CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY

(Approved by AICTE New Delhi, Affiliated to Anna University Chennai.

Zamin Endathur Village, Madurntakam Taluk, Kancheepuram Dist.-603311.)

MA6452 - STATISTICS AND NUMERICAL METHODS

QUESTION BANK

(YEAR/SEM: II/IV)

UNIT - I

TESTING OF HYPOTHESIS

PART – A (2 Marks)

1. Define Type –I and Type –II errors (Apr/May-2014/2011/2012/Nov2013)

2. State the conditions for applying (Apr/May-2014/2012)

3. It has been found that 2% of the tools produced by a certain machine are defective. What is the probability

that in a shipment of 400 such tools (i) 3% or more (ii) 2% or less will prove defective. (Apr/May-2013)

4. A random sample of 200 tine of coconut oil gave an average weight of 4.95Kgs. With a standard deviation

of 0.21kg. (Apr/May-2013)

5. Twenty people were attacked by a disease and only 18 survived. The hypothesis is set in such a way that

the survival rate is 85% if attacked by this disease. Will you reject the hypohthesis that it is more at

5%level (Nov/Dec-2014)

6. Write the formula for the Chi-square test of goodness of fit of a random sample to a hypothetical

distribution. (Nov/Dec-2014)

7. What are the application of t-distributions? (Apr/May-2011)

8. Mention the various steps involved in testing of hypothesis. (Apr/May-2010)

9. Define test of goodness of fit. (Apr/May-2010)

10. State Level of Significance (Nov/Dec-2013)

11. What are null and Alternate hypothesis? (Nov/Dec-2012)

12. Give the formula for the - test of independence for (Nov/Dec-2012) 13. What are parameters and Statistics in sampling? (Nov/Dec-2010).

14. Write any two application of -test. (Nov/Dec-2010).

15. What are the applications of t-distributions?

16. A coin is tossed 400 times and it turns up head 216 times. Discuss whether the coin may be unbiased

one at 5%level of significance.

17. In a large city A, 20 percent of a random sample of 900 school boys had a slight physical defect. In

another large city B, 18.5 percent of a random sample of 1600 school boys had some defect. Is the

difference between the proportions significant?

18. What do you mean by test of Hypothesis two tailed test?

19. What are the properties of “F” test.

20. Define a „F‟ variate.

21. Define errors in sampling and critical region.

22. In a large city A, 20 percent of a random sample of 900 school boys had a slight physical defect. In

another large city B, 18.5 percent of a random sample of 1600 school boys had some defect. Is the

difference between the proportions significant?

23. A sample of size 13 gave an estimated population variance of 3.0 while another sample of size 15 gave

an estimate of 2.5. Could both samples be from populations with the same variance?

24. Define normal distribution in single mean.

25. Define t distribution .

a b

c d

PART-B (16 Marks)

1. A manufacturer of light bulbs claims that an average of 2% of the bulbs manufactured by him are defective .

A random sample of 400 bulbs contained 13 defective bulbs. On the basis of the sample, can you support

the manufacturer‟s claim at 5% level of significance? (Apr/May-2014)

2. A Survey of 320 families with 5 children each revealed the following distribution (Apr/May-2014)

No.of boys: 5 4 3 2 1 0

No.of girls: 0 1 2 3 4 5

No.of families: 14 56 110 88 40 12

Is this result Consistent with the hypothesis that male and female births are equally probable?

3. In a random sample of 100 men taken from village A, 60 were found to be consuming alcohol. In another

sample of 200 men taken from village B, 100 were found to be consuming alcohol . Do the two villages

differ significantly in respect of the propotion of men who consume alcohol? (Apr/May-2014)

4. Two independent samples of sizes 9 and 7 from a normal population had the following values of the

variables. (Apr/May-2014)

Sample I 18 13 12 15 12 14 16 14 15

Sample II 16 19 13 16 18 13 15

Do the estimates of population variance differ significantly at 5% level of significance?

