## Friday, January 04, 2019

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) Ifbe 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 equationbe an exact differential equation? 1
(b) Write down the Bernoulli’s equation. 1
(c) Find the complementary function of the differential equation 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
1. , where

GROUP – C
(Numerical Analysis)

9. (a) Define interpolation. 1
(b) With usual notation, prove that 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) Determineby 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
Findby trapezoidal rule.
(b) Find by using Lagrange’s interpolation formula from the following table: 2
 : 0 1 2 5 : 1 3 12 147

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