires 2 units of wood & 1 units of board. A table requires 3 units of wood & 4 units of board. If he sells one chair, he gets a profit of Rs. 100 & if he sells one table he gets a profit of Rs. 200. In order to earn maximum profit, formulate above problem as L.P.P. 3. A company produce two types of presentation goods A & B that require gold & silver. Each unit of type A requires 3 gms of silver & 1 gm of gold while that of B requires 2 gms of gold & 2 gms of silver. The company can procure (purchase) 12 gms of silver & 8 gms of gold. If each unit of type A brings a profit of Rs. 35 & that of type B Rs. 50, formulate this problem as LPP to get the maximum profit. 4. Food A contains 6 units of vitamins & 7 units of minerals per gram & its cost 24 paisa per gram. Food B contains 8 units of vitamins & 12 units of minerals per gram & its cost 40 paisa per gram. The daily requirement of vitamins & minerals are 96 units & 128 units respectively. Formulate the above as LPP to minimize the costs. 5. A diet for a sick person must contain at least 3000 units of vitamins, 75 units of minerals & 1500 calories. Two foods A & B are available at cost of Rs. 4 & Rs. 8 per unit respectively. If one unit of A contains 150 units of vitamins, 1 unit of minerals and 30 calories & one unit of B contains 100 units of vitamins, 2 units of minerals & 35 calories. Form the LPP to minimize the cost. 6. A pineapple farm produced two products. Pineapple and juice. The amounts of materials, labour and equipment required to produce each product & weekly availability of each of these resources are given below: Product Labour Materials Equipments Juice 12 2 50 Pineapple 4 2 25 Availability 1800 500 10000

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7. A small factory produces two types of product requiring two process in melting and in machine shop. The number of man hours of labour required in production of each unit of these products & available time is given below. Product Melting Machine shop A 10 8 B 6 4 Available time 1000 hrs 600 hrs 8. A chemist has a compound to be made using three basic elements. A, B, C, so that it has atleast 10 litres of A, 12 litres of B & 20 litres of C. He makes this compound by mixing two compounds. Each unit of compound I has 4 litres of A, 3 litres of B & no C. Each unit of Compound II has 1 litres of A, 2 litres of B & 4 litres of C. The unit cost of the compounds I & II are Rs. 400 & Rs. 600. Formulate the problem to minimize the cost. 9. A Business firm produces two types of products P and Q. The average profit for the product P is Rs. 100 per ton and that for the product Q is Rs. 70 per ton. The plant consist of three production departments A, B and C. The equipment of each department can be used for 8 hours a day. Product P requires 2 hours in department A and 1 hour in department C per ton. Product Q requires 1 hour in department B and 1 hour in department C per ton. Formulate this problem as a linear programming problem for maximum profit. 10. A toy manufacturers produces bicycles and scooters, each of which must be processed through two machines A and B. The maximum availability of the machines A and B per day are 12 & 10 hours respectively. For manufacturing a bicycle require 4 hours in machine A and 2 hours in machine B, where as a scooter requires 3 hours in machine A and 6 hours in machine B, form the LPP and find the solution set graphically.

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MATHS – I CH. NO. 1. LOGIC Statements, truth values and truth tables A statement is an assertion that can be determined to be true or false. The truth value of a statement is T if it is true and F if it is false. For example, the statement ``2 + 3 = 5'' has truth value T. Statements that involve one or more of the connectives ``and'', ``or'', ``not'', ``if then'' and `` if and only if '' are compound statements (otherwise they are simple statements). For example, ``It is not the case that 2 + 3 = 5'' is the negation of the statement above. Of course, it is stated more simply as ``2 + 3 5''. Other examples of compound statements are: If you finish your homework then you can watch T.V. This is a question if and only if this is an answer. I have read this and I understand the concept. In symbolic logic, we often use letters, such as p, q and r to represent statements and the following symbols to represent the connectives. Connective

Symbol Formal name

Not

Negation

And

Conjunction

Or

Disjunction

If …… then

Conditional

….. if and only if ……

Bi – conditional

Note that the connective ``or'' in logic is used in the inclusive sense (not the exclusive sense as in English). Thus, the logical statement ``It is raining or the sun is shining '' means it is raining, or the sun is shining or it is raining and the sun is shining. If p is the statement ``The wall is red'' and q is the statement ``The lamp is on'', then is the statement ``The wall is red or the lamp is on (or both)'' whereas is the statement ``If the lamp is on then the wall is red''. The statement translates to ``The wall isn't red and the lamp is on''.

