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Dibrugarh University Arts Question Papers:MATHEMATICS (Matrices, Ordinary Differential Equations and Numerical Analysis)' (May)-2013


2013
(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
GROUP – A
(Matrices)


1. (a) If is a non-singular matrix of order, what will be its rank? 1
(b) Find the rank by row transformations of the following: 3
(c) Find two non-singular matricesandsuch thatis in normal form, where 3
2. (a) Define characteristic polynomial of square matrix.
(b) For what values ofthe equations
are consistent?
(c) (Solve):
Or
Show that every spare matrix satisfies its characteristic equation.
(d) Determine the eigenvalues and eigenvectors: 4
GROUP – B
(Ordinary Differential Equations)


3. (a) Define an exact equation. 1
(b) Solve any one: 3
(c) Solve any one: 3
(d) Using Wronskian, examine whether the functionsare linearly independent or dependent.
4. (a) Solve any one: 5
(b) Solve any one: 5
(c) Solve any one: 5
5. (a) Answer any one: 5
  1. Ifis a particular solution of, find the general solution.
  2. Removing the first-order derivative, solve the following: 5
(b) Answer any one: 5
  1. Transform the equation, whereare functions ofby changing the independent variable.
  2. Solve by changing the independent variable:
  3. Apply the method of variation of parameters to solve:


GROUP – C
(Numerical Analysis)


6. (a) Describe regula falsi method for solving an algebraic equation. 5
Or
Solvefor the root betweenandby bisection method.
(b) Describe Newton-Raphson method and interpret it geometrically. 5
Or
Find a real root of the following equation by iteration method:
(c) Solve by Gauss-Jordan method: 5
7. (a) Deduce Newton’s backward interpolation formula. 5
Or
The population of a country was as follows:
:
1901
1911
1921
1931
1941
Population in (thousand):
46
66
81
93
101


Estimate the population for the year 1925, using Newton’s backward interpolation formula.
(b) Deduce Lagrange’s interpolation formula for unequal intervals. 5
Or
Given,
Find,using Lagrange’s interpolation formula.
(c) Find the general quadrature formula for equidistant ordinates and deduce the trapezoidal rule. 3+2=5
Or
Find an approximate value ofby taking seven equidistant ordinates by using Simpson’s rule and trapezoidal rule.


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