5. Random samples drawn from two countries gave the following data relating to the heights of adult males. Is

the difference between standard deviation significant? (Apr/May-2013)

Country A Country B

Mean heights (in inches) 67.42 67.25

S.D (in inches) 2.58 2.50

Numbers in samples 1000 1200

6. 1000 student at college level were graded according to their I.Q and their economic conditions. What

conclusion can you draw from the following data: (Apr/May-2013)

Economic conditions I.Q. Level

High Low

Rich 460 140

Poor 240 160

7. The sales manager of a large company conducted a sample survey in states A and B taking 400 samples in

each case. The results were in the following table. Test whether the average sales in the same in the 2 states

at 1% level. (Apr/May-2013)

State A state B

Averages Sales Rs.2,500 Rs. 2,200

S.D Rs.400 Rs.550

8. Find if there is any association between extravagance in fathers and extravagance in sons from the following

data . Determine the coefficient of association also(Apr/May-2013)

Extravagant father Miserly father

Extrav. Sons Under 327 741

Miser. Sons 545 234

9. A dice is thrown 400 times and a throw of 3 or 4 is observed 150 times. Test the hypothesis that the dice is

fair. (Apr/May-2012)

10. Theory predicts that the proportion of beans in four groups A, B, C, D should be 9:3:3:1. In an experiment

among 1600 beans, the numbers in the four groups were 882, 313, 287 and 118. Does the experiment

support the theory? (Apr/May-2012)

11. The means of two large samples of 1000 and 2000 members are 67.5 inches and 68.0 inches respectively.

Can the samples be regarded as drawn from the same populations of standard deviation 2.5 inches?

(Apr/May-2012)

12. Two random samples gave the following results: (Apr/May-2012)

Sample Size Sample mean Sum of squares of deviation from the mean

1 10 15 90

2 12 14 108

Test whether the samples have come from the same normal population.

13. Explain clearly the procedure generally followed in testing of a hypothesis. (Nov/Dec-2014)

14. The demand for a particular spare part in a factory was found to vary from day-to- day. In a sample study

the following information was obtained. (Nov/Dec-2014)

Days: Mon Tues Wed Thurs Fri Sat

No.of spare parts demanded 1124 1125 1110 1120 1126 1115

Test the hypothesis that the number of parts demanded does not depend on the day of the

week.(

15. Explain briefly the procedure involved in testing the significance for difference of proportions in the case

of large samples. (Nov/Dec-2014)

16. The height of six randomly chosen sailors are (in inches):63, 65, 68,69,71 and 72. Those of 10 randomly

chosen soldiers are 61, 62, 65, 66, 69, 69,70, 71, 72 and 73. Discuss, the height that these data thrown on

the suggestion that sailors are on the average taller than soldiers .

(Nov/Dec-2014) 17. In a random sample of 1000 people from city A, 400 are found to be consumers of wheat. In a sample of

800 from city B, 400 are found to be consumers of wheat. Does this data give a significant difference

between the two cities as far as the proportion of wheat consumers is concerned?

(Apr/May-2011) 18. 4 Coins were tossed 160 times and the following results were obtained:

No. of heads 0 1 2 3 4

Observed frequencies 17 52 54 31 6

Under the assumption that the coins are unbiased, find the expected frequencies of getting 0,1,2,3,4

heads and test the goodness of fit. (Apr/May-2011)

19. The heights of 10 males of a given locality are found to be 70,67,62,68,61,68,70,64,64,66 inches. Is it

reasonable to believe that the average height is greater than 64 inches? (Apr/May-2011)

20. Test of the fidelity and selectivity of 190 radio receivers produced the result shown in the following table:

Selectivity Fidelity

Low Average High

Low 6 12 32

Average 33 61 18

High 13 15 0

Use the 0.01 level of significance to test whether there is a relationship between fidelity and

Selectivity. (Apr/May-2011)

21. A sample of 900 members has a mean 3.4cm and standard deviation 2.61cm. Is the sample from a large

population of mean 3.25cms and standard deviation of 2.61cms?(Test at 5% level of significance. The

value of z at 5%level is . (Apr/May-2010)

