Set No. 1
Code No: Z0224/R07
I B.Tech Supplementary Examinations, December 2012
MATHEMATICAL METHODS
( Common to Electrical & Electronics Engineering, Mechanical Engineering,
Electronics & Communication Engineering, Computer Science &
Engineering, Electronics & Instrumentation Engineering, Bio-Medical
Engineering, Information Technology, Electronics & Control Engineering,
Mechatronics, Computer Science & Systems Engineering, Electronics &
Telematics, Electronics & Computer Engineering, Production Engineering,
Instrumentation & Control Engineering and Automobile Engineering)
Time: 3 hours
Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
⋆⋆⋆⋆⋆
1. (a) Show that the system of equations x +2y + z = 3, 2x + 3y +2z = 5,
3x - 5y +5z = 2, 3x + 9y- z = 4 are consistent and solve them
(b) Write the following equations in matrix form AX = B and solve for X by
finding A−1: x + y - 2z = 3, 2x - y + z = 0, 3x +y - z = 8.
[8+8]
2. Verify Cayley-Hamiltion Theorem for the matrix A =
A−1.
3 4 1
2 1 6
−1 4 7
. Hence find
[16]
3. Reduce the quadratic form x2 + 2y2 + z2 - 2xy - 2yz to canonical form.
[16]
4. (a) Given u1 = 22, u2 = 30, u4 = 82, u7 = 106, u8 = 206, find u6. Use Lagrange’s
interpolation formula.
(b) Find a real root of x3-x-2=0.
[8+8]
5. (a) Fit a straight line of the form y= a +bx by using the method of least squares
and hence find the missing term.
x:
3 6 7
9 10
y: 168 - 120 72 73
∫π sin x
’s rule and Trape(b) Evaluate
x dx dividing into six equal parts using Simpson
0
zoidal rule.
[8+8]
6. (a) Use Eulers method to find y(0.1), y(0.2) given y′ = (x3 + xy2) e−x y(0)=1.
(b) Solve y = x2 + y2 given y(0)=0 using Picard’s method correct to three places
of decimals.
[8+8]
7. (a) Find the Fourier series for f(x) = 2lx- x2 in 0 < x < 2l and hence deduce
1
1
1
1- 22 + 32 − 42 + ..... .. = π212 .
(b) Find the Fourier sine series for f(x) = 2x- x2, in 0 < x < 3 and f(x+3)=f(x).
[8+8]
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Set No. 1
Code No: Z0224/R07
8. (a) Solve
p
x2 + yq2 = z
(b) Find the inverse Z-transform of
z2 +z
(z−1)(z2 +1)
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[8+8]
Set No. 2
Code No: Z0224/R07
I B.Tech Supplementary Examinations, December 2012
MATHEMATICAL METHODS
( Common to Electrical & Electronics Engineering, Mechanical Engineering,
Electronics & Communication Engineering, Computer Science &
Engineering, Electronics & Instrumentation Engineering, Bio-Medical
Engineering, Information Technology, Electronics & Control Engineering,
Mechatronics, Computer Science & Systems Engineering, Electronics &
Telematics, Electronics & Computer Engineering, Production Engineering,
Instrumentation & Control Engineering and Automobile Engineering)
Time: 3 hours
Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
⋆⋆⋆⋆⋆
1. (a) Discuss the consistency of x - y + z = - 1, x - 3y + 4z = -6, 4x + 3y -2z = -3,
7x -4y + 7z = -16 and solve
(b) Solve the system x + 2y +3z = 1; 2x + 3y + 2z = 2; 3x + 3y + 4z = 1 by
matrix method.
[8+8]
2. (a) Find the characteristic roots of the matrix A =
[
(b) If A =
]
1 0
0 3 , find A50.
2 2 1
1 3 1
1 2 2
[8+8]
3. Reduce the quadratic form 6x1 2+3x2 2+3x3 2−4x1x2−2x2x3+4x3x1 to the sum of
squares and find the corresponding linear transformation. Also find the index and
signature.
[16]
4. (a) If h is the interval of differencing( prove that
)
Δn sin(ax + b) = 2 sin(ah/2)n sin ax + bnah+nπ2
(b) The following data give I, the indicated HP and V, the speed in knots developed by a ship.
V
I
8
10
12
14
16
1000 1900 3250 5400 8950
Find I when V = 9, using Newton’s forward interpolation formula.
[8+8]
5. (a) From the following table, Find f ′(2.5).
x:
f(x)=ex:
0
1
2
3
4
1 2.72 7.39 20.09 50.60
(b) The rod is rotating in a plane. The following table gives the angle θ through
which the rod has turned for various values of time t. Calculate the angular
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Set No. 2
Code No: Z0224/R07
velocity and angular acceleration of rod at t=0.6 seconds.
t(in seconds): 0 0.2
0.4
0.6
0.8
1.0
θ(in radians): 0 0.12 0.49 1.12 2.02 3.20
[8+8]
6. (a) Solve the following using R - K fourth method y′ = y-x, y(0) = 2, h=0.2. Find
y(0.2).
7.
(b) Given y′ = x2 - y, y(0) =1, find correct to four decimal places the value of
y(0.1), by using Euler’s method.
[8+8]
{
x; 0 < x < π
(a) Expand f (x) =
2π − x; π < x < 2π
as a Fourier series of periodicity 2π, and deduce the value of
1
1
1
.. = π2
12 + 22 + 32 +
6 .
(b) Expand f(x)= -1, in (-2,0) and f(x)=1, in (0,2) as a Fourier series.
[8+8]
8. (a) Solve the difference equation, using Z-transform y(n+2)+3y(n+1)+2y(n)=0,
given y(0)=0, y(1)=1.
