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COURSE FILE


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TIRUMALA ENGINEERING COLLEGE

BOGARAM-R.R.DIST.DEPARTMENT OF HUMANITIES&SCIENCES

COURSE FILE

BY

ASSOC.PROF. : P.SHANTAN KUMAR

M.Sc.(Maths).,M.Phil.,B.Ed.,D.Ph.,

SUBJECT : MATHEMATICAL METHODSBRANCH : ECE-A & B

YEAR : I-B.TECH - A.Y. 2009 2010


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CONTENTS

ACADEMIC CALENDER

SYLLABUS

TEACHING SCHEDULE

LESSON PLAN

LECTURE NOTES

ASSIGNMENTS(UNIT WISE)

IMPORTANT QUESTIONS (UNIT WISE)

JNTU PREVIOUS YEARS QUESTION PAPERS


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ACADEMIC CALENDER2009---2010

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY

HYDERABAD

I -Year B.Tech Common to all Branches

Orientation Programme 07-09-09 12-09-09(1w)

I-Unit of Instructions 14-09-09 21-11-09(10w)

I-Mid Exams 23-11-09 28-11-09(1w)

II-Unit of Instructions 30-11-09 06-02-10(10w)

II-Mid Exams 08-02-10 13-02-10(1w)

III-Unit of Instructions 15-02-10 24-04-10(10w)

III-Mid Exams 26-04-10 01-05-10(1w)

Preparation & Practical exams 03-05-10 15-05-10(2w)

End Exams 17-05-10 29-05-10(2w)

Summer vacation 31-05-10 03-07-10(5w)

05-07-10 II-year-I-sem , III-year-I-sem , IV-year-I-sem Class work start


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SYLLABUS


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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY

HYDERABAD

I-Year B.Tech Common to all Branches T: 3+1 P: 0 C: 6

MATHEMATICAL METHODS

UNIT-I SOUTION FOR LINEAR SYSTEMS

Matrces and linear system of equations:Elementary row transformations-Rank- Echelon

form,Normal form-Solution of Linear Systems-Direct Methods-LU decomposition from

Gauss Elimination-Solution of Tridiagonal Systems-Solution of Linear Systems.

UNIT-II EIGEN VALUES & EIGEN VECTORS

Eigen values,eigen vectors-properties-cayley Hamilton theorem-inverse and powers of a

matrix by cayley Hamilton theorem-diagonalization of matrix.Calculation of powers of

matrix-modal and spectral matrices.

UNIT-III LINEAR TRANSFORMATIONS

Real matrices-symmetric ,skew-symmetric,orthogonal,linear transformation-orthogonaltransformation.complexmatrices:hermitian,skew-hermitian and unitary-eigen

values,eigen vectors of complex matrices and their properties.quadratic forms-reductionof quadratic form to canonical form-rank-positive,negative definite-semi definite-index-

signature-sylvester law,singular value decomposition.

UNIT-IV SOLUTION OF NON-LINEAR SYSTEMS

Solution of algebraic and transdental equation:Introduction-The bisection method-the

method of false position-the iteration method-newton-raphson method.

Interpolation:Introduction-errorsin polynomial interpolation-finite differences-forward

differences-backward differences-central differences-symbolic relations and separation of

symbols-differences of polynomials-newtons formulae for interpolation-central

difference interpolation formulae-gauss central difference formula-interpolation withequally spaced points-lagranges interpolation formula,B.spline-interpolation-cubic

spline.


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UNIT-V CURVE FITTING & NUMERICAL INTEGRATION

Curve fitting:Fitting a straight line-Second degree curve-exponential curve-power curveby method of least squares.Numerical differentiation -simpsons-3/8th rule,Gauss

integration,Evaluation of principal value integrals,Generalized Quadrature.

UNIT-VI NUMERICAL SOLUTION OF IVPS IN ODENumerical solution of ordinary differential equations:Solution by Taylors series-Picards

method of successive approximation-Eulers method-Runga-kutte methods-predictors-

correctors methods-Adams-Bashforth method.

UNIT-VII FOURIER SERIESFourier series:Determination of Fourier coefficients-Fourier series-even and oddfunctions-Fourier series in arbitrary interval-even and odd periodic continuation-half

range-Fourier sine and cosine expansions.