22. Before an increase in excise duty on tea, 800 persons out of a sample of 1,000 persons were found to

be tea drinkers. After an increase in duty 800 people were tea drinkers in a sample of 1,200 people. Using

standard error of proportion, stat whether there is a significant decrease in the consumption of tea after the

increase in excise duty? ( at 5% level 1.645, 1%level 2.33). (Apr/May-2010)

23. Out of 8000 graduates in a town 800 are females, out of 1600 graduate employees 120 are females.

Use to determine if any distinction is made in appointment on the basis of sex. Value of at 5% level

for one degree of freedom is 3.84. (Apr/May-2010)

24. An automobile company gives you the following information about age groups and the liking for

particular model of car which it plans to introduce. On the basis of this data can it be concluded that the

model appeal is independent of the age group ( 0.05(3)=7.815) (Apr/May-2010)

Age Group

Persons Who: Below 20 20-39 40-59 60 and above

Liked the car: 140 80 40 20

Disliked the car: 60 50 30 80

25. Time taken by workers in performing a job are given below; (Nov/Dec-2013)

Type I: 21 17 27 28 24 23 -

Type II 28 34 43 36 33 35 39

Test whether there is any significant difference between the variances of time distribution.

UNIT – II

DESIGN OF EXPERIMENTS

PART – A (2 Marks)

1. What ate the basic principles of experimental design? (Apr/May-2014/Nov-2012)

2. State any two advantages of completely Randomized Experimental Design (Apr/May-2014).

3. What do you understand by “Design of an experiment” (Apr/May-2013)

4. Write down the ANOVA table for one way classification? (Apr/May-2013)

5. State the assumptions involved in ANOVA. (Apr/May-2012)

6. What are the advantages of a Latin square design? (Apr/May-2012)

7. Explain the situations in which randomized block design is considered an improvement over a completely

randomized design. (Nov/Dec-2014)

8. State the advantages of a factorial experiment over a simple experiment. (Nov/Dec-2014/Apr-2012)

9. What are the conditions for the validity of (Apr/May-2011)

10. Write any two differences between RBD and CRD (Apr/May-2011)

11. Discuss the advantages and disadvantages of Randomized block design(Apr/May-2010)

12. What is the aim of the design of experiments? (Nov/Dec-2013)

13. Define factorial design. (Nov/Dec-2013)

14. Define RBD (Nov/Dec-2012)

15. Compare one-way classification model with two- way classification model. (Nov/Dec-2010).

16. What is meant by Latin Square? (Nov/Dec-2010).

17. Define Mean sum of Squares.

18. Write down the format the ANOVA table for two factors of classification

19. Why a 2 x 2 Latin square is not possible? Explain.

20. What is meant by tolerance limits?

21. Write down the format the ANOVA table for three factors of classification.

22. Define Local control.

23. Define Replication.

24. Define sum of Squares.

25. Define variance.

PART - B(16 Marks)

1. Four varieties A,B,C,D of a fertilizer are tested in a Randomized Block Design with four

replication. .The plot yields in pounds are as follows (Apr/May-2014/2012)

A12 D20 C16 B10

D18 A14 B11 C14

B12 C15 D19 A13

C16 B11 A15 D20

Analyses the experimental yield.

2. Analyze the variance in the Latin square of yields paddy where

denote the different methods of cultivation (Apr/May-2014)

S122

Q124

P120

R122

P121

R123

Q119

S123

R123

P122

S120

Q121

Q122

S125

R121

P122

Examine whether the different methods of cultivation have given significantly different yields .

3. The following data represent the number of units production per day turned out by

different workers using 4 different types of machines. (Apr/May-2013/2011Nov-2010)

Machine type

A B C D

1 44 38 47 36

2 46 40 52 43

3 34 36 44 32

4 43 38 46 33

5 38 42 49 39

Test whether the five men differ with respect to mean productivity and test whether

the mean productivity is the same for the four different machine types.