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Ex. No. 1 SENTENCE CONNECTIVES Negation (not) ( ) Disjunction (or) (v) Conjunction (and)(but)(as well as) ( ) If then ( ) If and only if ( ) 1. Express the following in the symbolic form i. Hari is either intelligent or hard working. ii. 2. Given p ≡ x is an irrational number. q ≡ x 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. “Kiran failed or Kiran passed as well as he is happy” 4. Write the following statements in symbolic form. i. Bangalore is a garden city and Mumbai is a metropolitan city. ii. Ram is tall or Shyam is intelligent. 5. Write the following statements in symbolically. 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 symbolic form. 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 the following statements into symbolic form. “A person is successful only if he is a politician or he has good connections”. 9. Express the following statements in verbal form:

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10. Let p: Rohit is tall. q: Rohit is handsome. Write the following statements in verbal form using p & q. a. b. 11. a. b.

Ex. No. 2 Truth Table 1. Truth Table for negation T F

F T

2. Truth table for Disjunction T T F F

q T F T F

T T T F

3. Truth table for Conjunction q T T T T F F F T F F F F 1. Construct the truth table and determine whether the statement is tautology, contradiction or neither. i. ( p → q) ⋀ (q ⋀ ~q) ii. [ p ⋁ (~ q ⋀ p)] → p iii. ~( p ⋀ q) iv. p → (q → p) v. p ⋁ (~q ⋀ p). vi. ~ ( p ↔ q). vii. [ p ⋁ (~ q ⋀ ~p)] → p viii. ( p → ~q) → (q ⋀ ~q) ix. ~( ~p ⋀ ~q ) x. ~( ~p ⋀ ~q )

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2. Do as directed. i. Prove that the following statements are logically equivalent: p → q ≡ ~q → ~p ii. Show that the statements p → q and ~( p ⋀ ~q) are equivalent. iii. Write the truth table for “Disjunction”. Write the disjunction 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 are equivalent: p ⋀ q and ~ (p → ~q). 3. Represent the following statements by Venn 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

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. 32

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Ex. No. 3. Write the negation of the following. Formula

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

4 is an irrational. No men is animal. All politicians are corrupt. Rajesh is intelligent or is not an irrational number. All students are hardworking. if and only if Everest is not in Nepal and Tokyo is in India. If 2+5=10 then 4+7 = 21 All students are sincere. I like Mathematics or English. If a quadrilateral is a rectangle then it is a parallelogram.

Ex.No. 4. Converse, Inverse & contra positive Suppose

is a statement then

Converse Inverse Contra positive 1. If weather is humid then it will rain. 2. If x is zero then we cannot divide by x. 3. If two numbers are equal then their squares are unequal.

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Ex. 5. Dual of the statements. Dual : In dual statements ( will be converted into and negation should be kept as it is. 1. 2. 3. I like ice – cream and cream – biscuits. 4. I don’t like fried rice or cold – drinks.

and (

will be converted into

Ex. 6. State whether the sentences are statements in logic or not also state their truth value. 1. 2. 3. 4. 5. 6. 7. 8. 9.

London is in Mumbai. 3 + 3 = 6. 3 + 9 = 18. You are intelligent. Sit down. X + 5 = 7. The sky is blue. 23 is a perfect square. All rational number are natural numbers.

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CH. NO. 2. LIMIT Ex. No. 1. [Algebraic Limits] 1.

9.

2.

10.

3.

11.

4.

12. 13.

5.

14.

6.

15.

7. 8. Ex No 2. [Algebraic Limits] 1.

6.

2.

7.

3.

8.

4. 5. Ex No 3

1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Ex No 4 [Rationalizing] 1.

6.