UNIT-VIII PARTIAL DIFFERENTIAL EQUATIONSIntroduction & Formation of partial differential equations by elimination of arbitrary

constants and arbitrary functions-solutions of first order linear (legrange)equation andnon-linear (standard type)equations.Method of separation of variables for second order

equations Two dimensional wave equation.

TEXT BOOKS:

1. P.B.BHASKARA RAO,RAMA CHARY,BHUJANGA RAOB.S.P.PUBLICATION

2. SURYANARAYANA RAO.SCITECH PULICATION

REFERENCES BOOKS:1. S.chand2. Grewal3. Himalaya publication4. Kreyszig5. Numerical analysis by s.s. sastry6. G.shanker rao by I.K.International publication


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Dept.of Humanities & Sciences

Teaching Schedule

Name of the faculty: P.Shantan kumar A.Y.:2009-10

Subject to be handled: MM Total No.of hours required:

Class: I-B.Tech Total No.of hours available:

Unit Topic to be covered No. of hours

required

Teaching aids

required if

any

Reference

books/materials

I Types of Matrices 1

Elementary Transforms 1

Types of Ranks 3

Linear Equations 3

LU & Tridiagonal

Methods

3

II Eigen values 1

Eigen vectors 1

Properties 3

Cayley Hamilton Method 3

Diagonalization Method 5


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III Real Matrices 2

Orthogonal Transforms 2

Complex Matrices 2

Properties 2

Quadratic forms 5

IV Solns in Algebraic &

Transdental eqns

4

Interpolation Methods 2

Newton & Gauss Methods 4

Legranges Method 5

V Curve Fitting Methods 5

Numerical D.E. Methods 3

Numerical Integration onDifferent Methods 2

Problems 2

VI Introduction 1

Numerical Solns of 2


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LESSON PLAN


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Dept.of Humanities & Sciences

LESSON PLANName of the faculty: P.Shantan kumar A.Y.:2009-10

Subject to be handled: MM Total No.of hours required:

Class: I-B.Tech Total No.of hours available:

S.No. Date Topic to be covered No. ofPeriods

Remarks

1 types of matrices 12 elementary

tranformations1

3Rank of a matrix

14 Rank of a matrixproblems

1

5 Rank of a matrixproblems

1

6 Homogeneous ens 17 Non-hom eqns 18 LU- decomposition 19 Tri diagonal systems 110 previous qn.paper prob 1

11 slip test 112 eigen values,eigen

vectors def1

13 Properties of eigen values 114 Problems of eigen values 115 Properties of eigen

vectors1

16 Problems of eigen vectors 117 caley-hamilton theorem 1

18 prob on ch-theorem 119 Digonalization 1

20 problems onDiagonalization

1


1


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1

23 previous qn.paper prob 1

24 slip test 1

25 Def of real matrices 1

26 Linear Transformation 1

27 orthogonaltransformation

1

28 def of complx matrices 1

29 properties of complexmatrices

1

30 properties of complexmatrices

1

31 reduction of qf tocanonical form

1

32 finding natures of qf 1

33 Singular valuedecomposition

1


35 Bisection methonds 1

36 False method &problems

1

37 Iteration method 1

38 Newton Raphsonmethod

1

39 Finite differences 1

40 forward,backward,centraldifferences

1


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41 Newton forward method 1

42

Newton forward method

1

43 Newton backwardmethod

1

44 Gauss forward 1

45 Gauss Backward 1

46 Central & sterlingsformula

1

47 Legranges interpolationformula

1

48 B.spline interpolation 149 Cubic s line 150 previous qn.paper prob 1

51 slip test 1

52 least squares & fitting ast.line

1

53 parabola 1

54 exoponential curves 1

55 power curve 1

56 problems on curve fitting 1

57 Derivatives usingforward method

1

58 Derivatives usingbackward method

1

59 simpson's 3/8 rule 1

60 Gaussian integration 161 Eval. of principal value

integrals

1

62 Generalized Quadrature 1


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64

slip test

1

65 Taylor's series 1

66 Picard's method 1

67 Eulers method 1

68 RK - method 1

69 RK - method 1

70 Pridictor-correctormethod

1

71 Adam's moulton'smethod

1


73 slip test 1

74 Fourier series & coeff 1

75 Even odd Neither fns 1

76 problems 1

77 problems 1

78problems 1

79Half range series 1

80problem 1

81previous qn.paper prob 1

82slip test 1

83Introduction on P.D.E. 1

84Formation of pde inorbitary const

1


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85 Formation of pde inorbitary functions 1

