MM COURSE FILE -2010-2011
Transcript of MM COURSE FILE -2010-2011
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COURSE FILE
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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
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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 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
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SYLLABUS
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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.
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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
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TEACHING SCHEDULE
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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
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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
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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
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LESSON PLAN
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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
21 15-11-10 problems on Diagonalization
1
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22 16-11-10 problems on Diagonalization
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
34 07-12-10 previous qn.paper prob 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
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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
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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
75 29-03-11 previous qn.paper prob 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
82 19-04-11 previous qn.paper prob 1
83 20-04-11 Introduction on P.D.E. 1
84 21-04-11 Formation of pde in orbitary const
1
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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
95 11-05-11 previous qn.paper prob 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.
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ASSIGNMENTS(UNIT WISE)
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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
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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.
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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)
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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)
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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.
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TIRUMALA ENGINEERING COLLEGEI-B.TECH - MATHEMATICAL METHODS
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
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IMPORTANT QUESTIONS (UNIT WISE)
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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=
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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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Code No. 07A1BS06 UNIT-4
MATHEMATICAL METHODS
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
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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 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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Code No. 07A1BS06 UNIT-5
MATHEMATICAL METHODS
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
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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)
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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)=
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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
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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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OBJECTIVE TYPE QUESTIONS (UNIT WISE)
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JNTU
PREVIOUS QUESTION PAPERS
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ALL THE BEST
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