09 a1bs04 mathematical methods

R09 SET-1Code.No: 09A1BS04
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
I B.TECH – REGULAR EXAMINATIONS, JUNE - 2010
MATHEMATICAL METHODS
(COMMON TO EEE, ECE, CSE, EIE, BME, IT, ETE, E.COMP.E, ICE)
Time: 3hours Max.Marks:80
Answer any FIVE questions
All questions carry equal marks
- - -
1.a) Find the Rank of the Matrix, by reducing it to the normal form
1 2 1 3
4 1 2 1
3 1 1 2
1 2 0 k
−⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥−
⎢ ⎥
⎣ ⎦
.
b) Solve the system of linear equations by matrix method.
x + y + z = 6, 2x + 3y - 2z = 2, 5x + y + 2z = 13. [8+7]
2. Verify Cayley Hamilton theorem and find the inverse of
1 0 3
2 1 1
1 1 1
⎡ ⎤
⎢ ⎥− −⎢ ⎥
⎢ ⎥−⎣ ⎦
[15]
3. Prove that the following matrix is Hermitian. Find the eigen values and the
corresponding eigen vectors of the matrix
4 1 3
1 3 7
i
i
−⎡ ⎤
⎢ ⎥+⎣ ⎦
[15]
4.a) Find a real root of the equation x3
- x - 4 = 0 by bisection method.
b) Use Newton's forward difference formula to find the polynomial satisfied by (0, 5),
(1, 12), (2, 37) and (3, 86). [8+7]
5.a) Derive the normal equation to fit the parabola y = a + bx + cx2
.
b) By the method of least squares, find the straight line that best fits the following data:
[7+8]
x 1 2 3 4 5
y 14 27 40 55 68
6. Using Taylor series method, find an approximate value of y at x=0.2 for the
differential equation for2 3 x
y y e′− = ( )0 0y = . [15]
7.a) Find the Fourier Series to represent the function f(x) given:
( ) 2
0 0
0
for x
f x
x for x
−Π ≤ ≤⎧
= ⎨
≤ ≤ Π⎩
b) Find the Fourier series in [ ],−Π Π for the function ( )
( )
( )
1
0
2
1
0
2
x for x
f x
x for x
−⎧
Π + −Π ≤⎪⎪
= ⎨
⎪
≤
Π − ≤
⎪⎩
≤ Π
[8+7]

8.a) Form a partial differential equation by eliminating a,b,c from
2 2 2
2 2 2
1
x y z
a b c
+ + = .
b) Form the partial differential equation by eliminating the constants from
(x - a)2
+ (y - b)2
= z2
cot 2
α where α is a parameter. [8+7]
-oOo-

- - -
1.a) Find the Rank of the Matrix,
2 3 7
3 2 4
1 3 1
⎡ ⎤
⎢ ⎥−⎢ ⎥
⎢ ⎥− −⎣ ⎦
by reducing it to the normal form.
b) Find all the non-trivial solutions of 2x - y + 3z = 0, 3x + 2y + z = 0, x - 4y + 5z = 0.
[7+8]
2. Find the eigen values and the corresponding eigen vectors of
1 3 7
1 2 3
1 2 1
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
. [15]
3.a) Prove that 1
2
1 1
1 1
i i
i i
+ − +⎡
⎢ + −⎣ ⎦
⎤
⎥ is unitary.
b) Prove that the eigen values of a real skew symmetric matrix are either zero or purely
imaginary. [8+7]
4.a) Find a real root of the equation 3x =ex
by bisection method.
b) Using Lagrange’s formula find y(6) given: [7+8]
x 3 5 7 9 11
y 6 24 58 108 74
5.a) Fit a straight line y = a + bx from the following data:
x 0 1 2 3 4
y 1 1.8 3.3 4.5 6.3
b) Fit a straight line to the form y = a + bx for the following data:
[7+8]x 0 5 10 15 20 25
y 12 15 17 22 24 30
6. Find y(0.1), y(0.2),z(0.1), z(0.2) given
dy
x z
dx
= + , 2dz
x y
dx
= − and ,
by using Taylor’s series method. [15]
( )0 2y =
( )0z =1
7.a) Express f(x)=x as a Fourier Series in ( ),−Π Π .
b) Expand the function f(x) = x2
as a Fourier series in( ),−Π Π . [8+7]