86Solns. Of first orderL.eqns

1

87Solns. Of first orderL.eqns

1

88Solns. Of first order nonLlinear.eqns

1

89type-1 1

90type-2 1

91type-3 1

92type-4 1

93Method of separation ofvariables

1

94Two dimensional wave eqns 1

95previous qn.paper prob 1

96slip test 1

97Grand test 1

98REVISION 1

99REVISION 1

Signature of the faculty Signature of the H.O.D.


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ASSIGNMENTS(UNIT WISE)


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I-B.TECH - MATHEMATICAL METHODS

ASSIGNMENT ON UNIT-I(COMMON TO ECE -A & B)

NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

1.Solve the system 2x-y+3z = 0 , 3x+2y+z = 0 , x-4y+5z = 0.

2.Show that the system of equations 3x+3y+2z = 1, x+2y = 4, 10y+3z = -2, 2x-3y-z = 5

is consistent and solve it.

3.i.Show that the system of equations x-4y+7z = 14, 3x+8y-2z = 13, 7x-8y+26z = 5 arenot consistent.

ii.Find the rank of for which the system of equations3x-y+4z = 3, x+2y-3z=-2,6x+5y + z = - 3 will have infinite number of solutions &solve with that value.

4.i.Solve by matrix method the given equations 3x+y+2z = 3, 2x-3y-z = -3, x+2y+z = 4.

ii.Find the non-singular matrices P & Q such that the normal form of A is PAQ , where

1 3 6 -1

A = 1 4 5 1 .Hence find its rank.

1 5 4 3

5.Find the rank of a matrix 2 -4 3 -1 0

1 -2 -1 -4 20 1 -1 3 1

4 -7 4 -4 5


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ASSIGNMENT ON UNIT-II(COMMON TO ECE -A & B)


1. Find the eigen values & eigen vectors of 8 - 6 2

- 6 7 - 4

2 - 4 3

2. Verify cayley-hamilton theorem for the matrix 1 0 20 2 1 . Hence find A-1

2 0 3

3.i. Find the eigen values & eigen vectors of 1 0 - 20 0 0

- 2 0 4

ii. Diagonalize the matrix 8 - 8 - 2

4 -3 -23 - 4 1

4. Find the eigen values & eigen vectors of 5 -2 0

-2 6 20 2 7

5. Show that the matrix 1 -2 2

1 2 3 satisfies its characterstic equation.Hence find A-1

0 -1 2 & A4


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ASSIGNMENT ON UNIT-III(COMMON TO ECE -A & B)


1. 8x2+7y2+3z2-12xy-8yz+4zx into sum of squares by an orthogonal method &

find nature.

2. 3x2-2y

2-z

2+12yz+8zx-4xy into canonical form by an orthogonal method &

find nature.

3. 2x2+2y2+2z2-2xy-2yz-2zx into canonical form by an orthogonal method.

4.i. Define hermitian , skew-hermitian , unitary , orthogonal matrices .

ii. Show that the eigen values of an unitary matrices is of unit modulus.

5.i. Show that A = i 0 0

0 0 i is a skew-hermitian matrix and also unitary.0 i 0 find eigen values and corresponding eigen

vectors of A

ii. Prove that the inverse of an orthogonal matrix is orthogonal and its transpose isalso orthogonal.


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ASSIGNMENT ON UNIT-IV(COMMON TO ECE -A & B)


1. i. Find a positive root of x4-x3-2x2-6x-4 = 0 by bisection method.

ii. Find a positive root of xlogx 1.2 = 0 by Regula false method.

2. i. Find a positive root of x3-6x-4 = 0 by bisection method.

ii. Find a positive root of x3-x-2 = 0 by Newton Raphson method.

