Mathematics Ii Nov2004 Rr 210101

Set No.

1

Code No: RR-210101 II B.Tech. I-Semester Regular Examinations, November-2004 MATHEMATICS-II (Common to all Branches) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks --1.a) Determine the rank of the matrix.  − 2 − 1 − 3 − 1   2 3 − 1  1 A=  by reducing it to normal form. 1 0 1 1    0 1 1 − 1  b) Determine the values of λ for which the following set of equations may posses non-trivial solution and solve them in each case. 3x1 + x2 - λx3 = 0; 4x1 – 2x2 – 3x3 = 0; 2λx1 + 4x2 + λx3 = 0. 2.a) b)

Show that if λ1, λ2 …… . λn are latent roots of a matrix A, then A3 has the latent roots λ 13, λ 23, …….. λ n3 and kλ1, kλ2, kλ3………kλn are latent roots of kA. Find the eigen values and the corresponding eigen vectors of the matrix  8 −6 2   − 6 7 − 4  .  2 − 4 3 

3.a) b)

Identify the nature, index, signature of the quadratic form 2 x1 x 2 + 2 x 2 x3 + 2 x3 x1 .10 Prove that transpose of a unitary matrix is unitary.

4.a)

Define a periodic function. Find the Fourier expansion for the function f ( x ) = x − x 2 ,−1 < x < 1 .

b)

5.a) b) c) 6.a) b)

Prove that the function f ( x ) = x, 0 ≤ x ≤ π can be expanded in a series of sines as  sin x sin 2 x sin 3 x  x = 2 − + − ....... . 2 3  1  Form the partial differential equation by eliminating the arbitrary constants from (x-a)2+ (y-b)2 + z2 = r2 Solve the partial differential equation x(y-z)p +y(z-x)q = z( x-y). Solve the partial differential equation y2 z p + x2 z q = xy2 Solve by separation of variables 3ux +2uy = 0 with u(x,0) = 4 e-x. Obtain the general solution of the one dimension wave equation

∂2 u/∂t2 = c2 ∂2u/ ∂x2. Code No. RR-210101 7.a) b) 8.a) b)

-2-

State and prove Fourier Integral Theorem. e ikx a< x b State and prove final value theorem Using Z-transform solve 4un-un+2= 0 given that u0=0, u1=2,

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Contd…(2) Set No.1

Set No. Code No: RR-210101 II B.Tech. I-Semester Regular Examinations, November-2004 MATHEMATICS-II (Common to all Branches) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks --1.a)

b)

2.a) b)

3.a) b) 4.a)

b) 5.a) b) c)

2

Reduce the matrix  1 − 1 2 − 3   4 1 0 2  A=  to the normal form and hence determine its rank. 0 3 0 4    0 1 0 2    Determine whether the following equations will have a non-trivial solution if so solve them. 3x + 4y – z - 6ω = 0; 2x + 3y + 2z - 3ω = 0 2x + y – 14z - 9ω = 0; x + 3y + 13z + 3ω = 0. Show that A and AT have same eigen values but different eigen vectors. Find the eigen values and the corresponding eigen vectors of  − 2 2 − 3  1 − 6 A=  2  − 1 − 2 0   a + ic Show that A =   b + id

− b + id   is unitary matrix if a2 + b2 + c2 + d2 = 1. a − ic  1 − 3i  4 . Find the eigen values of  7  1 + 3i Write the Dirichlet’s conditions for the existence of Fourier series of a function f(x) in the interval ( α, α + 2π ) .4.a). Find the Fourier series representing f ( x ) = x, 0 < x < 2 π . Find a half range sine series for f ( x ) = ax + b, in 0 < x < 1 . Form the partial differential equation by eliminating the arbitrary function from xyz = f( x2 + y2 + z2) . Solve the partial differential equation z2 (p2 + q2) = x2 + y2 Solve the partial differential equation p tanx + q tan y = tan z

Code No. RR-210101 6.a) b)

7.a)

-2-

Solve by the method of separation of variables ux = 2ut + u where u(x,0) = 6e-3x. Find the temperature in a thin metal rod of length L, with both the ends insulated πx and with initial temperature in the rod is sin . L 2 1 − x if x <1 Find the Fourier transform of f ( x) =  if x >1  0 ∞

Hence evaluate

x  x cos x − sin x   cos 2 dx. 2 x  0

∫ 

cos x 0 < x < a b) Find Fourier cosine transform of f ( x) =  x≥a  0 8.a) b)

Contd…(2) Set No.2

Find the Z transform of sin (3n+5) z  −1  Find Z  2   z + 11z + 24 

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Set No. Code No: RR-210101 II B.Tech. I-Semester Regular Examinations, November-2004 MATHEMATICS-II (Common to all Branches) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks --1.

