## Friday, January 04, 2019 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 matrices and such that is in normal form, where 3 2. (a) Define characteristic polynomial of square matrix.
(b) For what values of the 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
1. 2. (c) Solve any one: 3
1. 2. (d) Using Wronskian, examine whether the functions are linearly independent or dependent.
4. (a) Solve any one: 5
1. 2. 3. 4. (b) Solve any one: 5
1. 2. 3. (c) Solve any one: 5
1. 2. 3. 5. (a) Answer any one: 5
1. If is a particular solution of , find the general solution.
2. Removing the first-order derivative, solve the following: 5
1. Transform the equation , where are functions of by 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
Solve for the root between and by 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 of by taking seven equidistant ordinates by using Simpson’s rule and trapezoidal rule.

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