3. i. Solve x3= 2x+5 for a positive root by iteration method.

ii. Using Newton-Raphson method,find a positive root of cosx-xex = 0

4. i. If the interval of differencing is unity, P.T. [2x /x !] = [2x(1-x)] / (x+1)!

ii. Find the parabola passing through the points (0,1) , (1,3) ,(3,55) using

Lagranges Interpolation Formula.

5. i. Using Lagranges Interpolation ,Find y(10) from X : 5 6 9 1

Y : 12 13 14 16

ii. If the interval of differencing is unity ,

P.T. [x(x+1)(x+)(x+3)] = 4(x+1)(x+2)(x+3)


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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS

ASSIGNMENT ON UNIT-V(COMMON TO ECE -A & B)


1. i. Find by the method of least squares the straight line that best fits the following

data: x: 0 5 10 15 20y: 7 -11 16 20 26

ii. Find a second degree parabola to the following data: x : 0 1 2 3 4v : 1 1.8 1.3 2.5 6.3

2. i. Find a curve y = aebx to the data: x : 0 2 4

y : 5.1 10 31.1

ii. Using the table below , find f1(0) and f(x) dxx : 0 2 3 4 7 9

f(x) : 4 26 58 110 460 920

3. i. Using simpsons 3/8th

rule ,evaluate dx/(1+x2) by dividing the range into 6 equal

parts in between 0 to 6

ii. Evaluate e-x2 dx by dividing the range of integration into 4 equal parts in between0 to 1 using (a). Trapezoidal rule (b). Simpsons 1/3

rdrule

4.i. Find the curve of best fit of the type y = aebx

to the following data by the method of

least squares x : 1 5 7 9 12y : 10 15 12 15 21

ii. Evaluate dx / (1+x2) by taking h = 1/6 using a) Simpsons 1/3rd ruleb) Simpsons 3/8th rule

5.i. Fit a parabola y = a + bx + cx2 to the following data

x : 1 2 3 4 5 6 7

y : 2.3 5.2 9.7 16.5 29.4 29.4 35.5

ii. Evaluate dx / (1+x2) by taking h = .5 , .25 , .125 using Trapezoidal rule inbetween 0 to 1


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ASSIGNMENT ON UNIT-VI(COMMON TO ECE -A & B)

NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H &S)

1.i. Solve dy/dx = xy using R-K-method for x = .2 given y(0) = 1 , y1(0)=0 taking h = .2

ii. Use Eulers method to find y(.1) , y(.2) given y1 = (x 3+xy2)e-x , y(0) = 1

2.i. Obtain y(.1) given y1 = (y-x)/(y+x) , y(0) =1 by Picards method.

ii. find y(.1) , y(.2) &y(.3) using Taylors series method that dy/dx = l - y , y(0) = 0

3.i.Apply R-K- 4th order method to find y(.2),y(.4) and y(.6) , y1 = -xy2 , y(0) = 2using h = .2

ii. Tabulate the value of y(.2),y(.4),y(.6) ,y(.8) & y(1) using Eulers method given that

dy/dx = x2-y ,y(0)=1

4.i. Find y(.1),y(.2) using Taylors series method given that dy/dx = x2-y,y(0) = 1

ii. Tabulate the values of y at x = .1 to .3 , using Eulers Modified method given thatx+y = dy/dx & y(0)

5.i. Given y1

= x+siny , y(0) =1 compute y(.2),y(.4) with h = .2 using Eulers Modifiedmethod.

ii. Find the solution of dy/dx = x-y at x = .4 subject to the condition y = 1 at x= 0 and

h = .1 using Milnes method. Use Eulers Modified method to evaluate y(.1),y(.2) &

y(.3)


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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS

ASSIGNMENT ON UNIT-VII(COMMON TO ECE -A & B)


1.i. Obtain the fourier series expansion of f(x) given that f(x) = kx ( - x ) in0 < x < 2 where k is a constant.

ii. Obtain sine series f(x) = x x2 , 0 < x <

2.i. If f(x) = k x , 0 < x < /2k( - x) , /2 < x 1

Hence evaluate { (x cosx sinx) / x2} cosx/2 dx.


