R7210101 Mathematics Ii

1

Code No: R7210101

II B.Tech I Semester(R07) Regular/Supplementary Examinations, December 2009 MATHEMATICS-II (Common to Civil Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 

1 2

2

 1 2  3 1. (a) Find the rank of the matrix A =   −1 1 0 −1

−1 2 1 4

2

0 1 0 − 15

    

(b) Obtain the non tirvial solution of the system x + 3y − 2z = 0 2x − y + 4z = 0 x − 11y + 14z = 0

[8+8] 

4 2  2. Find the eigen values and eigen vectors of the matrix A =  −5 3 −2 4 Find the matrix P such that P−1 A P is diagonal.

 −2  2  1 [16]

3. (a) If A and B are square matrices of the same order and A is symmetric, then prove that BT AB is also symmetric.   0 2b c   (b) Determine a, b, c so that the matrix A =  a b −c  is orthogonal. a −b c (c) Determine the nature, index and signature of the quadratic form 2x1 x2 + 2x1 x3 + 2x2 x3 .

[4+6+6]

4. (a) Express f (x) = |x|, −π < x < π as fourier series (b) Find the half range cosine series of f(x)=x in 0 < x < π

[8+8] ¡ ¢¡ ¢ 5. (a) Form the partial differential equation by eliminating the arbitrary constants a and b from z = x2 + a y 2 + b (b) Solve the PDE : (y-z)p+(z-x)q = x-y (c) Solve the PDE : p2 + q 2 = x2 + y 2

[4+6+6]

6. Use the separation of variables technique to solve ∂u −y 4 ∂u − e−5y when x=0. ∂x + ∂y = 3u given u = 3e ( a2 − x2 if |x| < a 7. (a) Find the fourier transform of f(x) = 0 if |x| ≥ a (b) Find the fourier cosine transform of e−ax sin ax.

[16]

[8+8]

8. (a) Using the linearity property , find the Z- transforms of the following i. (n-1)2 ii. sin(3n+5) (b) Using Z-transform solve un+2 - 2un+1 + un = 2n with u0 = 2, u1 = 1. ?????

[8+8]

2

Code No: R7210101

II B.Tech I Semester(R07) Regular/Supplementary Examinations, December 2009 MATHEMATICS-II (Common to Civil Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 

1 1 2 1

  1. (a) Find the rank of the matrix A =  

2 3 4 1

3 3 3 1

1 2 3 1

    by reducing to Normal form 

(b) Test for consistency and solve, the equations 2x1 + 6x2 + 11 = 0 6x1 + 20x2 − 6x3 + 3 = 0 6x2 − 18x3 + 1 = 0

[8+8]

2. (a) If λ is an eigen value of a square matrix A, show that λ1 is the eigen value of A−1 .   1 3 7   (b) Show that the matrix A =  4 2 3  satisfies its characteristic equation and hence obtain A−1 . 1 2 1

[8+8]

3. Reduce the quadratic form 4x2 + 3y 2 + z 2 − 8xy − 6yz + 4xz to the cononical form. What is the matrix of the transformation? [16] 4. (a) A function f(x) is given by f (x) = 1 + 2x when −π ≤ x ≤ 0 π = 1 − 2x when 0 ≤ x ≤ π π Draw a rough sketch and obtain the fourier series expansion of f(x) (b) Find the fourier sine series of eax in (0,π).

[8+8]

5. (a) Obtain the partial differential equation by eliminating the arbitrary function f and g from z=f(2x+y) + g(3x-y) (b) Solve the PDE : x2 p2 + y 2 qz = z 2 2

[8+8] 2

∂ y 6. Solve completely the equation ∂∂t2z = c2 ∂x 2 representing the vibrations of a string of length L, fixed at both ends, given that y(0,t)=0; y(x,0)=f(x) and ∂y(x,0) = 0, 0 < x < L [16] ∂t

7. (a) Using fourier integral formula show that e−ax =

2a π

R∞ cos λ x 0

λ2 +a2 dλ , a

(b) If F(s) is the complex fourier transform of f(x), then prove that F{f(x)cosax} = 1/2 {F(s+a)+F(s-a)}

> 0, x ≥ 0

[8+8]

8. (a) Using damping rule, find the Z-transforms of the following: i. ean ii. nean iii. n2 ean (b) Use convolution theorem to evaluate Z

−1

½³

z z−1

´3 ¾

?????

