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Code No: R10107/R10
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Set No. 1
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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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β β β β β
- (a) Find rank of matrix using Echelon form A =
2 -4 5 2 -1 3 8 1 9 6 6 7
(b) Solve the equations using Gauss Jordan method x+5y+z=9, 2x+y+3z=12, 3x+y+4z=16 [7+8] - Using Cayley β Hamilton theorem find A8 if A =
[15]1 2 2 -1 - Reduce the quadratic form 6x12 + 3x22 + 3x32 - 4x1x2 + 4x1x3 - 2x2x3 to the sum of squares form by diagonalization and find the corresponding linear transformation. Also find the index and signature. [15]
- (a) Find a real root of the equation x + cosx=0, using Newton-Raphson's method
(b) Evaluate v12 and using fixed point iteration method. [8+7] - (a) If the interval of differencing is unity, prove the following: ?{f(x)/f(x+1)} = ?f(x) / f(x)f(x+1)
(b) Given that sin 45Β° = 0.7071, sin 50Β° = 0.8192, sin 60Β° = 0.8660, find sin 48Β°. - (a) Compute f'(1) using the given data:
X f(x) 1.0 2.7 1.5 10.675 2.0 32.4 2.5 78.375 3.0 162.1 --- Content provided byβ FirstRanker.com ---
(b) Using Simpson's 3/8th rule evaluate ?14 dx/(1+x2) by dividing the range into 6 equal parts [8+7] - (a) Solve y'= -xy2, y(0)=2 by modified Euler's method and hence find y(0.1), y(0.2)
(b) Solve dy/dx = (y2-x2)/(y2+x2), y(0)=1 by fourth order R-K method and hence find y(0.2), y(0.4) [8+7] - (a) Fit a least square parabola y= a+bx+cx2 to the data (-1,2),(0,1),(1,4)
(b) By the method of least squares fit a straight line to the following data
[8+7]X y 5 15 10 15 15 20 19 23 26 30
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β β β β β
Code No: R10107/R10
Set No. 2
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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
β β β β β
- (a) Find rank of matrix using Echelon form A =
1 1 -1 2 -3 4 3 -2 3
(b) Solve the equations using Gauss Jordan method x1+x2+x3=8, 2x1+3x2+2x3=19, 4x1+2x2+3x3=23 [7+8] - Verify Cayley β Hamilton theorem and find A-1if A =
[15]1 0 3 2 1 -1 1 -1 1 - (a) Find the nature of the quadratic form 5x2 + 14z2 + 2xy - 16yz - 8zx [8+7]
(b) If A =
then find A500 1 3 0 - (a) Apply Newton-Raphson's form to find the cube root of 5 correct up to three decimal places starting from 1
(b) Find a real root of f(x)= 3x + 1 = 0 correct up to three decimal places starting with x=1 by iterative method. [8+7] - The following table shows the population of a town during the last six censuses. Estimate, using Newton's interpolation formula, the increase in the population during the period 1986 to 1988.
[15]year 1911 1921 1931 1941 1951 1961 Population (in thousands) 12 15 20 27 39 52 - (a) Given the following data of X and Y
Find the first and second derivatives at x = 1.0X Y 1.0 2.72 1.2 3.32 1.4 4.06 1.6 4.96 1.8 6.05 2.0 7.39
(b) The table below shows the temperature f(t) as a function of time
Use Simpson's 1/3 method to estimate ?17 f (t) dt. [8+7]t 1 2 3 4 5 6 7 f(t) 81 75 80 83 78 70 60 - (a) Solve y'=3x2+1 by Euler's method and find y at x=2 by taking h=0.5 [8+7]
(b) Solve by fourth order R-K method y'=x-y, y(1)=0.4 and hence find y(1.2) - (a) Fit a curve of the type y= a+bx+cx2 to the following data
[8+7]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=abx to the following data by the method of least squares
[7+8]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
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β β β β β
Code No: R10107/R10
Set No. 3
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I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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
β β β β β
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- (a) Define rank and find the rank of matrix A using Echelon form A =
1 3 6 1 4 5 1 5 4 -1 1 1
(b) Find values of x,y and z using Gauss Jordon method 2x+y-z=1; x-y+z=2 ; 5x+5y-4z=3 [7+8] - Find Eigen Vectors of A =
[15]5 -2 0 -2 6 2 0 2 7 - (a) By Lagrange's reduction reduce the quadratic form XTAX to sum of squares form for A=
1 2 4 2 6 -2 4 -2 18
(b) Find the values of a, b, c if
is an orthogonal matrix [8+7]a a -2b a b -b b c -c - (a) Find a real root of the equation using Newton-Raphson's method Cos2x-x=0
(b) Find a root the equation x3e-x-1=0 by Bisection method. [8+7] - (a) (i) Solve ?(eaxlog bx ) (ii) Prove that ?6y5 = ?6y-1
(b) From the following table for find f(3.3) using gauss forward interpolation formula.
