2013
(May)
MATHEMATICS
(Major)
Course: 201
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions
(A) Matrices
(Marks: 20)
1. (a) Write the rank of 1
(b) Show that every elementary matrix is nonsingular. 2
(c) Reduce the matrix
into echelon from the hence find its rank.
Or
Reduce the matrix
to the normal form and hence find its rank.
2. (a) Write the number of linearly independent solutions ofhomogeneous linear equations invariables, where, . 1
(b) Show that the equations 2
(c) Define the terms, ‘characteristic polynomial’ and ‘eigenvector’. 2+1=3
(d) Show that the matrix
satisfies CayleyHamilton theorem. 6
Or
Find the characteristic roots of the matrix
and verify CayleyHamilton theorem for.
(B) Ordinary Differential Equations
(Marks: 30)
3. (a) Write the number(s) of integrating factors for an , whereandare functions of and . 1
(b) If
and ifwhen, expressin terms of. 2
(c) Solve (any one):
(d) Two solutionsandof the equation,,are linearly independent if and only if their Wronskian is not zero at some points . 4
Or
Show thatandare linearly independent solutions of. Find the solutionwith the property thatand.
4. (a) Define linear differential equation with constant coefficients. What is meant by auxiliary equation of a linear differential equation with constant coefficients? 1+1=2
(b) Solve (any two): 4x2=8
(c) Solve (any two): 5x2=10
(By removing firstorder derivative)
(By changing the independent variable)
(By the method of variation of parameters)
(C) Numerical Analysis
(Marks: 30)
5. (a) Write the order of convergence of the NewtonRaphson method. 1
(b) Write the condition of convergence for the iteration method of finding a real root of the equation. 1
(c) Using NewtonRaphson method, establishes an iteration formula to find the square root of a positive number and hence compute the value of so that the result is correct to two places of decimal. 3
(d) Explain the regula falsi method of finding real root of an equation.
Or
Solve by Gauss elimination method:
(e) Solve by GaussJordan method:
Or
Solve by GaussSeidel method:
6. (a) State ‘true’ or ‘false’: “Simpson’s 3/8th rule is more accurate than Simpson’s 1/3rd rule.” 1
(b) Prove that 1
(c) Evaluate
the interval of difference being unity.
(d) Show that where the symbols have their usual meaning. 2
(e) Deduce Newton’s forward interpolation formula for equidistant ordinates. 4
Or
Deduce the Simpson’s 1/3rd rule.
(f) Using Newton’s divided difference formula, find the value of from the following table: 5
:

4

5

7

10

11

13

:

48

100

294

900

1210

2028

Or
Calculate (up to 3 decimal places)
by dividing the range into eight equal parts.