4. The following is a Latin square of a design when 4 varieties of seeds are being tested. Set up the

analysis of variance table and state your conclusion. You may carry out suitable change of origin and

scale. (Apr/May-2013)

A 105 B 95 C 125 D 115

C 115 D 125 A 105 B 105

D 115 C 95 B 105 A 115

B 95 A 135 D 95 C 115

5. Compare and contrast the Latin square Design with the Randomised block Design. (Apr/May-2013)

6. Analyse the following of Latin square experiment (Apr/May-2013)

Column Row 1 2 3 4

1 A(12) D(20) C(16) B(10)

2 D(18) A(14) B(11) C(14)

3 B(12) C(15) D(19) A(13)

4 C(16) B(11) A(15) D(20)

7. A variable trial was conducted on wheat with 4 varieties in a Latin Square design. The plan of the

experiment and per plot yield are given below: D25 B23 A20 D20

A19 D19 C21 B18

B19 A14 D17 C20

D17 C20 B21 A15

Analyse the data. (Apr/May-2012)

8. A company wants to produce cars for its own use. It has to select the make of the care out of the four makes

A,B, C and D available in the market. For this he tries four cars of each make by assigning the cars to four

drivers to run on four different routes. The efficiency of cars is measured in terms of time in hours. The layout

and time consumed is as given below. (Nov/Dec-2014)

Drivers

Routes 1 2 3 4

1 18(C) 12(D) 16(A) 20(B)

2 26(D) 34(A) 25(B) 31(C)

3 15(B) 22(C) 10(D) 28(A)

4 30(A) 20(B) 15(C) 9(D)

9. Consider the results given in the following table for an experiment involving six treatment in four

randomized blocks. The treatments are indicated by numbers within parenthesis. (Nov/Dec-2014)

Blocks yield for a randomized block experiment treatment and yield

1 (1) (3) (2) (4) (5) (6)

24.7 27.7 20.6 16.2 16.2 24.9

2 (3) (2) (1) (4) (6) (5)

22.7 28.8 27.3 15.0 22.5 17.0

3 (6) (4) (1) (3) (2) (5)

26.3 19.6 38.5 36.8 39.5 15.4

4 (5) (2) (1) (4) (3) (6)

17.7 31.0 28.5 14.1 34.9 22.6

Test whether the treatments differ significantly. . .

10.The following are the number of mistakes made in 5 successive days by 4 technicians working for a

photographic laboratory test at a level of significance . Test whether the difference among the

four sample means can be attributed to chance. (Apr/May-2011)

Technician

I II III IV

6 14 10 9

14 9 12 12

10 12 7 8

8 10 15 10

11 14 11 11

11. (i) What are the basic assumptions involved in ANOVA? (Apr/May-2011)

(ii) In a Latin square experiment given below are the yields in quintals per acre on the paddy crop carried

out for testing the effect of five fertilizers A,B,C,D,E. Analyse the data for variations.

B25 A18 E27 D30 C27

A19 D31 C29 E26 B23

C28 B22 D33 A18 E27

E28 C26 A20 B25 D33

D32 E25 B23 C28 A20

12. A set of data involving four “four tropical feed stuffs A, B, C, D” tried on 20 chicks is given below. All the

twenty chicks are treated alike in all respects except the feeding treatments and each feeding treatment is

given to 5 chicks. Analysze the data. (Apr/May-2010)

Weight gain of baby chicks fed on different feeding materials composed of tropical feed stuffs.

Total

A 55 49 42 21 52 219

B 61 112 30 89 63 355

C 42 97 81 95 92 407

D 169 137 169 85 154 714

Grand Total G=1695

13. An experiment was planned to study the effect of sulphate of potash and super phosphate on the yield of

potatoes. All the combinations of 2 levels of super phosphate and 2 levels of sulphate of potash were

studied in a randomized block design with 4 replications for each. The yields (per plotP obtained are

given below. (Apr/May-2010)

Block Yields (lbs per plot)

I (1) k p kp

23 25 22 38

II p (1) k kp

10 26 36 38

III (1) k kp p

29 20 30 20

IV kp k p (1)

34 31 24 28

Analyze the data and comment on your findings .

