Download JNTU Kakinada B.Tech 1-1 2014 Feb R10 MATHEMATICAL METHODS Question Paper

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Code No: R10107/R10

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Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

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Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

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1. (a) Find value of K if rank of A is 3, if A = [ 1 2 -1 4 1 2 3 -1 1 2 0 K ]
   (b) Solve by Gauss elimination method 10x+y+z=12; 2x+10y+z=13; x+y+5z=7; [7+8]
2. (a) Prove that the Eigen values of a triangular matrix are diagonal elements of the matrix
   (b) Find eigen vectors of B=2A²-A + 3I where A = [ 8 -4 2 2 ] [5+10]

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3. Define the nature of the quadratic form. Identify the nature of the quadratic form x² + 4x₂² + x₃² - 4x₁x₂ + 2x₁x₃ - 4x₂x₃ [15]
4. (a) Evaluate the real root of the equation x² - 9x + 1 = 0 by Bisection method
   (b) Compute the real root of the equation x³- x² - 1 = 0 by the method of false position. [8+7]
5. (a) Compute the approximate value of e^-x when x= 1.7489 from the following table using the Gauss forward interpolation formula.

   X	e-x
   1.74	0.175520
   1.75	0.173774
   1.76	0.172045
   1.77	0.170333
   1.78	0.168638

   (b) Find the Parabola passing through the points (0, 1), (1,3) and (3,5), Using Lagrange's Interpolation formula. [8+7]
6. (a) Find the first and second derivatives of the function tabulated below at the point x = 1.5.

   X	1.5	2.0	2.5	3.0	3.5	4.0
   Y	3.375	7.0	13.625	24.0	38.875	59.0

   (b) Evaluate ∫_0.6^2.0 y dx using Trapizoidal, Simpsons 1/3 and Simpsons 3/8 rules.

   X	0.6	0.8	1.0	1.2	1.4	1.6	1.8	2.0
   y	1.23	1.58	2.03	4.32	6.25	8.38	10.23	12.45

   [8+7]

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7. (a) Solve y¹=3x+y/2, y(0)=1 by Taylor series method and hence find y(0.1), y(0.2)
   (b) Solve the equation dy/dx = xy + 1, y(0)=1 by Picard's method and hence find y(0.1) [8+7]
8. (a) Fit a least square parabola y= a+bx+cx² to the following data

   X	-3	-2	-1	0	1	2	3
   y	4.63	2.11	0.67	0.09	0.63	2.15	4.58

   (b) Fit a straight line of the form y= a+bx to the following data

   X	1	2	4	5	6	8	9
   y	2	5	7	10	12	15	19

   [7+8]

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Code No: R10107/R10

Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

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( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

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Answer any FIVE Questions

All Questions carry equal marks

?

1. (a) Find rank of A = [ 2 1 3 1 0 1 2 -2 4 0 2 6 ] using Normal Form
   (b) Solve by Gauss seidal method x+4y+15z=24, x+12y+z=26, 10x+y-2z=10 [7+8]

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2. (a) Find Eigen Vectors of A = [ 5 4 1 2 ]
   (b) If A is an Eigen value of A then prove that is an Eigen value of Adj. A [7+8]
3. Find the rank, signature and index of the quadratic form 2x₁²+x₂²-3x₃² + 12x₁x₂ + 4x₁x₃ - 8x₂x₃by reducing it to normal form. Also write the linear transformation which brings about the normal reduction [15]
4. (a) Using Newton- Raphson's method compute √41 correct to four decimal places.
   (b) Find a real root of the equation e^x = x+2in the interval [1, 1.4] using bisection method. [8+7]

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5. (a) Apply Gauss backward interpolation formula to find y when x = 26 from the following table

   X	Y
   20	2854
   24	3162
   28	3544
   32	3992

   (b) Using Lagrange's interpolation formula, find the value of y when x = 2 from the following data:

   X	y
   1	4
   3	40
   4	85
   6	259

   [8+7]
6. (a) Find the value of f'(x) at x=0.01 from the following table using Bessel's formula.

   X	f(x)
   0.01	0.1023
   0.02	0.1047
   0.03	0.1071
   0.04	0.1096
   0.05	0.1122
   0.06	0.1148

   (b) Find the area bounded by the curve y = e^x2, x-axis between x = 0 and x = 3 by using Simpson's 3/8 rule. [8+7]
7. (a) Solve y¹=x-y, y(0)=1 by modified Euler's method and find y(0.1), y(0.2)
   (b) Apply third order R-K method to find y(0.25) where y¹=1+xy, y(0)=1 [8+7]
8. (a) Fit a power curve y=ax^b to the following data

   X	5	6	7	8	9	10
   y	133	55	23	7	2	2

   (b) Fit a curve of the type y= a+bx+cx² to the following data

   X	0	1	2	3	4	5	6
   y	14	18	23	29	36	40	46

   [7+8]

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Firstranker's choice

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Code No: R10107/R10

Set No. 3

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I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

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Engineering)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?

