mm course file -2010-2011

COURSE FILE

TIRUMALA ENGINEERING COLLEGEBOGARAM-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 METHODS BRANCH : common to all branches YEAR : I-B.TECH - A.Y. 2010 – 2011

CONTENTS

ACADEMIC CALENDER

SYLLABUS

TEACHING SCHEDULE

LESSON PLAN

LECTURE NOTES

ASSIGNMENTS(UNIT WISE)

IMPORTANT QUESTIONS (UNIT WISE)

JNTU PREVIOUS YEARS QUESTION PAPERS

ACADEMIC CALENDER 2010---2011

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITYHYDERABAD

I -Year B.Tech Common to all Branches

Orientation Programme 04-10-10 08-10-10(1w)

I-Unit of Instructions 11-10-10 18-12-10(10w)

I-Mid Exams 20-12-10 23-12-10(4days)

II-Unit of Instructions 24-12-10 05-03-11(10w)

II-Mid Exams 07-03-11 10-03-11(4days)

III-Unit of Instructions 11-03-11 14-05-11(10w)

III-Mid Exams 16-05-11 19-05-11(4days)

Preparation & Practical exams 20-05-11 28-05-11(9days)

End Exams 30-05-11 11-06-11(2w)

Summer vacation 13-06-11 02-07-11(3weeks)

04-07-11 II-year-I-sem , III-year-I-sem , IV-year-I-sem Class work start

SYLLABUS

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITYHYDERABAD

I-Year B.Tech Common to all Branches T: 3+1 P: 0 C: 6MATHEMATICAL 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-orthogonal transformation.complexmatrices:hermitian,skew-hermitian and unitary-eigen values,eigen vectors of complex matrices and their properties.quadratic forms-reduction of 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-newton’s formulae for interpolation-central difference interpolation formulae-gauss central difference formula-interpolation with equally spaced points-lagrange’s interpolation formula,B.spline-interpolation-cubic spline.

UNIT-V CURVE FITTING & NUMERICAL INTEGRATION

Curve fitting:Fitting a straight line-Second degree curve-exponential curve-power curve by method of least squares.Numerical differentiation -simpson’s-3/8th rule,Gauss integration,Evaluation of principal value integrals,Generalized Quadrature.

UNIT-VI NUMERICAL SOLUTION OF IVP’S IN ODENumerical solution of ordinary differential equations:Solution by Taylor’s series-Picards method of successive approximation-Euler’s method-Runga-kutte methods-predictor’s-corrector’s methods-Adam’s-Bashforth method.

UNIT-VII FOURIER SERIESFourier series:Determination of Fourier coefficients-Fourier series-even and odd functions-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 and non-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 RAO… B.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

TEACHING SCHEDULE

TIRUMALA ENGINEERING COLLEGEDept.of Humanities & Sciences

Teaching ScheduleName of the faculty: P.Shantan kumar A.Y.:2010-11Subject 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 B.S.P.PUBLICATION

