Time : 3 hours
Maximum Marks : 60Answer question No.1 Compulsory
Answer ONE question from each Unit
1. Answer the following [12 x 1 = 12M]
a) Is the set of vectors (3, 2, 7), (2, 4, 1), (1,-2, 6) linearly dependent? Justify.
b) Find the Eigen values of the 2 x 2 matrix whose first row is (5, 4) and second row is (1, 2).
c) Define the Rank of a matrix.
d) For a given Hermitian matrix H whose first row is (0, i) and second row is (-i, 0) and the 1 x 2 matrix with the column (1, i) ,find the Hermitian form.
e) State the Rolle's theorem.
f) Find the critical points of the function f(x,y) = x3+y3-3axy.
g) What is the general solution of the Euler- cauchy equation x2y11+7xy1+13y=0.
h) Find the integrating factor for the differential equation y1-y/(x+1)=e3x(x+1).
i) State the necessary and sufficient condition for the differential equation Mdx + Ndy = 0 to be exact.
j) Find the solution of the equation y11+6y1+9y=0.
k) Find the particular integral of the differential equation (D2+5D+6)y=ex.
l) Define the order of a differential equation.
2. a) Using the Gauss. Jordan method, find the inverse of the 3 x 3 matrix with first row : (1, 1, 3) & second row : (1, 3 ,-3) & third row : (-2, -4, -4).
2. b) Test the consistency and solve the system of equations 2x-3y+7z=5, 3x+y-3z=13, 2x+19y-47z=32. (OR)
3. a) Find the rank of the 4 x 4 matrix with first row : (1, 2, 3, 0) & second row : (2, 4, 3, 2) & third row : (3, 2, 1, 3) & fourth row : (6, 8, 7, 5).
3. b) Find eigen values and eigen vectors of the 3 x 3 matrix with first row : (1, 1, 3) & second row : (1, 5, 1) & third row : (3, 1, 1).
4. a) What is the diagonal form after reduction of the 3 x 3 matrix with first row : (-1, 2, -2) & second row : (1, 2, 1) & third row : (-1, -1, 0).
4. b) Transform the quadratic form 7x12+6x1x2+7x22=0 to principle axes.Find the conic section represented by the quadratic form. (OR)
5. a) Using Maclaurin series expand tan(x) in a series of ascending powers of x as far as the term containing x5.
5. b) Find the maximum and minimum values of x3+y3-3x-12y+20.
6. a) Solve 3x(1-x2)y2y1+(2x2-1)y3=ax3.
6. b) Solve he differential equation (2x2+3y2-7)xdx-(3x2+2y2-8)ydy=0. (OR)
7. a) Solve the differential equation y1+xsin(2y)=x3cos2y.
7. b) If the air is maintained at 30ºC and the temperature of the body cools from 80ºC to 60ºC in 12 minutes, find the temperature of the body after 24 minutes.
8. a) Solve y11+2y1+4y=2x2+3e-x.
8. b) Solve by the method of variation of parameters y11-6y1+9y=2x2+3e-x. (OR)
9. a) Solve the differential equation y11-3y1+2y=xe3x+sin(2x).
9. b) Find the current I(t) in an RLC circuit with R=100 ohms, L=0.1 Henry, C=10-3 F which are connected to a source of voltage E(t)=155sin(377t) assuming zero charge and current when t=0.