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Answer EACH GROUP in a SEPARATE ANSWER SCRIPT.
GROUP A
(ALGEBRA III)
Answer ANY FOUR questions. (48=32)
1. (a) Let be a cyclic group of order where , are relatively
prime positive integers.Show that
m nG . (4)
(b)If and be two nontrivial subgroups of , , show that is also
nontrivial. Henceor otherwise show that , cannot be expressed as an
internal direct product of two
nontrivial subgroups. (4)
2. (a) Write the class equation of a finite group explaining all
the symbols used in the equation.Show that the center of a finite
group contains more than one element. (2+2)
(b)Show that every group of order 99 has an element of order 33.
(4)3. (a) Assuming Cauchys theorem for finite groups, establish the
converse of Lagranges
theorem for finite abelian groups. (4)
(b)Show that every group of order 35 is cyclic. (4)4. (a) Find
all irreducible cubic polynomials over 2 . (4)
(b)Obtain a field, with proper justification, a field of order
27. (4)5. (a) Define finite and algebraic extension. Show that
every finite extension is algebraic. (4)
(b)If , are algebraic over of degrees and respectively, where
and arerelatively prime, prove that , is of degree over . (4)
6. (a) State and prove the Cauchy Schwarz inequality in inner
product space. (4)(b)Find an orthonormal basis for 3 that contains
1 (1,2,0)
5. (4)
7. (a) Let KF
be a field extension and c K be algebraic over with minimal
polynomial .
Show that[ ]
[ ]( )
F xF c
m x . (4)
(b)Let be a polynomial of degree 2 or 3. Show that is
irreducible over ifand only if has no root in . Is the result true
for the polynomials of degree 4? Justify.
(4)
Of the questions attempted, the answers to only the first
required number of
questions (as stipulated in the question paper) will be
evaluated.
So, PLEASE DO NOT ATTEMPT EXTRA QUESTIONS.
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GROUP B
(PROGRAMMING IN C)
Answer ANY FOUR questions. (44=16)
8. (a) Subtract 2(01110001) from 2(11001101) using twos
complement method.(b)Convert
10
(35.3125) to binary form and16
( )ABC to decimal form. (2+1+1)
9. Draw a flowchart to evaluate the following infinite series 2
31 ...2! 3!
x xx+ + + + for a given value
of real correct upto 5 places of decimal. (4)
10.Write a C program to evaluate and print the roots of a
quadratic equation2 0, 0ax bx c a+ + = , where , , are taken as
user inputs. (3+1)
11.Let n nA be a given real matrix ( is taken as user input).
Write a C program to reduce it to adiagonal matrix and print it.
(3+1)
12.Write a C program to evaluate and print the standard
deviation of an array of real numberstaken as user input by writing
separate functions for calculating mean and standard
deviations.
(2+2)
13.(a) Write a C program to read two sets of coefficients (real
numbers) , , and , , andprint the solutions of the linear
system
ax by c
dx ey f
+ =
+ =
keeping provision for consistency test.
(b)Shown below is a Floyds triangle1
2 3
4 5 6
7 8 9 10
.
.
.79 . . . . . . 91
Write a C program to print this triangle. (2+2)
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