Elementary Transforms 1 B.S.P.PUBLICATION

Types of Ranks 3 B.S.P.PUBLICATION

Linear Equations 3 B.S.P.PUBLICATION

LU & Tridiagonal Methods

3 B.S.P.PUBLICATION

II Eigen values 1 SCITECH PULICATION

Eigen vectors 1 SCITECH PULICATION

Properties 3 SCITECH PULICATION

Cayley Hamilton Method 3 SCITECH PULICATION

Diagonalization Method 5 SCITECH PULICATION

III Real Matrices 2 B.S.P.PUBLICATION

Orthogonal Transforms 2 B.S.P.PUBLICATION

Complex Matrices 2 B.S.P.PUBLICATION

Properties 2 B.S.P.PUBLICATION

Quadratic forms 5 B.S.P.PUBLICATION

IV Solns in Algebraic & Transdental eqns

4 S.chand

Interpolation Methods 2 S.chand

Newton & Gauss Methods 4 S.chand

Legranges Method 5 S.chand

V Curve Fitting Methods 5Himalaya publication

Numerical D.E. Methods 3Himalaya publication

Numerical Integration on Different Methods

2Himalaya publication

Problems 2Himalaya publication

VI Introduction 1Himalaya publication

Numerical Solns of 2Himalaya publication

O.D.E. Himalaya publication

Taylor’s, Picard’s,Euler’s Methods

3 Himalaya publication

R-k, Predictor-Corrector,Adams & Bashforth Methods

3 Himalaya publication

VII Fouries Series Coeficients 1 Grewal

Even , Odd & Neither fns Methods

3 Grewal

Fourier series in arbitrary interval

2 Grewal

Half Range Series Problems

3 Grewal

VIII Formation of P.D.E. 4 B.S.P.PUBLICATION

Solns of P.D.E. in Linear eqns

5 B.S.P.PUBLICATION

Solns of P.D.E. in non-linear eqns

3 B.S.P.PUBLICATION

Two Dimential Wave Eqns

2 B.S.P.PUBLICATION

Grand Total

97

LESSON PLAN

TIRUMALA ENGINEERING COLLEGEDept.of Humanities & Sciences

LESSON PLANName of the faculty: P.Shantan kumar A.Y.:2010-11Subject to be handled: MM Total No.of hours required: 99Class: I-B.Tech Total No.of hours available:99

S.No. Date Topic to be covered No. of Periods

Remarks

1 11-10-10 types of matrices 12 12-10-10 elementary

tranformations1

3 13-10-10 Rank of a matrix 14 14-10-10 Rank of a matrix

problems1

5 18-10-10 Rank of a matrix problems

1

6 19-10-10 Homogeneous ens 17 20-10-10 Non-hom eqns 18 21-10-10 LU- decomposition 19 25-10-10 Tri diagonal systems 110 26-10-10 previous qn.paper prob 1

11 27-10-10 slip test 112 28-10-10 eigen values,eigen

vectors def1

13 01-11-10 Properties of eigen values 114 02-11-10 Problems of eigen values 115 03-11-10 Properties of eigen

vectors1

16 04-11-10 Problems of eigen vectors 117 08-11-10 caley-hamilton theorem 1

18 09-11-10 prob on ch-theorem 119 10-11-10 Digonalization 1

20 11-11-10 problems on Diagonalization

1


1


1

23 17-11-10 previous qn.paper prob 1

24 18-11-10 slip test 1

25 22-11-10 Def of real matrices 1

26 23-11-10 Linear Transformation 1

27 24-11-10 orthogonal transformation

1

28 25-11-10 def of complx matrices 1

29 29-11-10 properties of complex matrices

1

30 30-11-10 properties of complex matrices

1

31 01-12-10 reduction of qf to canonical form

1

32 02-12-10 finding natures of qf 1

33 06-12-10 Singular value decomposition

1


35 08-12-10 Bisection methonds 1

36 09-12-10 False method & problems

1

37 13-12-10 Iteration method 1

38 14-12-10 Newton Raphson method

1

39 27-12-10 Finite differences 1

40 28-12-10 forward,backward,central differences

1

41 29-12-10 Newton forward method 1

42 30-12-10 Newton forward method 1

43 03-01-11 Newton backward method

1

44 04-01-11 Gauss forward 1

45 05-01-11 Gauss Backward 1

46 19-01-11 Central & sterlings formula

1

47 20-01-11 Legranges interpolation formula

1

48 24-01-11 B.spline interpolation 149 27-01-11 Cubic spline 150 01-02-11 previous qn.paper prob 1

