Mathematics Ii Nov2004 Or210156

OR

Code no:210156

II B.Tech I-Semester Supplementary Examinations. November, 2004 MATHEMATICS-II (Common to all branches) Time:3 hours

1.a)

Max.Marks:70

Answer any FIVE questions All questions carry equal marks. --Find the rank of the matrix by reducing it to the normal form. 1 1 1 1  1 3 − 2 1    2 0 − 3 2   3 3 0 3 

b) 2.

3.a) b)

Find whether the following system of equations are consistent, if so solve them. 3x+y+2z=3, 2x-3y-z=-3, x+2y+z=4. Find the eigen values and the corresponding eigen vectors of the following matrix. −2 2 2 1 1 1   1 3 − 1  If λ1, λ2 - - - λn are eigen values of a matrix of A of order n, then prove that kλ1, kλ2- - -kλn are eigen values of the matrix KA. 2 2 2 Reduce the quadratic form 3 x1 + 3x 2 + 3 x3 + 2 x1 x 2 + 2 x1 x3 − 2 x 2 x3 to the cannonical form.

4.a) b)

Obtain the Fourier series for f(x) = x2 in [0,2π]. Obtain half range cosine series for f(x) = x in [0,π].

5.a)

Form the partial differential equation by eliminating the arbitrary constants.

b) 6. a) b)

z = (x-a)2 + (y-b)2 + 1 Solve the partial differential equation. z2 (p2 + q2 + 1) = 1 Solve the following partial differential equations. x2 p2 + xpq = z2 y2 z p + z x2 q = xy2 Contd…….2

Code no:210156

-2-

7.

Solve the following Linear partial differential equations.

a)

(D

2

+ 2 DD 1 + D 12 z = e 2 x +3 y

b)

(D

2

+ DD 1 + D 12 z = cos (2 x + 3 y )

8.

Solve the heat equation.

)

)

∂u ∂ 2u under the conditions = a2 ∂t ∂x 2 u(0,t)=0, u(π,t)=0, u(x,0)= x2, 0<x<π. ***********