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: Ifis a nonzero matrix, then rank. 1
(b) Define elementary matrix. Also find the rank of the matrixconsidering. 2
Reduce the following matrixto Echelon form and hence finds its rank:
2. (a) Write down the condition under which the system of equationspossesses 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 offor which the following equations have nonzero solution is 6: 3
(d) Show that the following system of equations is consistent and solve them completely: 2+4=6
State CayleyHamilton theorem. Show that the matrix
Satisfies CayleyHamilton 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
 , where
(d) Use Wronskian to show that the functionsare linearly independent. Determine the differential equation with these as independent solutions. 4
Or
Show that the Wronskian of the functionsand is nonzero. 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
Whereand are constant?
(b) Define linear homogeneous equation. 1
(c) Solve any two: 4x2=8
(d) Solve any two: 5x2=10
(By removing 1st order derivative)
(By changing the independent variable)
(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 NewtonRaphson method for solving an algebraic equation. 3
(d) Answer any two: 5x2=10
 Describe the regulafalsi method for obtaining a real root of an algebraic equation.
 By using NewtonRaphson method, find the root of, which is nearer to, correct to three decimal places by performing at least 3 iterative.
 Solve the following equations by Gauss elimination method:
6. (a) State True or False: Simpson’s onethird 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
 Deduce Lagrange interpolation formula.
 Estimate the missing term in the following table:
:

0

1

2

3

4

:

1

3

9

?

81

 Show that by dividing the range into 10 equal parts.