## Friday, January 04, 2019 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 matrix is ? 1 (b) Show that the rank of a skew-symmetric matrix cannot be one. 2
(c) Reduce the matrix to its normal form and hence find its rank: 5 Or
Find non-singular matrices and such that is in the normal form where 2. (a) Under what condition, a system of homogeneous linear equations in unknowns has only trivial solution? 1
(b) What is the eigenvalues of , if eigenvalues of matrix is . 1
(c) Investigate for what values of and , the simultaneous equations. Have (i) no solution, (ii) a unique solution and (iii) an infinite number of solutions. 4
(d) State and prove Cayley-Hamilton 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 equation exact? 1
(b) Find the integrating factor of the differential equation 2
(c) Solve (any one): 3
1. 2. (d) If and are any two solutions of , then prove that the linear combination , where and are constants, is also a solution of the given equation.    4
Or
Show that linearly independent solutions of are  and  . 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 second-order linear differential equation? 1
(b) What is the value of if? 1
(c) Solve (any two): 4x2=8
1. 2. 3. (d) Solve (any two): 5x2=10
1. (by removing first-order derivative)
1. (by changing the independent variable)
1. (by applying the method of variation of parameters)

C: Numerical Analysis
(Marks: 30)

5. (a) Fill in the blank: 1
If is continuous in the interval and if and are of opposite signs then the equation will have ____ real root between and . 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
1. Describe Newton-Raphson method for obtaining the real roots of the equation .
2. Apply Gauss-Jordan method, to find the solution of the following system: 1. Solve by Gauss-Seidel method 6. (a) State Trapezoidal rule. 1
(b) Show that , where the symbols have their usual meanings. 2
(c) Evaluate 2
1. Deduce the Simpson’s one-third rule.
2. 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.
1. Evaluate by trapezoidal rule.

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