14. Analyse the following RBD and find your conclusion (Nov/Dec-2013)

Treatments

T1 T2 T3 T4

B1 12 14 20 22

B2 17 27 19 15

Blocks B3 15 14 17 12

B4 18 16 22 12

B5 19 15 20 14

15.The following is a Latin square of a design when 4 varieties of seeds are being tested. Set up the

analysis of variance table and state your conclusion. You may carry out suitable change of origin and

scale. (Nov/Dec-2013)

A 110 B 100 C 130 D 120

C 120 D 130 A 110 B 110

D 120 C 100 B 110 A 120

B 100 A 140 D 100 C 120

UNIT – III

SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS

PART-A(2 Marks)

1. Stage the order of convergence and condition for convergence of Newton –Raphson method. (Apr/May-

2014/2011/Nov-2012-2010)

2. What the procedure involved in Gauss elimination method. (Apr/May-2014)

3. Find the real positive root of by Newton‟s method correct to 6 decimal places.

(Apr/May-2013)

4. Solve the equations using Gauss elimination method. (Apr/May-2013)

5. Arrive a formula to find the value of , where , using Newton-Raphson method.

(Apr/May-2012) 6. Solve the following system of equations using Gauss-Jordan elimination method

. (Apr/May-2012)

.Find a real toot of the equation , using Newton-Raphson method. (Nov/Dec-2014)

8. Write down the iterative formula of Gauss- Seidal method. (Nov/Dec-2014)

9. Find the dominant eigenvalue of the matrix by power method (Apr/May-2011)

10. Using Newton-Raphson method, find the iteration formula to compute . (Apr/May-2010)

11. Explain the power method to determine the eigenvalue of matrix. (Apr/May-2010)

12. State the principle used in Gauss-Jordan method. ( Nov/Dec-2013)

13. Solve using Gauss elimination method.

(Nov/Dec-2010). 14. Compare Gauss elimination and Gauss Jacobi methods. (Nov/Dec-2012/2010)

15. Solve by Gauss – Jordan method the following system of equation , .

16. Solve the system of equations by Gaussian elimination method.

17. Solve the system of equations by Gaussian elimination method

18. Solve by Gauss – Jordan method the following system of equation , .

19. Find a real toot of the equation , using Newton-Raphson method.

20. Using Newton-Raphson method find the value of .

21. Find the dominant eigenvalue of the matrix by power method

22. Find the dominant eigenvalue of the matrix by power method.

Solve the equations using Gauss 23. elimination method

24. Find the real positive root of by Newton‟s method

25. Solve by Gauss – Jordan method the following system of equation , .

PART-B (16 MARKS)

1. Solve the equation using Newton-Raphson method. (Apr/May-2014)

2. By Gauss Jordan elimination method. Find the inverse of the matrix (Apr/May-2014)

3. Solve the following set of equations using Gauss –Seidal iterative procedure

. (Apr/May-2014)

4. Find the numerically largest eigen values of by using power method.

(Apr/May-2014/2012)

5. Solve the system of equation by Gauss – Jordan method

(Apr/May-2013)

6. Solve by Gauss – seidal method the following system

. (Apr/May-2013)

7. Solve by Gauss- Elimination method .

(Apr/May-2013)

8. Using Power method, find all the eigen values of (Apr/May-2013/Nov 2014)

9. Using Newton-Raphson method, solve taking the initial value x0 as 10

(Apr/May-2012)

10. Using Gauss Jordon method, find the inverse of (Apr/May-2012/2010)

11.Solve the following system of equations using Gauss-Seidal iterative method

(Apr/May-2012) 12.Find the solution to three decimals, of the system using Gauss-Seidal method

and . (Nov/Dec-2014)

13. Find the inverse of the matrix using Gauss-Jordan method. (Nov/Dec-2014)

14. Solve the system of equations using Gauss-elimination method.

and . (Nov/Dec-2014)

15. Find the real positive root of by Newton‟s method correct to 6 decimal places.

(Apr/May-2011/Nov2013)

16. Solve, by Gauss – Seidel method, the following system(Apr/May-2011)

correct to 3 decimal places. 17. By 17.