51 02-02-11 slip test 1

52 03-02-11 least squares & fitting a st.line

1

53 07-02-11 Parabola 1

54 09-02-11 exoponential curves 1

55 10-02-11 power curve 1

56 14-02-11 problems on curve fitting 1

57 15-02-11 Derivatives using forward method

1

58 16-02-11 Derivatives using backward method

1

59 17-02-11 simpson's1/ 3rd rule 1

60 18-02-11 trapezoidal rule 161 21-02-11 simpson's 3/8 rule 1

62 22-02-11 Gaussian integration 163 23-02-11 Eval. of principal value

integrals

1

64 24-02-11 Generalized Quadrature 165 28-02-11 previous qn.paper prob 1

66 01-03-11 slip test 1

67 02-03-11 preparation problems 168 15-03-11 Taylor's series 1

69 16-03-11 Picard's method 1

70 21-03-11 Eulers method 1

71 22-03-11 RK – method 1

72 23-03-11 RK – method 1

73 24-03-11 Pridictor-corrector method

1

74 28-03-11 Adam's moulton's method

1


76 30-03-11 slip test 1

77 31-03-11 Fourier series & coeff 1

78 06-04-11 Even odd Neither fns 1

79 07-04-11 problems 1

80 11-04-11 Half range series 1

81 13-04-11 Problem 1


83 20-04-11 Introduction on P.D.E. 1

84 21-04-11 Formation of pde in orbitary const

1

85 25-04-11 Formation of pde in orbitary functions

1

86 26-04-11 Solns. Of first order L.eqns

1

87 27-04-11 Solns. Of first order L.eqns

1

88 28-04-11 Solns. Of first order non Llinear.eqns

1

89 02-05-11 type-1 1

90 03-05-11 type-2 1

91 04-05-11 type-3 1

92 05-05-11 type-4 1

93 09-05-11 Method of separation of variables

1

94 10-05-11 Two dimensional wave eqns 1


96 12-05-11 slip test 1

97 13-05-11 Grand test 1

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

ASSIGNMENTS(UNIT WISE)

TIRUMALA ENGINEERING COLLEGEI-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 are not consistent.

ii.Find the rank of λ for which the system of equations 3x-y+4z = 3, x+2y-3z=-2,6x+5y + λ 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 2 0 1 -1 3 1 4 -7 4 -4 5


ASSIGNMENT ON UNIT-II(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

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 2 0 2 1 . Hence find A-1

2 0 3

3.i. Find the eigen values & eigen vectors of 1 0 - 2 0 0 0 - 2 0 4

ii. Diagonalize the matrix 8 - 8 - 2 4 -3 -2 3 - 4 1

4. Find the eigen values & eigen vectors of 5 -2 0 -2 6 2 0 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


ASSIGNMENT ON UNIT-III(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

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

2. 3x2-2y2-z2+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 is also orthogonal.


ASSIGNMENT ON UNIT-IV(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

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 Lagrange’s Interpolation Formula.

5. i. Using Lagrange’s 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)


ASSIGNMENT ON UNIT-V(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

1. i. Find by the method of least squares the straight line that best fits the following data: x: 0 5 10 15 20 y: 7 -11 16 20 26 ii. Find a second degree parabola to the following data: x : 0 1 2 3 4 v : 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 f 1(0) and ∫ f(x) dx x : 0 2 3 4 7 9 f(x) : 4 26 58 110 460 920

3. i. Using simpson’s 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 between 0 to 1 using (a). Trapezoidal rule (b). Simpson’s 1/3rd rule

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 12 y : 10 15 12 15 21

ii. Evaluate ∫ dx / (1+x2) by taking h = 1/6 using a) Simpson’s 1/3rd rule b) Simpson’s 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 in between 0 to 1


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 Euler’s method to find y(.1) , y(.2) given y1 = (x3+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 Taylor’s 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) = 2 using h = .2

ii. Tabulate the value of y(.2),y(.4),y(.6) ,y(.8) & y(1) using Euler’s method given that dy/dx = x2-y ,y(0)=1

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

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

5.i. Given y1 = x+siny , y(0) =1 compute y(.2),y(.4) with h = .2 using Euler’s Modified method.

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 Milne’s method. Use Euler’s Modified method to evaluate y(.1),y(.2) & y(.3)


ASSIGNMENT ON UNIT-VII(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

1.i. Obtain the fourier series expansion of f(x) given that f(x) = kx ( ∏ - x ) in 0 < 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 < ∏/2 k(∏ - x) , ∏/2 < x < ∏ find the half range sine series.

ii. Obtain the fourier series expansion of f(x) given that f(x) = (∏- x )2 , 0 < x < 2 ∏ & deduce the value of 1/12 + 1/22 + 1/32+………. = ∏2 /6

3.i. Evaluate ∫ dx / (a2+x2) (b2+x2) using transform.