(b) Solve x2 (z − y) p + y2 (x − z) q = z2 (y − x)
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[8+8]
Set No. 3
Code No: Z0224/R07
I B.Tech Supplementary Examinations, December 2012
MATHEMATICAL METHODS
( Common to Electrical & Electronics Engineering, Mechanical Engineering,
Electronics & Communication Engineering, Computer Science &
Engineering, Electronics & Instrumentation Engineering, Bio-Medical
Engineering, Information Technology, Electronics & Control Engineering,
Mechatronics, Computer Science & Systems Engineering, Electronics &
Telematics, Electronics & Computer Engineering, Production Engineering,
Instrumentation & Control Engineering and Automobile Engineering)
Time: 3 hours
Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
⋆⋆⋆⋆⋆
1. (a) Express the following system in matrix form and solve by Gauss elimination
method.
2x1 + x2 + 2x3 + x4 = 6; 6x1 - 6x2 + 6x3 + 12x4 = 36,
4x1 + 3x2 + 3x3 - 3x4 =- 1; 2x1 + 2x2 - x3 + x4 = 10.
(b) Show that the system of equations 3x + 3y + 2z = 1; x + 2y = 4;
10y + 3z = - 2; 2x - 3y - z = 5 is consistent and hence solve it.
2. For the matrix A =
diagonal matrix.
4 1 0
1 4 1
0 1 4
[8+8]
, determine matrix P such that P−1 AP is a
[16]
3. Reduce the quadratic form to canonical form by an orthogonal reduction and state
the nature of the quadratic form. x1+2x2 2+2x3 2−2x1x2+2x1x3−2x2x3.
[16]
4. (a) If h is the interval of differencing( prove that
Δn sin(ax + b) = 2 sin(ah/2)n sin ax + bnah+nπ2
)
(b) The following data give I, the indicated HP and V, the speed in knots developed by a ship.
V
I
8
10
12
14
16
1000 1900 3250 5400 8950
Find I when V = 9, using Newton’s forward interpolation formula.
[8+8]
5. (a) Fit a straight line of the form y= a +bx by using the method of least squares
and hence find the missing term.
x:
3 6 7
9 10
y: 168 - 120 72 73
∫π sin x
’s rule and Trape(b) Evaluate
x dx dividing into six equal parts using Simpson
0
zoidal rule.
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[8+8]
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Set No. 3
Code No: Z0224/R07
6. (a) Solve y′= x+y2 +1, given y(0) =0 by using Picard’s method.
(b) Using Taylor series method, compute the value of y(0.2) correct to 4 decimal
places from y′= 1-2xy given that y(0)=0.
[8+8]
kx; 0 < x < π
2
k(π − x); π
2 <x<π
Find the half-range sine series.
(b) Find the Fourier expansion of f(x)= x cosx; 0 < x < 2π.
7. (a) If f (x) =
[8+8]
8. (a) Find the inverse Z-transform of
(
3− 5z−1 6
)(
)
1− z−1
1− z−1
4
3
if |z| > 1
3
(b) Solve q2 = z2(1 − p2)p2.
[8+8]
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Set No. 4
Code No: Z0224/R07
I B.Tech Supplementary Examinations, December 2012
MATHEMATICAL METHODS
( Common to Electrical & Electronics Engineering, Mechanical Engineering,
Electronics & Communication Engineering, Computer Science &
Engineering, Electronics & Instrumentation Engineering, Bio-Medical
Engineering, Information Technology, Electronics & Control Engineering,
Mechatronics, Computer Science & Systems Engineering, Electronics &
Telematics, Electronics & Computer Engineering, Production Engineering,
Instrumentation & Control Engineering and Automobile Engineering)
Time: 3 hours
Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
⋆⋆⋆⋆⋆
1. (a) Show that the system of equations x +2y + z = 3, 2x + 3y +2z = 5,
3x - 5y +5z = 2, 3x + 9y- z = 4 are consistent and solve them
(b) Write the following equations in matrix form AX = B and solve for X by
finding A−1: x + y - 2z = 3, 2x - y + z = 0, 3x +y - z = 8.
[8+8]
2. For the matrix A =
diagonal matrix.
4 1 0
1 4 1
0 1 4
, determine matrix P such that P−1 AP is a
[16]
3. Write down the quadratic form corresponding to the matrix A=
reduce it to the canonical form.
1 2 5
2 0 3
5 3 4
and
[16]
4. (a) Solve tan x + tanh x =0 by iteration method:
(b) Find a root of 4x - ex = 0 that lies between 2 and 3.
[8+8]
5. (a) It is known that x, y are related by y =ax + bx and the experimental values
are given below:
x:
1
2
4
6
8
y: 5.43 6.28 10.32 14.86 19.5
Obtain the best values of a and b.
(b) Find the first two derivatives of the function tabulated below at x=0.6
x:
0.4
0.5
0.6
0.7
0.8
y: 1.5836 1.7974 2.0442 2.3275 2.6511
6. Given y′= 1-y, and y(0)=0, find
(a) y(0.1) by Euler’s method
(b) y(0.2) by Picard’s method
(c) y(0.3) by R- K method of fourth order and
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[8+8]
Set No. 4
Code No: Z0224/R07
(d) y(0.4) by Milne’s method.
{
x; 0 < x < π
7. (a) Expand f (x) =
2π − x; π < x < 2π
as a Fourier series of periodicity 2π, and deduce the value of
1
1
1
.. = π26 .
12 + 22 + 32 + ......
(b) Expand f(x)= -1, in (-2,0) and f(x)=1, in (0,2) as a Fourier series.
8.
[16]
[8+8]
z2
(a) Using Convolution theorem, find the inverse-Z transform of
(b) Solve z=px+qy+p2 + q2.
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(z−a)(z−b)
[8+8]