[BA 2nd Sem Question Papers, Dibrugarh University, 2012, Mathematics, General, Matrices, Ordinary Differential Equations, Numerical Analysis]
2012 (May)
MATHEMATICS (General)
Course: 201
(Matrices, Ordinary Differential Equations, Numerical Analysis)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
1. (a) Define rank of a matrix. 1
(b) If
be an
-rowed square matrix; then write the characteristic equation for the Matrix
. 1
be an-rowed square matrix; then write the characteristic equation for the Matrix. 1
2. (a) Find the rank of the following matrix by reducing it to normal form: 3
(b) Prove that the rank of the transpose of a matrix is same as that of the original matrix. 3
3. (a) Show that the following equations are consistent and solve:
(b) State Cayley-Hamilton theorem and verify it with the following matrix: 2+5=7
Or
Determine the characteristic roots and corresponding characteristic vectors of the following matrix:
GROUP – B
(Ordinary Differential Equations)
4. (a) What is the condition that the equation
be an exact differential equation? 1
be an exact differential equation? 1
(b) Write down the Bernoulli’s equation. 1
(c) Find the complementary function of the differential equation
1
1
5. Solve any two: 6x2=12
6. Solve any one: 5
7. Apply the method of variation of parameters to solve the following equation: 5
8. Solve: 2+3=5
, where
GROUP – C
(Numerical Analysis)
9. (a) Define interpolation. 1
(b) With usual notation, prove that
1
1
(c) State True or False: The iterative methods are based on the principle of successive approximation. 1
10. Apply Gauss-Jordan method to solve: 7
Or
Discuss the bisection method for solution of an algebraic equation.
11. (a) Determine
by using Newton-Raphson method, using four iterations. 7
by using Newton-Raphson method, using four iterations. 7
(b) Deduce Newton’s forward interpolation formula. 7
12. (a) Derive Simpson’s one-third rule for numerical integration. 4
Or
Find
by trapezoidal rule.
by trapezoidal rule. Also Read: Dibrugarh University Question Papers
(b) Find 
by using Lagrange’s interpolation formula from the following table: 2
by using Lagrange’s interpolation formula from the following table: 2 : |
0
|
1
|
2
|
5
|
 : |
1
|
3
|
12
|
147
|
***
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