2015
(May)
MATHEMATICS
(Major)
Course: 201
(Matrices, Ordinary Differential Equations, Numerical Analysis)
Full Marks: 80
Pass Marks: 32/24
Time: 3 hours
The figures in the margin indicate full marks for the questions
A: Matrices
(Marks: 20)
1. (a) Under what condition, the rank of the following matrixis ? 1
(b) Show that the rank of a skewsymmetric matrix cannot be one. 2
(c) Reduce the matrixto its normal form and hence find its rank: 5
Or
Find nonsingular matricesandsuch thatis in the normal form where
2. (a) Under what condition, a system ofhomogeneous linear equationsinunknowns has only trivial solution? 1
(b) What is the eigenvalues of, if eigenvalues of matrixis. 1
(c) Investigate for what values ofand, the simultaneous equations.
Have (i) no solution, (ii) a unique solution and (iii) an infinite number of solutions. 4
(d) State and prove CayleyHamilton theorem. 6
Or
Find the characteristic roots and associated characteristic vectors for the matrix.
B: Ordinary Differential Equation
(Marks: 30)
3. (a) Is the differential equationexact? 1
(b) Find the integrating factor of the differential equation 2
(c) Solve (any one): 3
(d) Ifandare any two solutions of, then prove that the linear combination, whereand are constants, is also a solution of the given equation. 4
Or
Show that linearly independent solutions ofareand. What is the general solution? Find the solution with the property.
4. (a) If auxiliary equations has tow equal pairs of imaginary roots, then what is the general solution of the secondorder linear differential equation? 1
(b) What is the value of if? 1
(c) Solve (any two): 4x2=8
(d) Solve (any two): 5x2=10
(by removing firstorder derivative)
(by changing the independent variable)
(by applying the method of variation of parameters)
C: Numerical Analysis
(Marks: 30)
5. (a) Fill in the blank: 1
Ifis continuous in the intervaland ifandare of opposite signs then the equation will have ____ real root betweenand. 1
(b) What is the length of the subinterval which contains after bisections?
(c) Using regula falsi method, find the first approximate value of the root of the equation that lies between 2.5 and 3. 3
(d) Answer (any two): 5x2=10
 Describe NewtonRaphson method for obtaining the real roots of the equation.
 Apply GaussJordan method, to find the solution of the following system:
 Solve by GaussSeidel method
6. (a) State Trapezoidal rule. 1
(b) Show that, where the symbols have their usual meanings. 2
(c) Evaluate 2
(d) Answer (any two): 5x2=10
 Deduce the Simpson’s onethird rule.
 The population of a town is as follows:
Year (x):

1941

1951

1961

1971

1981

1991

Population (in lakhs)(y):

20

24

29

36

46

51

Estimate the population increase during the period 1946 to 1976.
 Evaluate by trapezoidal rule.
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