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ASSIGNMENT ON UNIT-VIII(COMMON TO ECE -A & B)


1.i. Solve (x2-yz) p + (y2-zx) q = z2 xy

ii. Find the z- transform of the sequence {x(n)} , where x(n) is i. n.2n ii. An2+bn+c

2.i. Form the P.D.E. i. z = f(x2+y

2) ii. Z= y f(x) + x g(y)

ii. Z-1 [ (z2-3z)/(z+2)(z-5) ]

3.i. Solve the P.D.E. x2p2 + y2q2 = 1

ii. Solve the D.E. use z-transform y(n+2) +3y(n+1) +2y(n) = 0 given that y(0) =0 ,

y(1) = 1

4.i. Solve (x+y)p +(y+z)q = z+x

ii. Form the P.D.E. by eliminating the arbitrary constants a,b fromz = ax + by + a/b - b

iii. Find z-1[ z / (z2+11z+24)]

5.i. Solve the P.D.E. x2(z-y)p +y2(x-z) q = z2 (y-x)

ii. Form the P.D.E. by eliminating arbitrary functions z = f(y) + g (x+y)

iii. Solve the P.D.E. z4p2 + z4 q2 = x2y2


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IMPORTANT QUESTIONS

(UNIT WISE)


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Code No. 07A1BS06 UNIT-3


1. Show that any square matrix can be writher as sum of a symmetric matrix and askew-symmetric matrix.

Express8 6

10 2

as a sum of a symmetric matrix and a skew-symmetric

matrix.

2. Verify wheter the matrix A =

1 1 1 1

1 1 1 11

1 1 1 12

1 1 1 1

A

=

is orthogonal.

3. Define orthogonal matrix.

Verify whether the matrix

8 4 11

1 4 89

4 7 4

A

=

is orthogonal

4. If A is any square matrix, prove that A+A*, AA*.A*A are all Hermition and A-

A* is skew Hermition.

5. Show that the complex matrixa ic b i

Ab id a i

+ + =

+ is unitary if a

2b

2+c

2+d

2= 1

6. Show that the complex matrix2 3 4

3 4 2

iA

i

+ =

is Hermition.

Find the eigen values and eigenvectors.

7. Show that the eigen values of a skew Hermition matrix are purely imaginary orreal.

8. Define an orthogonal matrix.

1 2 21

2 1 23

2 2 1

A

=

is orthogonal.

9. Find the eigen values and eigen vectors of the unitary matrix A=

1

2 2

1

2 2

i

Ai

=


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10. Find the eigen values and the eigen vectors of the complex matrix

3 2 1

2

iB

i i

+ =

+

11. Reduce the Quadratic form 2x1x2+2x2x3+2x3x1 into conomonical form andclassify the quadratic form.

12. Reduce 3x2+3z2+8xz+8yz into canonical form. Give the rank, index and signatureof the Quadratic form.

13. Reduce the Quadratic form 2x2+2y2+3x2+2xy-4yz-4xz to conomical form. Findthe rank,index and signature.

14. Determine the nature, index and signature of the Quadratic form2x

2+2y

2+3z

2+2xy-4xz-4yz.

15. Show that the linear transformationY1=2x1+x2+x3 ; y2= x1+x2+2x3; y3=x1 2x3 is regular. Write down the inverse

transformation.

16. Find the nature, index and signature of the Quadratic form2 2 2

1 2 3 1 2 2 3 3 15 2 , 2 6x x x x x x x x x+ + + + + .

17. Find the nature, index and signature of the Quadratic form2 2 2

1 2 3 1 2 2 3 3 13 3 7 6 6 6x x x x x x x x x .

18. Reduce the Quadratic form 2 21 1 2 2

2( ) x x x x+ + to canonical form.

19. Reduce the Quadratic form 5x26xy+5y

2to sum of squares.

20. Reduce the Quadratic form 2 21 1 2 26 16 6 x x x x+ to sum of squares.

-oOo-


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1. Find a root of the equation x3-4x-9 = 0 using bisection method correct to threedecimal places.

