## Friday, January 04, 2019 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 non-singular. 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 of homogeneous linear equations in variables, 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 Cayley-Hamilton theorem. 6
Or
Find the characteristic roots of the matrix and verify Cayley-Hamilton theorem for .

(B) Ordinary Differential Equations
(Marks: 30)

3. (a) Write the number(s) of integrating factors for an , where and are functions of and .  1
(b) If and if when , express in terms of . 2
(c) Solve (any one):
1. 2. (d) Two solutions and of the equation , , are linearly independent if and only if their Wronskian is not zero at some points . 4
Or
Show that and are linearly independent solutions of . Find the solution with the property that and .
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
1. 2. 3. (c) Solve (any two): 5x2=10
1. (By removing first-order derivative)
1. (By changing the independent variable)
1. (By the method of variation of parameters)

(C) Numerical Analysis
(Marks: 30)

5. (a) Write the order of convergence of the Newton-Raphson method. 1
(b) Write the condition of convergence for the iteration method of finding a real root of the equation .     1
(c) Using Newton-Raphson 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 Gauss-Jordan method:   Or
Solve by Gauss-Seidel 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.

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