ii. Find the fourier series of periodicity 3 for f(x) = 2x – x2 , 0 < x < 3

4.i. Using fourier integral theorem , prove that e-ax – e-bx = 2(b2-a2) / ∏ ∫ λsin λx/ (λ2+a2) (λ2 + b2) dλ

ii. Obtain the fourier series for the function f(x) = x2 , - ∏ < x < ∏ . Hence show that 1/12 + 1/22 + 1/32 + …….. = ∏2 / 6

5.i. Show that fourier transform of e-x2/2 is reciprocal.

ii. Find the fourier transform of f(x) = 1-x2 , if │ x │ < 1 0 , if │ x │ > 1

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


ASSIGNMENT ON UNIT-VIII(COMMON TO ECE -A & B)NAME OF THE FACULTY-P.SHANTAN KUMAR(Dept. H & S)

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+y2) 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 from z = 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

IMPORTANT QUESTIONS (UNIT WISE)

Code No. 07A1BS06 UNIT-3

MATHEMATICAL METHODS

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

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

matrix.

2. Verify wheter the matrix A = is orthogonal.

3. Define orthogonal matrix.

Verify whether the matrix 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 matrix is unitary if a2b2+c2+d2 = 1

6. Show that the complex matrix is Hermition.

Find the eigen values and eigenvectors.

7. Show that the eigen values of a skew – Hermition matrix are purely imaginary or real.

8. Define an orthogonal matrix. is orthogonal.

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

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

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

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

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

14. Determine the nature, index and signature of the Quadratic form 2x2+2y2+3z2+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 form .

17. Find the nature, index and signature of the Quadratic form .

18. Reduce the Quadratic form to canonical form.19. Reduce the Quadratic form 5x26xy+5y2 to sum of squares.20. Reduce the Quadratic form to sum of squares.

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

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

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

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

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

6. Using Regular falsi method; compute the real root of the equation x ex = 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 to three 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 , 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 31Estimate the number of students whose weights are between 4 and 45.

12. Obtain the relations between the operators.

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

X: 20 25 30 35 40 45

F(x): 354 332 291 260 231 204

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

X: 2 4 6 8 10

Y: -14 22 154 430 89815. 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 Newton’s Backward interpolation formula.

17. Use stirling’s 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 170Viscosity 10.2 .7.9 5.1 4.4Apply Lagrange’s formula to estimate viscosity of the lubricant at 130 degrees of

temperature.

20. Apply Lagrange’s formula to estimate y (3) from the following detaX: 0 1 2 4Y: 2 3 12 78.

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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 9Y : 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 76Fit 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 4Y : 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 5Y : 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 8Y : 1 3 2 5 4

8. Find the best fitting exponential curve y = aebx to the following deta.X : 2 3 4 5 6Y: 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 5Y: 0.5 2 4.5 8 12.5

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

rule.

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 0.4Y: 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 4Y: 6.98 7.40 7.78 8.12 8.45.

14. Explain how the thepezridel rule is obtained from Newton – cote’s general quedreture formula.

15. Given the following table of values of x and y, first, first and second derivatives at x = 1.25

X : 1.10 1.15 1.20 1.25 1.30.Y : 1.05 1.07 1.09 1.12 1.14

16. Evaluate using Simpson’s the rule.

17. Find 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 5F(x) 8 15 7 6 2

19. Evaluate using (i) simpsous rule (ii)simpsous

X 4.0 4.2 4.4 4.6 4.8 5.0 5.2Logx 1.38 1.44 1.48 1.53 1.57 1.61 1.65

20. Evaluate by symposiums rule, using 11ordinates and compare with

actual value of the integral.