2.

7.

3.

8.

4.

9.

5.

10. 35

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11.

14.

12.

15.

13. Ex. No. 5 Trigonometric Limits

1.

9.

16.

2.

10.

17.

3. 4. 5. 6. 7. 8. Ex. 6. [Logarithmic Limits]

18.

11.

19.

12.

20.

13.

21.

14. 15.

1.

8.

15.

2.

9.

16.

3.

10.

17.

4.

11.

18.

5.

12.

19.

6.

13.

20.

7. Ex. 7. Exponential Limits

1.

14.

5.

2. 3. 4.

6. 7. 8. 36

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9.

14.

10.

15.

11. 12.

8TH YEAR

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= 10

16. 17.

=1

18.

13. Ex. 8. Trigonometric Limits 1.

13.

2.

14.

3.

15.

4.

=(

16.

5. 6.

17.

7.

18.

8.

19.

9.

20.

10.

21.

11.

22.

12.

Ex. 9. Using first principle find 1. 2. 3. 4.

or Find 6. 7. 8. 9. 10.

5.

11. 12. 13. 14.

Ex. 10. 1. 2. 3. 4. 37

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5. 6.

Continuity Ex. No. 1. Discuss the continuity for the following functions and if the function discontinues, determine whether the discontinuity is removable.

1. 2. 3. 4. 5.

6. 7.

8.

9.

10. 11. 12. 13.

38

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Ex. No. 2. 1. 2.

3. 4. 5.

6. 7.

8.

9.

39

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CH. NO. 3. DIFFERENTIATION (DERIVATIVES) EX. NO. 1. 1. 2. 3.

5.

8.

6.

9.

4.

7.

10.

18.

30.

19.

31.

20.

32.

21.

33.

22.

34.

23.

35.

24.

36.

25.

37.

26.

38.

EX. NO. 2. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

.

27.

39.

28.

40.

29.

41. If

EX. NO. 3. 1.

8.

2.

9.

3.

10. 11. 12. 13. 14. 15. 16.

4. 5. 6. 7.

40

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17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

8TH YEAR

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. If 53. If

EX. NO. 4. INVERSE [FORMULAE] Formulae 1. 2. 3. 4. 5. 6. 7. 8. 9.

Standard substitutions. 1. 41

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2. 3. 4. 5. 6. 7.

EX. NO. 4. INVERSE [SUMS] 1.

18.

34.

3.

19.

35.

4.

20.

36.

5.

21.

2.

6. 7. 8. 9.

38.

22. 23.

39.

24.

40.

10.

25.

11.

26.

12.

27.

13.

28.

14.

37.

41. 42. 43. 44. 45.

29.

46.

30. 15.

47.

31.

16.

32.

17.

33.

48. 49. 50.

EX. NO. 5. LOGARITHMIC FUNCTIONS 1.

2.

42

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3.

5.

18. 19. 20. 21.

6.

22.

7. 8.

23.

9. 10. 11.

25.

4.

24.

26.

12. 13. 14. 15. 16. 29. If

8TH YEAR

27. 28.

then show that

EX. NO. 6. IMPLICIT FUNCTION 11. 12. 13. 14.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. If 20. If

15. 16. 17. 18.

show that

19.

, Show that

21. If 22. If 23. If 24. If 25. If 26. If

show that show that show that show that show that 43

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27. If

8TH YEAR

show that

28. If

show that

29. If

show that

30. If

show that

31. If

show that

32. If

show that

33. If

show that

34. 35. If

show that

36. If

, show that

37. If

Prove that

38. If

Prove that

39. If 40. If

show that show that

EX. NO. 7. PARAMETER FUNCTIONS In the following problems

are parameters

1. If 2. If 3. Differentiate 4. Differentiate 5. If 6. If

w.r.to w.r.to and show that

7. If 8. If 9. If 10. If 11. If 12. If 13. If

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8TH YEAR

with respect to

15. Differentiate

with respect to

16. Differentiate

with respect to

17. If 18. If 19. Differentiate

with respect to

20. If

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CH. 4. APPLICATION OF DERIVATIVES

Ex. No. 1 Approx.

