2014
(May)
MATHEMATICS
(General)
Course: 201
(Matrices, Ordinary Differential Equations and 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 elementary matrix. 1
(c) Show that 3
rank= rank
2. (a) Show that the following equations are consistent and solve them: 3
(b) Define characteristic roots and characteristic vectors of a square matrix. 1+1=2
(c) Verity CayleyHamilton theorem for the following matrix: 5
Or
Show that the inverse of a nonsingular matrix can be computed with the help of the CayleyHamilton theorem.
GROUP – B
(Ordinary Differential Equations)
(Marks: 30)
3. (a) Write the standard form of linear equation of firstorder differential equation. 1
(b) Solve any one: 3
(c) Solve any one: 3
(d) Prove thatandare solutions of the differential equationand these solutions are linearly independent. 3
4. (a) Solve any two: 3x2=6
(b) Solve any one: 4
5. (a) Removing the firstorder derivative, solve the following equation: 5
Or
Solve by changing the independent variable:
(b) Apply the method of variation of parameters to solve the equation. 5
, where,andare functions of.
Or
If is a particular solution of
GROUP – C
(Numerical Analysis)
(Marks: 30)
6. (a) Write the condition of convergence of NewtonRaphson method. 1
(b) Describe bisection method for solving an algebraic equation. 4
Or
Find one root of the following equation by iterative method:
(c) Apply NewtonRaphson method to find the cube root of 25. 5
Or
Find the real root of the equation, by regula falsi method, correct to three places of decimal.
(d) Describe the solution of system of linear equations by Gauss elimination method. 5
Or
Solve by GaussJordan method:
7. (a) Show that 2
(b) Deduce Newton’s forward interpolation formula. 5
Or
Given
:

1

2

3

4

5

6

7

8

:

1

8

27

64

125

216

343

512

Then find
(c) Obtain the value ofby using trapezoidal rule from the relation: 5
Or
Given
:

1

3

4

6

10

:

0

18

48

180

900

Then find the value ofby using Lagrange’s interpolation formula.
(d) Deduce Simpson’srule. 3