ANU B.Tech 1st Semester Mathematics January 2013 Question Paper

University: Acharya Nagarjuna University I/IV B.Tech Degree Examinations, January 2013 First Year First Semester Mathematics

Time : 3 hours

Maximum Marks : 60

Answer 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)=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 y¹¹+6y¹+9y=0.
k) Find the particular integral of the differential equation (D²+5D+6)y=e^x.
l) Define the order of a differential equation.

UNIT - I [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).

UNIT - II [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 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.

UNIT - III [1 x 12 = 12M]
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.

UNIT - IV [1 x 12 = 12M]
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=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. 

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