Time : 3 hours

Maximum Marks : 60

Answer question No.1 CompulsoryAnswer 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) = x

^{3}+y

^{3}-3axy.

g) What is the general solution of the Euler- cauchy equation x

^{2}y

^{11}+7xy

^{1}+13y=0.

h) Find the integrating factor for the differential equation y

^{1}-y/(x+1)=e

^{3x}(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 y

^{11}+6y

^{1}+9y=0.

k) Find the particular integral of the differential equation (D

^{2}+5D+6)y=e

^{x}.

l) Define the order of a differential equation.

**[1 x 12 = 12M]**

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).

**[1 x 12 = 12M]**

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 7x

_{1}

^{2}+6x

_{1}x

_{2}+7x

_{2}

^{2}=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 x

^{5}.

5. b) Find the maximum and minimum values of x

^{3}+y

^{3}-3x-12y+20.

**[1 x 12 = 12M]**

6. a) Solve 3x(1-x

^{2})y

^{2}y

^{1}+(2x

^{2}-1)y

^{3}=ax

^{3}.

6. b) Solve he differential equation (2x

^{2}+3y

^{2}-7)xdx-(3x

^{2}+2y

^{2}-8)ydy=0.

**(OR)**

7. a) Solve the differential equation y

^{1}+xsin(2y)=x

^{3}cos

^{2}y.

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.

**[1 x 12 = 12M]**

8. a) Solve y

^{11}+2y

^{1}+4y=2x

^{2}+3e

^{-x}.

8. b) Solve by the method of variation of parameters y

^{11}-6y

^{1}+9y=2x

^{2}+3e

^{-x}. (OR)

9. a) Solve the differential equation y

^{11}-3y

^{1}+2y=xe

^{3x}+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.

good post

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