1. Find approximately, the value of 2. Find approximately, the value of three decimal place. 3. Find approximately, the value of four decimal places. 4. Find approximately, the value of 5. Find approximately, the value of 6. Find approximately, the value of 7. Find approximately, the value of 8. Find approximately, the value of 9. Find approximately, the value of 10. Find approximately the value of

12. Find approximately, the value of to

13. Find approximately, the value of to

&

14. Find approximately, the value of 15. Find approximately, the value of 16. Find approximately, the value of given 17. Find approximately, the value of

18. Find approximately, the value of cos(89030’), given 19. Find approximately, the value of cos(30030’), given

11. Find approximately, the value of given

Ex. No. 2 Error

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1. Radius of the sphere is measured as 12 cm with an error of 0.06cm. Find a. Approximate error b. Relative error c. Percentage error in calculating the volume. 2. Radius of a sphere is measures as 25 cm with an error of 0.01cm. Find a. Approximate error b. Relative error c. Percentage error in calculating the volume. 3. Radius of a sphere is found to be 24cm with the possible error of 0.01cm. Find approximately a. Consequent error b. Relative error c. Percentage error in the surface area of the sphere. 4. The side of a square is 5 meter is incorrectly measured as 5.11 meters. Find up to one decimal place the resulting error in the calculation of the area of sphere. 5. If an edge of a cube is measured as 2m with 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 area of the cube having an edge of 3m. If an error of 2cm is made in measuring the edge. Also find the percentage error. 7. The volume of a cone is found by measuring its height and diameter of base as 7 cm and 5 cm respectively. It is

8.

9.

10.

11.

12.

13.

14.

found that the diameter is measured incorrectly to the extent of 0.06 cm. Find the consequent error in the volume. The diameter of a spherical ball is found to be 2cm with a possible error of 0.082mm. Find approximately the possible error in the calculated value of the volume of the ball. Side of an equilateral triangle is measured as 6cm with a possible error of 0.4mm. Find approximate error in the calculated value of its area. Find the approximate % error in calculating the volume of a sphere, if an error of 2% is made in measuring its radius. If an error of 0.3% in the measurement of the radius of spherical balloon, find the %error in its volume. If the radius of a spherical balloon increases 0.1%. Find the approximate % increase in its volume. Under ideal conditions a perfect gas satisfies the equation PV = K; where P = Pressure, V = Volume and K = Constant. If K = 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. Time (T) for completing certain length (L) is given by the equation 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%. 47

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Ex. No. 3. MAXIMA AND MINIMA 1. Examine each of the function for Maximum and Minimum. i. ii. iii. 2. Output

, is given by

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 that 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. ii. 6. Product of two natural numbers is 36. Find them when their sum is minimum. 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 maximum ii. 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. Find its 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. Find its 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 6 mts.

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CH. NO. 5. INDEFINITE INTEGRATION Ex. No. 1 Integrate the following functions 1. 2.

18. 19.

32. 33.

3.

20.

34.

4.

21.

5. 6.

22.

7.

23.

37.

8.

24.

38.

9.

25.

39.

10.

26.

40.

11. 12. 13. 14. 15. 16. 17.

27.

41.

28.

42.

29.

43.

30.

44.

35. 36.

45.

31.

Ex. No. 2. Integrate the following functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

25.

15.

26.

16.

27.

17.

28.

18. 19. 20. 21. 22. 23.

29. 30.

24.

34.

31. 32. 33.

49

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35. 36.

8TH YEAR

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37. 38.

Ex. No. 3. Integrate the following functions Note: - Whenever the degree (Highest Power of a polynomial equation) of the numerator is greater than or equal to the degree of the denominator then divide the numerator by denominator. 1.

5.

9.

2.

6.

10.

3.

7.

11.

8.

12.

4.

Ex. No. 4. Integration by Substitution 1.

18.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

19. 20. 21. 22. 23. 24. 25. 26. 27.

12.

28.

13.

29.

14.

30.

15. 16.

31.

17.

32.

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 50

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49.

59.

69.

50.

60.

70.

51.

61.