2. Find a root of the equation x3-2x2-4 = 0 using bisection method correct to threedecimal places.

3. Find a real root equation f(x) = x2+x-3 = 0 correct to three decimal places usingBisection method.

4. Find a real root of the equation cosx = 3x-1, correct to three decimal places usingthe method of false position.

5. Find a real root of the equation x3-8x-40 = 0 in [4,5] correct to three decimalplaces using the method of false position.

6. Using Regular falsi method; compute the real root of the equation x e x = 1 in [0,1]correct to three decimal places.

7. Find a real root of the equation x3+x2-1 = 0 by using interative method, correct tothree decimal places.

8. Find by the method of interation a real root of the equation x = .21 sin(0.5+x)starting with x = 0.12 xorrect to three decimal places.

9. Using Newton Raphson method compute the root of equation x sin x + cos x = 0

which lies between ,2

, correct to three decimal places.

10. Find the double root of the equation x3

-3x+2 = 0 starting with x0 =1.2 by Newton Raphson method.

11. Following table gives the weights in pounds of 190 high school students.

Weight 30-40 40-50 50-60 60-70 70-80

(in pounds)

No.of students 31 42 51 35 31

Estimate the number of students whose weights are between 4 and 45.

12. Obtain the relations between the operators.1

1 1 1 12 2 2 2

( ) 1 ( ) 11

( ) ( ) ( )2

i E ii E

iii E E iv E E

= =

= = +

13. Estimate f(22) from the following data with the help of an appropriateinterpolation formula.

X: 20 25 30 35 40 45

F(x): 354 332 291 260 231 204


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14. Estimate y(3) from the following data, using an appropriate interpolation formula.

X: 2 4 6 8 10

Y: -14 22 154 430 898

15. Using an interpolation formula estimate y(4.1) from the following data.

X: 0 1 2 3 4

Y: 1 1.5 2.2 3.1 4.616. Given that f(45) = 0.7071,f(50) = 0.6427, f(55) =0.5735,f(0) = 0.5,f(65) = 0.4226,

find f(63) using Newtons Backward interpolation formula.

17. Use stirlings formula to find y(35), given that y(20) = 512,y(30) = 439, y(40)

=346, y(50)= 243.

18. Given that y(20) = 24, y(24) = 32, y(28) = 35, y(32) = 40. Find y25 central

interpolation formula.

19. The following table gives the viscosity of a lubricant as a function of temperature.

Temperature : 100 120 150 170

Viscosity 10.2 .7.9 5.1 4.4

Apply Lagranges formula to estimate viscosity of the lubricant at 130 degrees of

temperature.

20. Apply Lagranges formula to estimate y (3) from the following deta

X: 0 1 2 4Y: 2 3 12 78.

-oOo-


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1. Find the stright line of the form y = a + bx that best bits the following data, by

method of least sequences.

X: 1 2 3 4 5Y: 12 25 40 50 65.

Estimate y (2,5).

2. Find a second degree parabola y = a + bx + cx2

to the given deta, by method of

least sequences.

X : 1 3 5 7 9

Y : 2 7 10 11 9

3. In an experiment the measurement of electric resistance R of a meal at various

temperatures t0c lirted as.T : 20 24 30 35 42R: 85 82 80 79 76

Fit a relation of the form R = a +bt, by method of least sequences.

4. Fit a second degree parabola of the form y =a + bx +cx2

to the following data.

X : 0 1 2 3 4

Y : 1 1.8 1.3 2.5 6.3.

5. Fit the following deta to an exponential curve of the form y = aebx

.

X : 1 3 5 7 9Y : 100 81 73 54 43

6. For the deta given below find a best flitting curve of the form y = axb.X : 1 2 3 4 5

Y : 2.98 4.26 5.21 6.10 6.8

7. What is least squares principle ?

Fit a stright line y = a + bx to the following deta.X : 0 1 3 6 8

Y : 1 3 2 5 48. Find the best fitting exponential curve y = aebx to the following deta.

X : 2 3 4 5 6

Y: 3.72 5.81 7.42 8.91 9.68

9. Fit a parabola y = ax2 +bx + c which best bits with the observations.

X : 2 4 6 8 10Y: 3.07 12.85 31.47 57.38 91.29.

