## Friday, January 04, 2019 2015
(May)
MATIEMATICS
(General)
Course: 201
(Matrices, Ordinary Differential Equations and Numerical Analysis)
Full Marks: 80
Pass Marks: 32/24
Time: 3 hours
The figures in the margin indicate full marks for the questions
GROUP – A
(Matrices)
(Marks: 20)

1. (a) Define nullity of a matrix. 1
(b) Prove that the rank of transpose of a matrix is same as that of the original matrix. 3
(c) Find the rank of the following matrix by reducing it to echelon form: 4 2. (a) Show that the following equations are consistent and find their solutions: 5 Or
Solve: (b) Find the characteristic polynomial of the following square matrix: 2 (c) Show that every square matrix satisfies its own characteristic equation. 5
Or
Determine the characteristic roots and corresponding characteristic vectors of the following matrix:       2+3=5 GROUP – B
(Ordinary Differential Equations)
(Marks: 30)

3. (a) Write the necessary condition for the equation to be an exact differential equation. 1
(b) Write the integrating factor of the equation 1
(c) Define Wronskian of functions. 2
(d) Solve any one:
1. 2. (e) Solve any one:
1. 2. 4. (a) Solve any two: 3x2=6
1. 2. 3. Given, (b) Solve any one: 4
1. 2. 5. Answer either [(a) and (b)] or (c):
(a) If the equation reduces to by removing the first-order derivative, then write the value of 1
(b) Removing the first-order derivative, solve the following equation: 4 (c) Apply the method of variation of parameter to solve the following equation: 5 6. Transform the equation 5 By changing the independent variable; where , andare the functions of .
Or
If is a particular solution of find its general solution.

GROUP – C
(Numerical Analysis)
(Marks: 30)

7. (a) Write True or False: 1
In solving an equation by Newton-Raphson method, the derivative of the function should not be zero.
(b) Find a real root of the following equation by bisection method correct to two places of decimal: 5 Or
Describe iteration method for solving an algebraic equation.
(c) Obtain a formula to compute the square root of a number using Newton-Raphson method. 3
(d) Solve by Gauss elimination method: 6 Or
Describe the solution of system of linear equations by Gauss-Seidel method.
8. (a) Define interpolation. 1
(b) With usual notations, show that 2
(c) Deduce Newton’s backward interpolation formula. 5
Or
Given: Find , by using any method of interpolation.
9. (a) Find the general quadrature formula for equidistant ordinates and deduce the trapezoidal rule. 3+2=5
Or
Find the value of by Simpson’s.
(b) Find by using Lagrange’s interpolation formula from the following table: 2 : 0 1 2 5 : 2 3 12 147

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