## Saturday, January 05, 2019 2016
(November)
MATHEMATICS
(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 rank of a matrix. 1
(b) Find the rank of the following matrix by reducing it to normal form: 4 (c) Show that – 3
rank 2. (a) Show that the following equations are consistent and find their solution: 5 (b) Define characteristic roots and characteristic vectors. 2
(c) State and prove Cayley-Hamilton theorem. 5
Or
Determine the characteristic roots and corresponding characteristic vectors of the following matrix: 5 GROUP – B
(Ordinary Differential Equations)
(Marks: 30)
3. (a) Write the standard form of second-order linear differential equation. 1
(b) Solve any one: 3
1. 2. (c) Solve any one: 3
1. 2. , where (d) Using the transformation , reduce the following equation to a linear differential equation: 3 4. (a) Solve any two: 3x2=6
1. 2. 3. (b) Solve any one: 4
1. 2. 5. Answer any two questions: 5x2=10
1. Removing the first-order derivative, solve the following equation: 1. Solve by changing the independent variable: 1. Apply the method of variation of parameter to solve the equation where , and are the functions of .
GROUP – C
(Numerical Analysis)
(Marks: 30)
6. (a) Write True or False: 1
“Bisection method is always convergent”.
(b) Describe Newton-Raphson method for solving an algebraic equation. 4
Or
Find one root of the equation by using iteration method. 4
(c) Use of Regula Falsi method to obtain a root of correct to three decimal places 4
Or
Find the real root of the equation using bisection method. 4
(d) Solve by Gauss-Jordan method: 6 Or
Describe Gauss elimination method to solve the system of linear equations. 6
7. (a) Define operator . 1
(b) Evaluate: 2 (c) Show that – 2 (d) Deduce Lagrange’s interpolation formula. 5
Or
Given – : 1 2 3 4 5 6 7 8 : 1 8 27 64 125 216 434 512
Then find (e) Derive Simpson’s one-third rule for numerical integration. 5
Or
Find by trapezoidal rule.

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