Gauss Jordan method , find the inverse of (Apr/May-2011)

18.Find the numerically largest eigenvalue of and the corresponding eigenvector.

(Apr/May-2011) 19. Solve the following system of equation by, Gauss – elimination method

– (Apr/May-2010)

20. Solve and using Gauss seidal method.

(Apr/May-2010)

21. Determine the largest eigenvalue and the corresponding eigenvector of the matrix with

the initial vector . (Apr/May-2010/Nov-2010)

22. Solve the system of equations by Gauss-Elimination method ,

. (Nov/Dec-2013)

23. Find the inverse of the matrix by Gauss-Jordan method. (Nov/Dec-2013)

24.Find the largest eigen value of the matrix by power method. Also find its corresponding

eigen vector. (Nov/Dec-2013)

25. Solve the following equation by Gauss-Seidal method:

(Nov/Dec-2012)

UNIT IV

INTERPOLATION, NUMERICAL DIFFERENTIATION AND

NUMERICAL INTEGRATION

PART- A (2 Marks)

1. State any two properties of divided differences. (Apr/May-2014)

2. What is inverse interpolation? (Apr/May-2014)

4. Use Lagrange‟s formula to fit a polynomial to the data and find . (Apr/May-2013)

X -1 0 2 3

Y -8 3 1 12

5. Show that the divided difference of second order can be expressed as the quotient of two determinants

of third order. (Apr/May-2013)

6. Form the divided difference table for the following data: (Apr/May-2012)

x : 5 15 22

y : 7 36 160

7. Evaluate Trapezoidal rule, dividing the range into 4 equal parts (Apr/May-2012)

8. Write the Lagrange‟s formula for interpolation and state its uses. (Nov/Dec-2014)

9. Evaluate , correct to three decimal places using trapezoidal rule with

(Nov/Dec-2014)

10.Using Trapezoidal rule, evalute with . Hence obtain an approximate value of .

(Apr/May-2011)

11. State the formula to find the second order derivative using the forward differences. (Apr/May-2011)

12. Write down the Lagrange‟s interpolating formula. (Apr/May-2010/Nov-2013)

13.Write down the Simpson‟s 1/3 – Rule in numerical integration. (Apr/May-2010)

14. Evaluate using Trapezoidal rule, taking (Nov/Dec-2013)

15. What is the need of Newton‟s and Lagrange‟s interpolation formulae? (Nov/Dec-2012)

16. Find the area under the curve passing through the points and

(6,1.5) (Nov/Dec-2012)

17. Create a forward difference table for the following data and state the degree of polynomial for the same.

(Nov/Dec-2010).

18. Compare Simpson‟s 1/3 rule with Trapezoidal method. (Nov/Dec-2010).

19. Show that .

20. State Newton‟s backward difference interpolation formula.

21. State Newton‟s formula to find using the forward difference.

22. Evaluate by Trapezoidal rule dividing the range into 4 equal parts.

23. Find the polynomial for the following data by Newton‟s backward difference formula.

x 0 1 2 3

f(x) -3 2 9 18

24.State Simpson‟s 1/3 rule formula to evaluate .

25. What is „inverse interpolation‟.

PART - B(16 Marks)

1. Find Polynomial by using Lagrange‟s formula and hence find for (Apr/May-2014)

x: 1 3 5 7

f(x) 24 120 336 720

2. Evaluate by using Simpson‟s one third-rule and hence deduce the value of

(Apr/May-2014) 3. Construct Newton‟s forward interpolation polynomial for the following data: (Apr/May-2014)

x: 1 2 3 4 5

f(x) 1 -1 1 -1 1

And hence find .

x 0 1 2 3

y=f(x) -1 0 3 8

4. The Velocity v of a particle at a distance s from appoint on its path is given as follows:

(Apr/May-2014)

s in meter 0 10 20 30 40 50 60

v m/sec 47 58 64 65 61 52 38

5. By dividing the range into ten equal parts, evaluate by Trapezoidal and Simpson‟ s rule . Verify

your answer with integration. (Apr/May-2013)

