## Friday, January 04, 2019 2014
(May)
MATHEMATICS
(Major)
Course: 201
(Matrices, Ordinary Differential Equations, Numerical Analysis)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions

1. (a) State True or False: If is a non-zero matrix, then rank . 1
(b) Define elementary matrix. Also find the rank of the matrix considering . 2
(c) Find the rank of the following matrix reducing it into normal form: 5 Reduce the following matrix to Echelon form and hence finds its rank: 2. (a) Write down the condition under which the system of equations possesses a unique solution.    1
(b) Show that a characteristic vector of a matrix cannot correspond to more than one characteristic value of .     2
(c) Show that the only real value of for which the following equations have non-zero solution is 6:     3 (d) Show that the following system of equations is consistent and solve them completely: 2+4=6 State Cayley-Hamilton theorem. Show that the matrix Satisfies Cayley-Hamilton theorem. 1+5=6

(B) Ordinary Differential Equations
(Marks: 30)
3. (a) Write True or False:
“The singular solution of a differential equation in Clairaut’s from contains only one arbitrary constant.” 1
(b) Find the integrating factor of the differential equation. 2 (c) Solve any one: 3
1. , where 2. (d) Use Wronskian to show that the functions   are linearly independent. Determine the differential equation with these as independent solutions. 4
Or
Show that the Wronskian of the functions and is non-zero. Can these functions be independent solutions of an ordinary differential equation? If so, determine this differential equation.
4. (a) What is the auxiliary equation of the differential equation. 1 Where and are constant?
(b) Define linear homogeneous equation. 1
(c) Solve any two: 4x2=8
1. 2. 3. (d) Solve any two: 5x2=10
1. (By removing 1st order derivative)
1. (By changing the independent variable)
1. (By the method of variation of parameters)

(C) Numerical Analysis
(Marks: 30)
5. (a) State True or False: The bisection method always converges. 1
(b) Write the basic difference between the bisection method and method of false position. 1
(c) Explain the geometrical interpretation of the Newton-Raphson method for solving an algebraic equation.     3
1. Describe the regula-falsi method for obtaining a real root of an algebraic equation.
2. By using Newton-Raphson method, find the root of , which is nearer to , correct to three decimal places by performing at least 3 iterative.
3. Solve the following equations by Gauss elimination method: 6. (a) State True or False: Simpson’s one-third rule is better than the trapezoidal rule. 1
(b) Evaluate the interval of differencing being . 2
(c) Show that , where the symbols have their usual meanings. 2
(d) Answer any two of the following: 5x2=10
1. Deduce Lagrange interpolation formula.
2. Estimate the missing term in the following table: : 0 1 2 3 4 : 1 3 9 ? 81
1. Show that by dividing the range into 10 equal parts.

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