10. Fit a least sequence curve y =axb to the following detaX : 1 2 3 4 5

Y: 0.5 2 4.5 8 12.5

11. Evaluate1

0

(1 sin 4 )xe x dx

+ . Taking h =1 /4 by (i) Trapezoidal (b) simpsous

1

3rule.


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12. Find first and second derivation of the function tabulated below, as the point

x =1.X : 0.0 0.1 0.2 0.3 0.4

Y: 1.0000 0.9975 0.9900 0.9776 0.9604

13. Find first and second derivations of the function telruleted below, at the point

x =1.

X: 0 1 2 3 4

Y: 6.98 7.40 7.78 8.12 8.45.

14. Explain how the thepezridel rule is obtained from Newton cotes general

quedreture formula.

15. Given the following table of values of x and y, first, first and second

derivatives at x = 1.25X : 1.10 1.15 1.20 1.25 1.30.

Y : 1.05 1.07 1.09 1.12 1.14

16. Evaluate6

2

01

dx

x+using Simpsons

3

8the rule.

17. Find4

0

xe dx by simpson rule of numerical integration.

18. Find the first and second derivatives. Of the function tabulated below at the

point 1.5.

X : 1 2 3 4 5

F(x) 8 15 7 6 2

19. Evaluate5.2

4

logxdx using (i) simpsous1

3rule (ii)simpsous

3

8

X 4.0 4.2 4.4 4.6 4.8 5.0 5.2

Logx 1.38 1.44 1.48 1.53 1.57 1.61 1.65

20. Evaluate2

sino

xdx

by symposiums1

3rule, using 11ordinates and compare with

actual value of the integral.

-oOo-


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Code No. 07A1BS06 UNIT-VI


1. Using taylors series method, solve the equation 2 2dy

x ydx

= + for x = .4, given

that y=0 when x=0

2. Using taylor series method, find an approximate value of y at x=0.2 for the

differential equation 2 3 x y y e = , y(0)=0

3. Solve 2 , y x y = y(0)=1 using Taylors series method and y(0.1), y(0.2) correct

to 4 decimal places

4. Give the differential equ 2 2 y x y = + , y(0)=1 obtain y(0.25) and y(0.5) by

Taylors series method

5. Solvedy

dx-1=xy and y(0)=1 using Taylors series method and compute y(0.1)

6. Solve 2 , (0) 1 y x y y = = using Taylors series method Tabulate for x=0.1, 0.2

7. Given 2 1, (0) 1dy

x y ydx

= = . Compute y(0.1) by Taylors series method

8. Find the value of y for x=0.4 by picards method given that 2 2dy

x ydx

= + , y(0)=0

9. Solve 2dy

x ydx

= , y(1)=3 by picards method

10. Solve 2dy

x y

dx

= , y(0)=1 and Compute y(0.1) Correct to four decimal places by

picards method

11. Given 2 2dy

x ydx

= + , y(0)=0. find y(0.2) and y(1) by picords method

12. Solve 1 2dy

xydx

= + , y(0)=0 by picards method

13. Find the solution of 2 y x y = + , y(0)=1 by Picords method

14. Solve y x y = + , y(0)=1 by eulers method

15. Given , (0) 0xy y e y = + = . Find y(0.2) by Eulers modified method

16. Solve 2dy

x ydx

= + , y(0)=1 by modified Eulers method

17. Find the solutiondy

x ydx

= , y(0)=1 of x=0.1

18. Given thatdy

xydx

= , y(0)=1, Find y(0.1) using Eulers method

19. Solve by Eulers method2y

yx

= given y(1)=2 and find y(2)


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20. Find y(1.2) by modified Eulers method given log( ) y x y = + , y(0)=2 taken h =