6. If , show that Where r is any positive integer.

(Apr/May-2013)

7. The population of a certain town is given below. Find the rate of growth of the population in

1931,1941,1961,1971. (Apr/May-2013) Year (x) 1931 1941 1951 1961 1971

Population in thousands (y) 40.62 60.80 79.95 103.56 132.65 8. Using Newton‟s divided difference formula find the value of f(2) f(8) and f(15) from the following data

(Apr/May-2013/2011/Nov-2013)

X 4 5 7 10 11 13

f(x) 48 100 294 900 1210 2028

9. Using Lagrange‟s interpolation, find the value of f (3) , from the following table:

(Apr/May-2012/Nov-2013)

X 0 1 2 5

f(x) 2 3 12 147

10. Evaluate correct to three decimals dividing the range of integration into 8 equal parts using

Simpson‟s rule. (Apr/May-2012)

11. Using Newton‟s forward interpolation formula, find the polynomial satisfying the following data

hence evaluate f(x) at x=5. (Apr/May-2012)

x 4 6 8 10

f(x) 1 3 8 16

12. Compute and from the following data: (Apr/May-2012)

x 0 1 2 3 4

f(x) 1 2.718 7.381 20.086 54.598

13.Given the table of values

x 50 52 54 56

3.684 3.732 3.779 3.825

Use Lagrange‟s formula to find (Nov/Dec-2014)

14. Given the set of tabulated points and obtain the value of y when x=2 using

Newton‟s divided difference formula. (Nov/Dec-2014)

15.The velocities of car (running on a straight road) at intervals of 2 minutes are given below.

Time in minutes 0 2 4 6 8 10 12

Velocities in km/hr: 0 22 30 27 18 7 0

Apply Simpson‟s rule to find the distance covered by the car. (Nov/Dec-2014)

16. Find the first and second derivatives of at if (Nov/Dec-2014)

x 1.5 2.0 2.5 3.0 3.5 4.0

f(x) 3.375 7.000 13.625 24.000 38.875 59.000

17. From the following table of half-yearly premium for policies maturing at different ages, estimate the

premium for policies maturing at age 46 and 63. (Apr/May-2011)

Age x 45 50 55 60 65

Premium y 114.84 96.16 83.32 74.48 68.48

18. Using Lagrange‟s interpolation formula, find from the following table. (Apr/May-2011)

x 5 6 9 11

y 12 13 14 16

19. The table below gives the velocity V of a moving particle at time t seconds. Find the distance covered

by the particle in 12 seconds and also the acceleration at seconds, using Simpson‟s rule.

(Apr/May-2011)

t 0 2 4 6 8 10 12

V 4 6 16 34 60 94 136

20.Use the Newton divided difference formula to calculate and from the following

table: (Apr/May-2010)

21. From the following table of values of x and y, obtain and for (Apr/May-2010)

x 1.0 1.2 1.4 1.6 1.8 2.0 2.2

y 2.7183 3.3201 4.0552 4.9530 6.0496 7.3891 9.0250

22. A rocket is launched from the ground. Its acceleration is registered during the first 80 seconds and is in the

table below. Using trapezoidal rule and Simpson‟s 1/3 rule, find the velocity of the rocket at

(Apr/May-2010)

t(sec) 0 10 20 30 40 50 60 70 80

f:( cm/sec) 30 31.63 33.34 35.47 37.75 4.033 43.25 46.69 40.67

23. Evaluate the length of the curve from , using Simpson‟s 1/3 rule using 8 sub-

intervals (Apr/May-2010)

24. Evaluate using Trapezoidal rule. (Nov/Dec-2013)

25. Taking , evaluate by Simpson‟s 1/3 rule. (Nov/Dec-2013)

26. Evaluate by trapezoidal and Simpson‟s 1/3 rules by dividing the range into 10 equal parts.

(Nov/Dec-2012/2010)

UNIT – V

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL

EQUATIONS

PART – A(2 Marks) 1. State the advantages of Runge –Kutta method over Taylor Series method. (Apr/May-2014/2012)