Find the values of λ for which the equations. (λ - 1)x + (3λ + 1)y + 2λz = 0 (λ - 1)x + (4λ - 2)y + (λ +3)z = 0 2x + (3λ + 1)y + 3(λ - 1) z = 0 will have a nontrivial solution and solve them for each value of λ.

2.

Verify cayley Hamilton theorem for the matrix 8 − 8 − 2   A =  4 − 3 − 2 3 − 4 1 

3

 i 0 0   Show that A= 0 0 i  is a skew-Hermitian matrix and also umitary 0 i 0 Find eigen values and the corresponding eigen vectors of A.

3.

4.a)

Find the Fourier series for f ( x ) ; if f ( x ) is defined in - π < x < π as − π, − π < x < 0 f ( x) =  0<x<π x, π2 1 1 1 = 2 + 2 + 2 + ........ 8 1 3 5 Find the half range cosine series f ( x ) = x ( 2 - x ) , in 0 ≤ x ≤ 2 and hence find the 1 1 1 1 sum of series 2 − 2 + 2 − 2 + ........ 1 2 3 4 Deduce that

b)

5.a) b) c) 6.a) b)

Form the partial differential equation by eliminating the arbitrary functions Z = f (x) + ey g (x). Solve the partial differential equation p x + q y = z Solve the partial differential equation x2p (y-z) + y2q (z – x) = z2 (x – y). 4ux + uy = 3u given u = 3 e-y –e-5y when x=0. Find the general solution of one-dimensional heat equation. Contd…(2)

Code No. RR-210101 7.a)

-2-

1 Find the Fourier transform of f ( x) =  0 ∞

sin x dx and and hence evaluate ∫ x 0 b)

8.a)

b)

Find Fourier sine transform of

for for

Set No.3 x a > 0

∞

sin as . cos xs ds . s −∞

∫

1 . x

z2 + z If Z(n )= , find Z(n3) and Z(n4) ( z − 1) 3 2

  z2   -1  ( z − 4)( z − 5)  Using convolution theorem find Z

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Set No.

4

Code No: RR-210101 II B.Tech. I-Semester Regular Examinations, November-2004 MATHEMATICS-II (Common to all Branches) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks --1.a) Find the non singular matrices P and Q such that PAQ is in the normal form of 2 − 1 3   the matrix and find the rank of matrix A= 1 1 1 1 − 1 1 b)

2.

For what values of η the equations x + y + z = 1; x + 2y + 4z = η; x + 4y + 10z = η2. are consistent and solve them completely in each case. Diagnolize the matrix  − 1 2 − 2 1 2 1   − 1 − 1 0 

3.a) b) 4.a) b)

5.a) b) c) 6.a) b)

1 1 − 1 1  1 −1 1 1   Show that the matrix is orthogonal.  1 1 −1 1    1 − 1 1 1 Reduce the quadutic from 3x2-2y2-z2-4xy+12yz+8xz to the canomical from. 1 ( π − x ) 2 , 0 < x < 2π . 4 Find the half range cosine series for the function f ( x ) = x 2 , in 0 ≤ x ≤ π and hence 1 1 1 1 find the sum of the series 2 − 2 + 2 − 2 + ........ 1 2 3 4 Obtain the Fourier series to represent f ( x ) =

Find the differential equation of all spheres of fixed radius having their centers on the xy-plane. Solve the partial differential equation (mz –ny) p +(ny-lx) q =lx-mx. Solve the partial differential equation z = p2 + q2. Solve the following equation by the method of separation of variables zxx – 2zx+zy = 0. Find the general solution of Laplace equation ∂2u/∂x2 + ∂2u/∂y2 = 0. Contd…(2)

Code No. RR-210101 7.a) b)

-2-

State and prove Parseval’s identity. Using Paseval’s identity prove that ∞

π  sin t  2 dt = .  t  2 0

∫ 

8.a) b)

Find Z ( n cos n ϕ ) if Z (n) =

z

( z − 1) 2 Solve the difference equation, using Z – transforms un+2-3un+1+2un=0 given that u0=0 u1=1 ^^^

Set No.4