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1. (a) Find rank using Normal Form A = [ 1 2 3 6 0 2 4 8 3 2 1 7 3 6 8 5 ]
   (b) Solve Homogeneous equations x₁+2x₂+3x₃=0,2x₁+3x₂+x₃=0, 4x₁+5x₂+4x₃=0,x₁+x₂-2x₃=0 [7+8]
2. (a)Find Eigen values and Eigen vectors of A = [ 8 4 2 2 ]
   (b) If A is an Eigen value of A then prove that λ^-1 is an Eigen value of A^-1 if it exists [7+8]
3. Find the rank, signature and index of the quadratic form 2x₁²+x₂²-3x₃² + 12x₁x₂ + 4x₁x₃ - 8x₂x₃by reducing it to normal form. Also write the linear transformation which brings about the normal reduction [15]

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4. (a) Find out square root of 25 using Bisection method Given x₀=2, X₁=7
   (b) Solve the equation x² + 10x = 20 by iteration method [8+7]
5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) = 0.2707, f(30) = 0.28027, f(35) = 0.3386, f(40) = 0.3794.
   (b) Find the interpolating polynomial f(x) from the table given below.

   X	0	1	4	5
   f(x)	4	3	24	39

   [8+7]
6. (a) Using the table below, find f' (0)

   X	-2	0	2	3	4	7
   f(x)	460	-110	-58	-26	4	526

   (b) Evaluate ∫₀¹√1 + x³ dx taking h = 0.1 using Simpson's 3/8th rule. [8+7]

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7. (a) Solve y¹=x+y subject to the condition y(0)=1 by Taylor series method and hence find y(0.2), y(0.4)
   (b) Solve y¹=x-y, y(0)=1 by Picard's method and hence find y at x=0.2 [8+7]
8. (a) Fit a curve of the type y= a+bx+cx² to the following data

   X	10	15	20	25	30	35
   y	35.3	32.4	29.2	26.1	23.2	20.5

   (b) Fit a curve of the type y=ab^x to the following data by the method of least squares

   X	1	2	5	10	20	30	40	50
   Y	98.2	91.7	81.3	64	36.4	32.6	11.3	7.1

   [7+8]

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Firstranker's choice

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Code No: R10107/R10

Set No. 4

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICAL METHODS

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( Common to Civil Engineering, Electrical & Electronics Engineering,

Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and Automobile

Engineering)

Time: 3 hours Max Marks: 75

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Answer any FIVE Questions

All Questions carry equal marks

?

1. (a) Find rank of matrix using Normal form A = [ 1 2 3 3 0 4 -2 3 1 ]
   (b) Solve system of equations, if consistent 2x-y-z=2,x+2y+z=2, 4x-7y-5z=2 [7+8]

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2. Verify Cayley - Hamilton theorem and find A^-1 if A = A = [ -1 2 -1 1 -1 2 -1 2 -1 ] [15]
3. Reduce the quadratic form to canonical from by orthogonal reduction and state the nature of the quadratic form 5x² + 26y² + 5z² - 4zy + 4yz + 14zx. Also find its rank signature and index. [15]
4. (a) Using Newton-Raphson's method find the square root of a number and hence find the square root of 24.
   (b) Find a real root of the equation x=e^-x, using Bisection method [8+7]
5. (a) Apply Gauss's forward formula to find f(x) at x = 3.5 from the table below.

   X	2	3	4	5
   F(x)	2.626	3.454	4.784	6.986

   (b) Find sin 45° using Gauss's backward interpolation formula given that sin 20° = 0.342, sin 30° = 0.502, sin 40° =0.642, sin 50° = 0.766, sin 60° =0.866, sin 70° = 0.939, sin 80° = 0.984.

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6. (a) Given the following table. Find f'(1) and f" (3)

   X	0	2	4	6	8
   f(x)	7	13	43	145	367

   (b) Find approximate value of ∫₁^1.04 f(x)dx using the following table.

   X	1	1.01	1.02	1.03	1.04
   f(x)	3.953	4.066	4.182	4.300	4.421

   [8+7]
7. (a) Given that dy/dx = (1+2x)², y(0)=1, y(0.1)=1.06, y(0.2)=1.12, y(0.3)=1.21 then evaluate y(0.4) by Milne‘s predictor corrector method
   (b) Solve dy/dx = (y+x), y(0) = 1 estimate y(0.1) and y(0.2) using Euler's method in 5 steps [8+7]
8. (a) Fit a least square parabola y= a+bx+cx² to the following data

   X	1	2	3	4	5
   y	5	12	25	44	69

   (b) Fit a straight line of the form y= a+bx to the following data

   X	1	2	3	4	5
   y	5	12	26	60	90

   [8+7]

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