2. Convert the differential equation into finite difference equivalent form.

(Apr/May-2014)

3. Using Taylor series method, find y at given . (Apr/May-2013)

4. Compute by Modified Euler method given . (Apr/May-2013)

5. Write the central difference approximations for (Apr/May-2012)

x 0 1 2 4 5 6

f(x) 1 14 15 5 6 19

6. Use the Runge-Kutta fourth order method to find the values of y when given that when

and that . (Nov/Dec-2014)

7.Given . Determine using Euler‟s modified method. (Nov/Dec-2014)

8. Given and , determine the value of by Euler‟s method.

(Apr/May-2011/2010)

9. Write the Milne‟s Predictor –Corrector formula. (Apr/May-2011/2010)

10.Using taylor‟s method find y at given , (Nov/Dec-2013)

11. Obtain the finite difference scheme for differential equation . (Nov/Dec-2013)

12. Bring out the merits and demerits of Taylor series method. (Nov/Dec-2012)

13. Find by Euler‟s method, if (Nov/Dec-2012).

14. Using Taylor‟s series find for . (Nov/Dec-2010).

15. Solve (Nov/Dec-2010).

16. Write down Euler algorithm to the differential equation .

17. Write the merits and demerits of the taylor method of solution.

18. What is meant by initial value problem and give an example.

19. By Taylor series method find given .

20. Write down the Runge-Kutta formula of fourth order to solve with .

21. Write down the Runge-Kutta formula of second order to solve with .

22.Solve .at .

23. The modified Euler method is based on the average of points.

24. Find by Euler‟s method, if

25. Find by Euler‟s method, if

PART – B (16 Marks)

1. Apply Taylor series method to find and approximate value of y when given that

(Apr/May-2014)

2. Solve the BVP using finite difference method, taking h=0.25.

(Apr/May-2014)

3. Using Milne‟s predictor corrector method find given given

. (Apr/May-2014)

4. Evaluate and correct to three decimal places by the modified Euler method, given that

taking . (Apr/May-2014)

5. Solve (Apr/May-2013)

6. Using Runge- Kutta method of fourth –order, sovle at .

(Apr/May-2013/2011/2010/Nov2012)

7. Given find the value of by using Runge-Kutta method of

fourth order. (Apr/May-2013/Nov2013)

8. Consider the initial value problem using the modified euler method find

(Apr/May-2012)

9. Using Milne‟s method find given given , ,

, (Apr/May-2012)

10. Find given that by using R-K method of order 4 taking

(Apr/May-2012)

11. Solve the BVP with using finite difference method with h=0.2

(Apr/May-2012)

12. Compute and given that

. Using Milne‟s Predictor –corrector method. (Nov/Dec-2014)

13. Solve by Taylor‟s method to find an approximate value of y at for the differential equation

. Compare the numerical solution with the exact solution. Use first three non-

zero terms in the series. (Nov/Dec-2014)

14. Consider the initial value problem Compute by Euler‟s method and

modified Euler‟s method. (Nov/Dec-2014)

15. Solve , using fourth order Runge-Kutta method.

(Nov/Dec-2014)

16. Using Modified Euler method, find given

(Apr/May-2011/Nov-2010)

17. Using Taylor method, compute d correct to 4 decimal places given and

by taking (Apr/May-2011)

18. Using Adam‟s method find given

, (Apr/May-2011)

19. Given where when find and , using Taylor series method.

(Apr/May-2010)

20. Given and evaluate

by Adam-Bashforth predictor- corrector method. (Apr/May-2010)

21. Using finite differences solve the boundary value problem

with . (Apr/May-2010)

22. Using Milne‟s predictor- corrector method, find , given that

(Nov/Dec-2013)

23. Solve by Euler‟s method, the equation choose and compute and

(Nov/Dec-2013/2012)

24. Given find using Taylor‟s series

method. (Nov/Dec-2013)

25. Use Runge-Kutta method fourth order to find the if ,

(Nov/Dec-2010)

ALL THE BEST