[8+8]

3

Code No: R7210101

II B.Tech I Semester(R07) Regular/Supplementary Examinations, December 2009 MATHEMATICS-II (Common to Civil Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 

4 1 K 9

  1. (a) Find the value of K such that A =  

 −3 1 −1 0    may have a rank ‘3’. 2 2  K 3

4 1 2 9

(b) Test for consistency and solve the equations. x+y+z=6 -x + 2y - 4z = -9 2x + 3y - 2z = 2.

[8+8] "

2. (a) Verify Cayley Hamilton theorem for the matrix A = Hence Find A−1 .

Ã

(b) Diagonalize the matrix A =

0 1 3 2

1 2

2 −1

# .

! . Hence Find A3 .

[8+8]

3. (a) If A is orthogonal matrix, Prove that |A| = ±1 (b) Reduce the quadratic form 2x1 x2 + 2x2 x3 + 2x3 x1 into cononical form. Classify the quadratic form. [8+8] 4. (a) Find the fourier series of the function f (x) = x2 when 0 ≤ x ≤ π = −x2 when −π ≤ x ≤ 0 (b) Represent the following function by fourier sine series f (x) = 1 when 0 < x < 2l = 0 when 0 < x < l

[8+8] h

5. (a) Form the partial differential equation by eliminating the arbitrary constants a, b from Z = a log

b(y−1) 1−x

i

(b) Solve ³ 2 ´the partial differential equation y z p + x zq = y 2 x (c) Find the complete integral of p + q = p q 2

[5+6+5]

2

6. Solve ∂∂xu2 + ∂∂yu2 = 0 which satisfies the conditions u(0, y) = u(l, y) = u(x, 0) = 0 and u(x, a) = f (x) 7. Find the fourier sine transform and fourier cosine transform of f(x) = xa−1 (0
[16] √1 . x

is self [16]

8. (a) Find i. Z ii. Z

©1ª nn

1 (n+1)

(b) Find Z −1

n

o

z (z−1)(z 2 +1)

o [8+8] ?????

4

Code No: R7210101

II B.Tech I Semester(R07) Regular/Supplementary Examinations, December 2009 MATHEMATICS-II (Common to Civil Engineering and Bio-Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 

2 3 4 5

  1. (a) Find the rank of the matrix A =  

3 −1 −5 2

−2 2 6 0

5 0 −5 5

1 4 7 5

    

by applying elementery transformations. (b) Solve the system of non-homogenous equations x1 + x2 + x3 + x4 = 0 x1 + 3x2 + 2x3 + 4x4 = 0 2x1 + x3 − x4 = 0

[8+8] Ã

2. (a) Explain eigen value problem of a square matrix A. Verify whether à ! 1 4 the matrix 3 2

1 1

!

à and

1 2

! are eigen vectors of

(b) Find the eigen values  and the corresponding eigen vectors of the matrix −3 −7 −5   A= 2 4 3  1 2 2

[8+8]

2

2

∂ y 3. Solve completely the equation ∂∂t2z = c2 ∂x 2 representing the vibrations of a string of length L, fixed at both ends, given that y(0,t)=0; y(x,0)=f(x) and ∂y(x,0) = 0, 0 < x < L [16] ∂t

4. Find the fourier series to represent the function f (x) = |sin x| valid for −π < x < π. Hence find the sum of the 1 1 1 infinite series 1.3 + 3.5 + 5.7 + ... [16] 5. (a) Form the PDE by eliminating the arbitrary function from the relation ¡ ¢ z = x2 + y 2 + f x2 − y 2 (b) Solve the PDE : p3 + q 3 = 3pqz

[8+8]

6. If the string of length l is initially at rest in equilibrium position and each of its points is given the velocity ¡ ¢ ¡ ¢ uo sin 3πx cos 2πx where 0 < x < l at t=0. Determine the displacement function y(x ,t). [16] l l 7. (a) Find thefourier integral representation of the function  o , x < 0  1 F (x) = ,x = 0 2   −x e ,x rel="nofollow"> 0 (b) If F(s) complex fourier transform of f(x), then prove that F{f(ax)} = 1/a F(s/a), a 6= 0. 8. (a) If u(z) =

2z 2 +3z+12 (z−1)4 ,

[10+6]

find the value of u2 and u3 .

(b) Find inverse Z-transform of

4z 2 −2z z 3 −5z 2 +8z−4

[8+8] ?????