71.

52.

62.

72.

53.

63.

73.

54.

64.

74.

55.

65.

56.

66.

76.

57.

67.

77.

58.

68.

78.

75.

Ex. No. 5. Integration of the type or 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14.

27.

15.

28.

16.

29.

17.

30.

18.

31.

19.

32.

20.

33.

21.

34.

22.

35.

23.

36.

24. 25. 26.

37. 38. 39. 51

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Ex. No. 6. Integration of the type OR 1.

5.

9.

2.

6.

10.

3.

7.

4.

8.

Ex. No. 7. Integration of the type OR 1.

7.

12.

2.

8.

13.

3.

9.

14.

4.

10.

5.

11.

6.

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

OR

1.

5.

9.

2.

6.

10.

3.

7.

4.

8.

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

OR

1.

3.

5.

2.

4.

6.

Ex. No. 10. [Important] Integration of the type 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Ex. No. 11. [Important] Integrate the following . 1.

5.

9.

2.

6.

10.

3.

7.

4.

8.

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

1. 2.

8. 9.

14. 15.

3. 4. 5. 6. 7.

10.

16.

11. 12. 13. 53

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

1.

5. 6. 7.

2. 3. 4.

Ex. No. 1.

8.

DEFINITE INTEGRATION

1.

11.

20.

2.

12.

21.

13.

22.

14.

23.

3. 4. 5. 6.

15.

7.

16.

8.

17.

9.

18.

10.

24. 25. 26. 27.

19.

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Ex. No. 2. [Important] PROPERTIES 1. 2. 3. 4. 5. 6.

7.

–

1.

7.

2.

8.

3.

9.

4.

10.

5.

11.

6.

14. 15. 16. 17. 18.

12. 13.

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CH. NO. 6. DIFFERENTIAL EQUATION EX. NO. 1. A. Form the differential equations by eliminating the arbitrary constant. 1. 2. 3. 4. 5.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

(Note: Important sum use the condition for consistency)

31.

EX. NO. 2. 1. Solve 2. Solve the differential equation 3. Solve the differential equation 4. Solve 5. Solve 6. Find the particular solution of the differential equation 7. Solve the differential equation 8. Solve

when

by substituting

by using substitution

9. Solve 10. Find the particular solution of the differential equation

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11. Solve 12. Solve the D.E. 13. Solve the D.E. 14. Solve

Hence find the particular solution if

15. Solve the equation 16. Verity that

is a solution of

17. Verify that

is the general solution of the differential equation

18. Find the particular solution of the differential equation: 19. Solve the differential equation

when

, by taking

20. Find the order and degree of the D.E. 21. Determine the order and degree of the differential equation. 22. Determine the order and degree of the D.E.

.

23. Determine the order and degree of the differential equation

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CH. NO. 7. MATRICES Definition: - A rectangular arrangement of number of rows and number of column which is enclosed between two brackets is called ‘Matrix’.

Eg. 1. Representation of elements

Let ‘a’ is used to denote the different element of Matrix A then, a11 = 4, a12 = 5, a21 =8, a22=7 2. Row matrix:- A matrix having only one row is called as a row matrix. Eg. 3. Column matrix: -A matrix having only one column is called as a column matrix. Eg. 4. Square matrix: - A matrix whose number of rows is equal to the number of columns is called a square matrix. Eg. 5. Diagonal matrix: - A square matrix in which non – diagonal element is equal to zero and diagonal element may be or may not be equal to each other. Eg. 6. Scalar matrix:- A square matrix in which non – diagonal element is equal to zero and diagonal element are equal to each other is called a scalar matrix. Eg. 7. Unit matrix (identity matrix): - A square matrix having every diagonal element equal to one and every non – diagonal element equal to zero is called an unit matrix. It is denoted by ‘I’. Eg.

