(A) Matrices
(Marks: 20)
1. (a) Write the rank of 1
(b) Show that every elementary matrix is non-singular. 2
(c) Reduce the matrix
into echelon from the hence find its rank.
Or
Reduce the matrix
to the normal form and hence find its rank.
2. (a) Write the number of linearly independent solutions of
homogeneous linear equations in
variables, 
where, 
. 1
homogeneous linear equations invariables, where, . 1
(b) Show that the equations 2
(c) Define the terms, ‘characteristic polynomial’ and ‘eigenvector’. 2+1=3
(d) Show that the matrix
satisfies Cayley-Hamilton theorem. 6
Or
Find the characteristic roots of the matrix
and verify Cayley-Hamilton theorem for
.
.
(B) Ordinary Differential Equations
(Marks: 30)
3. (a) Write the number(s) of integrating factors for an 
, where
and
are functions of 
and 
. 1
, whereandare functions of and . 1
(b) If
and if
when
, express
in terms of
. 2
when, expressin terms of. 2
(c) Solve (any one):
(d) Two solutions
and
of the equation
,
,
are linearly independent if and only if their Wronskian is not zero at some points 
. 4
andof the equation,,are linearly independent if and only if their Wronskian is not zero at some points . 4
Or
Show that
and
are linearly independent solutions of
. Find the solution
with the property that
and
.
andare linearly independent solutions of. Find the solutionwith the property thatand.
4. (a) Define linear differential equation with constant coefficients. What is meant by auxiliary equation of a linear differential equation with constant coefficients? 1+1=2
(b) Solve (any two): 4x2=8
(c) Solve (any two): 5x2=10
(By removing first-order derivative)
(By changing the independent variable)
(By the method of variation of parameters)
(C) Numerical Analysis
(Marks: 30)
5. (a) Write the order of convergence of the Newton-Raphson method. 1
(b) Write the condition of convergence for the iteration method of finding a real root of the equation
. 1
. 1
(c) Using Newton-Raphson method, establishes an iteration formula to find the square root of a positive number and hence compute the value of 
so that the result is correct to two places of decimal. 3
so that the result is correct to two places of decimal. 3
(d) Explain the regula falsi method of finding real root of an equation.
.
Or
Solve by Gauss elimination method:
(e) Solve by Gauss-Jordan method:
Or
Solve by Gauss-Seidel method:
6. (a) State ‘true’ or ‘false’: “Simpson’s 3/8th rule is more accurate than Simpson’s 1/3rd rule.” 1
(b) Prove that 
1
1
(c) Evaluate 
the interval of difference being unity.
(d) Show that 
where the symbols have their usual meaning. 2
where the symbols have their usual meaning. 2
(e) Deduce Newton’s forward interpolation formula for equidistant ordinates. 4
Or
Deduce the Simpson’s 1/3rd rule.
Also Read: Dibrugarh University Question Papers
(f) Using Newton’s divided difference formula, find the value of 
from the following table: 5
from the following table: 5 : |
4
|
5
|
7
|
10
|
11
|
13
|
 : |
48
|
100
|
294
|
900
|
1210
|
2028
|
Or
Calculate (up to 3 decimal places)
by dividing the range into eight equal parts.
***
Post a Comment
Kindly give your valuable feedback to improve this website.