# Dibrugarh University Arts Question Papers: MATHEMATICS (Matrices, Ordinary Differential Equations and Numerical Analysis)' (May) - 2014

### 2014 (May)MATHEMATICS (General)Course: 201(Matrices, Ordinary Differential Equations and Numerical Analysis)Full Marks: 80Pass Marks: 32Time: 3 hoursThe figures in the margin indicate full marks for the questions

1. (a) Define elementary matrix. 1
(b) Find the rank of the following matrix by reducing it to normal form: 4
(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 Cayley-Hamilton theorem for the following matrix: 5
Or
Show that the inverse of a non-singular matrix can be computed with the help of the Cayley-Hamilton theorem.

GROUP – B
(Ordinary Differential Equations)
(Marks: 30)

3. (a) Write the standard form of linear equation of first-order 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 first-order 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 Newton-Raphson 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 Newton-Raphson 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 Gauss-Jordan 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

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