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Code No. 07A1BS06 UNIT-VIMATHEMATICAL METHODS

1. Using taylor’s series method, solve the equation for x = .4, given

that y=0 when x=02. Using taylor series method, find an approximate value of y at x=0.2 for the

differential equation , y(0)=03. Solve y(0)=1 using Taylor’s series method and y(0.1), y(0.2) correct

to 4 decimal places4. Give the differential equ , y(0)=1 obtain y(0.25) and y(0.5) by

Taylor’s series method

5. Solve -1=xy and y(0)=1 using Taylors series method and compute y(0.1)

6. Solve using Taylor’s series method Tabulate for x=0.1, 0.2

7. Given . Compute y(0.1) by Taylors series method

8. Find the value of y for x=0.4 by picard’s method given that , y(0)=0

9. Solve , y(1)=3 by picard’s method

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

picard’s method

11. Given , y(0)=0. find y(0.2) and y(1) by picord’s method

12. Solve , y(0)=0 by picard’s method

13. Find the solution of , y(0)=1 by Picord’s method

14. Solve , y(0)=1 by euler’s method

15. Given . Find y(0.2) by Euler’s modified method

16. Solve , y(0)=1 by modified Euler’s method

17. Find the solution , y(0)=1 of x=0.1

18. Given that , y(0)=1, Find y(0.1) using Euler’s method

19. Solve by Euler’s method given y(1)=2 and find y(2)

20. Find y(1.2) by modified Euler’s method given , 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) from ,

y(2)=2Taking h=0.25

26. Use Milne’s method to find y (0.4) from , y(0)=1

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

28. Calculate y(0.6) by milne’s predictor-corrector method given , y(0)=1 with h=0.2

29. Given 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) from

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Code No. 07A1BS06 UNIT-VIIMATHEMATICAL METHODS

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

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

3. Find a0, bn for f(x)=ex from x=0 to x=2 4. Find the Fourier series for f(x)=x, 0<x<2

5. Find Fourier a0 and ax for f(x)=

6. Find Fourier bn for f(x)=x sinx, 0<x<2 7. Find Fourier series for f(x)=-K for- <x<0

=k for 0<x<8. Find Fourier bn for f(x)=0, - x 0

=

9. Find Fourier a0, ax for f(x)=0 for -= sinx for 0< x<

10. Find Fourier a0 and ax for f(x)=

11. Find the Fourier series to

F(x)=

12. Define a periodic function13. Write the dirihlets condition for the existence of Fourier series of a function f(x)

in 14. Find the Fourier series of 15. Find the Fourier series of f(x) = -x in (0,2 )16. Obtain the Fourier series for the function f(x)= ex-1 in (0,2 )17. Find Fourier series for f(x)=e-x in (0, 2 )18. Define even and odd functions with 3 Examples19. Express f(x)=x as a Fourier series in (-20. Find the Fourier series to represent the function f(x)=x sinx, 21. Find the Fourier series for f(x) =si x, 22. Obtain Fourier series for

F(x)=

23. f(x)=

If so find the Fourier series for the function.

24. Expand the function f(x)=x3 as a Fourier series in the interval 25. Find the Fourier series for f(x)= x cosx, 26. Find the half range sine series for f(x)= 27. Obtain the half range sine series for ex in 28. find the Fourier series to represents (1-x2) in

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

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

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Code No. 07A1BS06 UNIT-VIIIMATHEMATICAL METHODS

1. Form the partial differential equation for 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 from

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+ when a, b are

parameters

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

7. Form the partial differential equ from z=(x-a)2+(y-b)2+1 where a, b are parameters8. Form the differential equation by eliminating a and b from log (az-1)=x+ay+b9. Find the differential equation of all planes passing through the origin10. Form the differential equation of all planes having equal intercepts on x and y axis11. Form a partial differential equ by eliminating the arbitrary functions from z = f(x2-y2)12. Eliminate the arbitrary function from z= 13. Find the differential equ from 14. Form the partial differential eqn by eliminating the arbitrary function f from

z=(x+y) f (x2-y2)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+y2+z2)17. Form the partial differential equation by eliminating the arbitrary function f from

f (x2+y2, x2-z2)=018. 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

20. Find the general solution of p+q=121. Solve px+Qy=z22. Solve p Tan x+q Tan y=Tanz23. Find the general solution of y2zp+x2zq=y2x24. Solve (y-z)p+(x-y)q=z-x25. Solve x(y-z)p+y(z-x)q=z(x-y)26. Solve p+3q=5z+Tan (y-3x)27. Find the integral surface of

X(y2+z)p-y(x2+z)q=(x2-y2)z

28. Solve

29. Solve p2+q2=npq30. Solve z=p2+q2

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