[8+7]X 1 2 3 4 5 Y 15.30 15.10 15.00 14.50 14.00 - (a) A curve is expressed by the following values of x and y. Find the slope at the point x = 0.5.
Calculate the angular velocity and the angular acceleration of the rod when t = 0.3 seconds.X 0.4 0.5 0.6 0.7 0.8 y 1.58 1.80 2.04 2.33 2.65 --- Content provided byο»Ώ FirstRanker.com ---
(b) Evaluate ?13 dx/(1+x) , by Trapezoidal rule and Simpson's rule. [8+7] - Solve y'=x-y, y(0)=1 ,h=0.1 by Milneβs predictor corrector method to find y(0.4).Use Euler's modified method to evaluate y(0.1), y(0.2) ,y(0.3) [15]
- (a) Fit a power curve y=axb to the following data
X 1 2 3 4 5 Y 7.1 27.8 62.1 110 161
(b) Fit a least square parabola y= a+bx+cx2 to the following data
[7+8]X 0 1 2 3 4 y 1 5 10 22 38
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β β β β β
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Code No: R10107/R10
Set No. 4
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
MATHEMATICAL METHODS
( Common to Civil Engineering, Electrical & Electronics Engineering,
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Computer Science & Engineering, Electronics & Instrumentation
Engineering, Aeronautical Engineering, Bio-Technology and Automobile
Engineering)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
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All Questions carry equal marks
β β β β β
- (a) Find rank of matrix A =
1 2 3 2 -2 0 3 1 4
using Normal form.2 -2 1
(b) Solve system of equations, if consistent x+y+2z=4,2x-y+3z=9, 3x-y-z=2 [7+8] - Show that matrix A =
satisfies Cayley β Hamilton theorem [15]0 c -b -c 0 a b -a 0 - Find the rank, signature and index of the quadratic form 2x12+x22 β 3x32+12x1x2 - 4x1 x3 - 8x2x3by reducing it to normal form Also write the linear transformation which brings about the normal reduction [15]
- (a) Find a real root of the equation x sinx +cosx=0, using Newton-Raphson's method
(b) Evaluate v12 and using fixed point iteration method. [8+7] - (a) Evaluate the following, interval of differencing being unity.
(i) ? (e2log tan-1 ax) (ii) ? (e2log 32)--- Content provided byβ FirstRanker.com ---
(b) Find y(25), given that y20 = 24, y24 = 32, y28 = 35, y32 = 40, Using Gauss forward difference Interpolation formula. [8+7] - (a) For the function y = f(x) given by the following Table, find y' at x = 0.04 using the Bessel's formula.
X y 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) Evaluate ?01tetdt by using the Simpson's 3/8th rule, by dividing the interval into 3 equal parts. [8+7] - (a) Solve y'=3x2+1 by Euler's method and find y at x=2 by taking h=0.5 [8+7]
(b) Solve by fourth order R-K method y'=x-y, y(1)=0.4 and hence find y(1.2) - (a) Fit a least square parabola y= a+bx+cx2 to the following data
X 1 2 3 4 5 y 2 3 5 8 10
(b) Fit a straight line of the form y= a+bx to the following data
[8+7]X -1 0 1 2 3 4 5 6 y 10 9 7 5 4 3 0 -1
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