0.2

21. Explain First order Range-kutta method.

22. Explain Second order Range-kutta method23. Explain Third order Rannge-kutta method

24. Explain fourth order Range-kutta method

25. Using Range-kutta method of second order, compute y(2.5) fromdy x y

dx x

+= ,

y(2)=2

Taking h=0.25

26. Use Milnes method to find y (0.4) from2

y xy y = + , y(0)=1

27. Find y(0.1) and y(0.2) using 2 y xy y = + , y(0)=1

28. Calculate y(0.6) by milnes predictor-corrector method given y x y = + , y(0)=1

with h=0.2

29. Given1

2

dy

dx= xy and y(0)=1, y(0.1)=1.0025, y(0.2)=1.0101, y(0.3)=1.0228,

Compute y(0.4) by Adams-Bashforth method

30. Use Adam-Bashforth-mpulton method to find initial value y(1.1) from2 2 , (1) 1

dyx y x y

dx = =

-oOo-


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Code No. 07A1BS06 UNIT-VII


1. Find Fourier a0 and an when f(x)=x2 is (0, 2 )

2. find the Fourier series of f(x)= 2 21

(3 6 2 ) (0,2 )12

x x in +

3. Find a0, bn for f(x)=ex

from x=0 to x=2 4. Find the Fourier series for f(x)=x, 0


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23. f(x)=2

1 , 0 ?x

x even

+

21 ,0 ?

x x even

If so find the Fourier series for the function.

24. Expand the function f(x)=x3 as a Fourier series in the interval x < < 25. Find the Fourier series for f(x)= x cosx, x < <

26. Find the half range sine series for f(x)= ( ) 0 x x in x < <

27. Obtain the half range sine series for ex

in (0, )

28. find the Fourier series to represents (1-x2) in 1 1x

29. find the Fourier series of f(x)= 0 22

xin x

< <

30. Find the half-Range cosine series expansion of f(x)=x in [0, 2]

-oOo-


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Code No. 07A1BS06 UNIT-VIII


1. Form the partial differential equation fora

z ax by bb= + + by eliminating a and

b2. Eliminate h, k from (x-h)2+(y-k)2+z2=a2

3. Form a partial differential equation by eliminating a, b, c from2 2 2

2 221

x y zb ca

+ + =

4. Find the differential en of all spheres of radius 5 having their centres in the xy

plane

5. Form the partial differential equation from z=axey+ 2 2

1

2

ya e b+ when a, b are

parameters

6. Form the partial differential equ by eliminating a and b from z=a log( 1)

1

b y

x

7. Form the partial differential equ from z=(x-a)2+(y-b)

2+1 where a, b are

parameters

8. Form the differential equation by eliminating a and b from log (az-1)=x+ay+b

9. Find the differential equation of all planes passing through the origin

10. Form the differential equation of all planes having equal intercepts on x and y axis

11. Form a partial differential equ by eliminating the arbitrary functions from z =

f(x2-y

2)

12. Eliminate the arbitrary function 2, and from z= 1 2( ) ( ) x iy x iy + +

13. Find the differential equ from 2 2 2( , ) 0x y z x y z + + + + =

14. Form the partial differential eqn by eliminating the arbitrary function f from

z=(x+y) f (x2-y

2)

15. Form the partial differential equation by eliminating the arbitrary function f from

Z = eax+by

f (ax-by)

16. Form the differential equation by eliminating the arbitrary function f from

xyz=f(x2+y

2+z

2)

17. Form the partial differential equation by eliminating the arbitrary function f from

f (x2+y

2, x

2-z

2)=0

18. Form the partial differential equation by eliminating the arbitrary in f from z

=xy+f(x2+y2)19. For the partial differential equation by eliminating the arbitrary function

2 2 2 2( , 2 ) 0 from x y z z xy + + =

20. Find the general solution of p+q=1

21. Solve px+Qy=z

22. Solve p Tan x+q Tan y=Tanz

23. Find the general solution of y2zp+x

2zq=y

2x

24. Solve (y-z)p+(x-y)q=z-x

25. Solve x(y-z)p+y(z-x)q=z(x-y)

26. Solve p+3q=5z+Tan (y-3x)


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27. Find the integral surface of

X(y2+z)p-y(x

2+z)q=(x

2-y

2)z

28. Solve 2 2 ( )

dx dy dz

x y z x y= = +

29. Solve p2+q2=npq

30. Solve z=p2+q2

-oOo-


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ALL THE BEST


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JNTU

I-MID-DESCRIPTIVE

&

OBJECTIVE TYPEQUESTIONS

ON UNIT I&II

course file mm

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