Dibrugarh University Arts Question Papers: MATHEMATICS (Major) (Matrices, Ordinary Differential Equations, Numerical Analysis)' (May)

[BA 2nd Sem Question Papers, Dibrugarh University, 2015, Mathematics, Major, Matrices, Ordinary Differential Equations, Numerical Analysis]
2015 (May)
MATHEMATICS (Major)
Course: 201
(Matrices, Ordinary Differential Equations, Numerical Analysis)
Full Marks: 80
Pass Marks: 32/24
Time: 3 hours
The figures in the margin indicate full marks for the questions

A: Matrices

(Marks: 20)

1. (a) Under what condition, the rank of the following matrix

? 1

(b) Show that the rank of a skew-symmetric matrix cannot be one. 2

to its normal form and hence find its rank: 5

Find non-singular matrices

and

such that

is in the normal form where

2. (a) Under what condition, a system of

homogeneous linear equations

unknowns has only trivial solution? 1

(b) What is the eigenvalues of

, if eigenvalues of matrix

. 1

and

, the simultaneous equations.

Have (i) no solution, (ii) a unique solution and (iii) an infinite number of solutions. 4

(d) State and prove Cayley-Hamilton theorem. 6

Find the characteristic roots and associated characteristic vectors for the matrix.

B: Ordinary Differential Equation

(Marks: 30)

3. (a) Is the differential equation

exact? 1

(b) Find the integrating factor of the differential equation

(d) If

and

are any two solutions of

, then prove that the linear combination

, where

and

are constants, is also a solution of the given equation. 4

Show that linearly independent solutions of

are

and

. What is the general solution? Find the solution

with the property

4. (a) If auxiliary equations has tow equal pairs of imaginary roots, then what is the general solution of the second-order linear differential equation? 1

(b) What is the value of