8.a) Form the partial differential equation by eliminating a and b from log(az-1) = x + ay + b
b) Find the differential equation of all spheres whose centres lie on z-axis with a given
radius r. [7+8]
-oOo-

- - -
2 1 3 5
4 2 1 3
8 4 7 13
8 4 3 1
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
− −⎣ ⎦
b) Find whether the following system of equations are consistent. If so solve them.
x + y + 2z = 9, x - 2y + 2z = 3, 2x - y + z = 3, 3x - y + z = 4. [8+7]
2. Verify Cayley Hamilton theorem and find the inverse of
1 3 7
1 2 3
1 2 1
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
[15]
3. Reduce the quadratic form to the canonical form 2 2 2
2 2 4 4x y z xy zx y+ + − + + z [15]
4.a) Find a real root of the equation ex
sin x=1using Newton Raphson method.
b) Find y(10), Given that y(5) = 12, y(6) = 13, y(9)=14, y(11) = 16 using Lagrange’s
formula. [8+7]
5.a) Using the method of least squares find the constants a and b such that y = aebx
fits the
following data:
x 0 0.5 1 1.5 2 2.5
y 0.10 0.45 2.15 9.15 40.35 180.75
b) Obtain a relation of the form y=abx
for the following data by the method of least
squares. [7+8]
x 2 3 4 5 6
y 8.3 15.4 33.1 65.2 127.4
6. Solve 1
dy
xy
dx
+= and ( )0y =1 using Taylor’s series method and compute y(0.1). [15]
7.a) If f(x)=cosh ax expand f(x) as a Fourier Series in ( ),−Π Π .
b) Expand the Function f(x) = x3
as a Fourier Series in x−Π < ≤ Π . [7+8]
8.a) Solve (z2
-2yz-y2
)p + (xy + zx)q = xy - zx.
b) Find the integral surface of x (y2
+ z) p-y(x2
+ z) q = (x2
+ y2
) z. [7+8]
-oOo-

- - -
1 3 4 5
1 2 6 7
1 5 0 10
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
b) Find whether the following system of equations are consistent. If so solve them.
x + 2y + 2z = 2, 3x - 2y - z = 5, 2x - 5y + 3z = - 4, x + 4y + 6z = 0. [7+8]
2. Find the eigen values and the corresponding eigen vectors of
1 0 1
1 2 1
2 2 3
−⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
[15]
3. Reduce the quadratic form to the canonical form [15]2 2 2
3x 2 3 2 2y z xy y+ + − − z
4.a) Prove that the newton’s method has quadratic convergence.
b) Find y(5) given that y(0)=1, y(1)=3, y(3)=13, and y(8) = 123 using Lagrange’s
formula. [8+7]
5.a) Find
dy
at x=7.5 from the following table.
dx
x 7.47 7.48 7.49 7.5 7.51 7.52 7.53
y .193 .195 .198 .201 .203 .206 .208
b) Find the first two derivative at x=1.4 from the following data: [8+7]
x 1.0 1.2 1.4 1.6 1.8 2.0
y 0 .128 .544 1.296 2.432 4.0
6. Using Euler’s method, solve for y at x=2 from 2
3 1
dy
x
dx
= + , ( )1 2y = taking step size:
a) h = 0.5
b) h = 0.25. [8+7]
7.a) Expand f(x)= cosx for 0 < x < Π in half range sine series.
b) Find cosine and sine series for f(x) = xΠ − in [0,Π ]. [7+8]
8.a) Solve (mz - ny) p + (nx - lz)q = (ly - mx).
b) Solve (x2
- y2
- yz) p + (x2
- y2
-zx) q = z(x - y). [7+8]
-oOo-

09 a1bs04 mathematical methods

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09 a1bs04 mathematical methods