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8. Null matrix or zero matrix : - A matrix having every element as zero is called as null or zero matrix. It is denoted by ‘0’ Eg. 9. Triangular matrix: a. Upper triangular matrix: - A square matrix in which all the elements below the diagonal are zero is called upper triangular matrix. Eg.

b. Lower triangular matrix: - A square matrix in which all the elements above the diagonal are zero is called lower triangular matrix. Eg. 10. Transpose Matrix: - A transpose matrix is obtained by interchanging row and column from a given matrix. If A is a given matrix then transpose of matrix A is represented by A’. Eg. 11. Determinants of matrix: - If the given matrix is a square matrix then only we can find out determinant value of a given matrix. If A is a given matrix then determinant of matrix

is represented by ,

12. Singular matrix: - A matrix in which the determinant value is equal to zero is called as singular matrix. 13. Non - singular matrix: - A matrix in which the determinant value is not equal to zero is called as non – singular matrix. 14. Equality of two matrices: - Two matrices are said to be equal only if they are of the same order and their corresponding elements are equal.

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Ex: 1 A.

1. Consider the Matrices ,

E=

,

, F=

,

D=

,

, G=

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. Find which of the following matrices are singular and non – singular.

3. If

is a singular matrix, find a.

4. If

is a singular matrix, find k.

B. 1. Consider the matrices.

Answer the following questions. i. ii. iii. 2. If 3. Find

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Ex: 2 1. Answer the following. a. Can you find, i. A + B; ii. A + C;, iii. B’ +D; iv. A + A’ b. If A + F = 0, find b. c. If C – E = I, Find a. 2. Verify the following. a. A + (B + C) = (A + B) + C b. 3(A + B – C) = 3A +3B – 3C c. (A + B)’ = A’ + B’. 3. If 4. Find 5. If 6. If 7. If

find the matrix B. Find the matrix C such that A + B + C is a zero matrix.

8. Find the matrix ‘X’ such that 9. Find the values of x and y satisfying the matrix equation.

10. Find x, y & z if

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Ex: 3 1. Find the following products:

2. Find x in the following cases.

Ex: 4 1. Find AB and BA whenever they exist in each of the following cases. 1. 2. 3. 4.

2.

Then verify the following

3. If

4. If 5. If

verify the following.

show that

is a scalar matrix.

,

6. (A) Find the values of a and b from the matrix equation: 6. (B) Find the values of x and y. 6. (C). Find x, y, z values in each of the following cases. i.

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7. Find x, y, z, a, b, c if

8. If

Find the values of x, y, z if

9. If

Find the Matrix AB and without computing the Matrix BA,

show that AB

BA.

10. If

Verify that AB BA.

11. If 12. If 13. If

show that AB is a Non singular matrix. Show that

is a null matrix.

14.

show that

14.

Show that A satisfies the Matrix Equation

.

Ex: 5 1. If

show that AB = 0.

2. If

show that BA = CA.

3.

show that AB = BA. Show that

4. 5. 6. If

show that show that show that

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7. If

8. If

and

such that

Ex: 6 I. Write down the following equation in the Matrix Form and hence find values of x, y, z using Matrix method. 1. . 2. 3. II. Solve the following equation by the methods of reduction. 1. 2. 3.

Ex: 7 A. Find the inverse of each of the following Matrices by using elementary transformations.

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ALGEBRAIC FORMULAE 1. 2. 3. 4.

5. 6. 7. 8.

TRIGONOMETRIC FORMULAE 1. 2. 3.

10.

4. 5.

11. 12.

6.

13.

7.

14. 15. 16. 17. 18. 19. 20. 21.

8. 9.

1.

5.

2.

6.

3.

7.

4.

8.

1.

4.

2.

5.

3.

6.

65

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2.

4. 5.

3.

6.

1. 2.

6.

10.

7.

11.

3. 4. 5.

8. 9.

1. 2. 3.

7.

4.

9.

5.

10.

6.

11.

12. 13. 14. 15.

8.

DERIVATIVES FORMULAE

1. 8. 2. 9. 3. 10. 4. 11. 5. 12. 6. 13. 7. 66

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14. 17. 15. 18. 16. 19.

Integration formulae 1. 2. 3.

15. 16. 17.

4.

18.

5.

19.

6.

20.

7. 8. 9.

21. 22. 23.

10. 11.

24.

12.

25.

13. 14.

26. 27.

67

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68