SECfiON'A'
Q. l. Answer any five of the following:
M and N are normal subgroups of a group G such that M n N show that every element of M commutes with every element of N.
Show that is a prime element in the ring R of Gaussian integers.

If are the roots of the equation in lamda and x/a+lamda+y/b+lamda+z/c+lamda=1, evaluate <img src='./qimages/1144-1c.jpg'>

Evaluate <img src='./qimages/1144-1d.jpg'> The integral being extended over all positive values of z such that

If u+iv is an analytic function of the complex variable z and u-v =ex(cosy-siny),determine in terms of z
(f)Put the following program in standard form:
Minimize z 25x1 30X2
subject to 4x1 7x2 l
8x1 5x2
6x1 9x2
and hence obtain an initial feasible solution.
Let H and K be two subgroups of a finite group G such that root|G| and root|G|. Prove that H intersection K not equal to

If f is an isomorphism, prove that the order of a belongs to G is equal to the order of
Prove that any polynomial ring F over a field F is a

Q. If and exist for every x belongs to and if does not vanish anywhere in(a,b),show that there exists c in such that
<img src='./qimages/1144-3a.jpg'>

Show that <img src='./qimages/1144-3b.jpg'> is an improper integral which converges for n>0.
Q.4.(a)expand in Laurent's series which is valid for
3
and

Use simplex method to solve the following Maximize Z 5x1 2x2 subject to 6x1 x2 6
4x 1 3x2 12
X1 2X2>=4
2 4
and x1,x2 0.
SECTION'B'
Q.5.Answer any jive of the following

Formulate partial differential equation for surfaces whose tangent planes form a tetrahedron of constant volume with the coordinate planes.
Find the particular integral of
x p y q z
which represents a surface passing through x=y=z.

(c)Use appropriate quadrature formulae out of the Trapezoidal and Simpson's rules to numerically integrate <img src='./qimages/1144-5c.jpg'> with h 0.2 Hence obtain an approximate value of pie. Justify the use of a particular quadrature formula.
Find the hexadecimal equivalent of (41819)10 and decimal equivalent of(111011.10)2
A rectangular plate swings in a vertical plane about one of its corners. If its period is one second, find the length of its diagonal.
(f)Prove that the necessary and sufficient condition for vortex lines and stream lines to be at right angles to each other is that <img src='./qimages/1144-5f.jpg'> where mew and are functions of z and t.

Q.6.(a) The ends A and B of a rod 20 em long have the temperature at 30°C and at 80°C until steady state prevails. The temperature of the ends are changed to 40°C and 60°C respectively. Find the temperature distribution in the rod at time t.
Obtain the general solution of
<img src='./qimages/1144-6b.jpg'>

Q.7.(a) Find the unique polynomial of degree 2 or less such that 27, 64. Using the Lagrange interpolation formula and the Newton's divided difference formula, evaluate P(1.5).
Draw a flow chart and also write a program in BASIC to find one real root of the non linear equation x pie(x) by the fixed point iteration method. Illustrate it to find one real root, correct up to four places of decimals, of x3 5 0.
Q.8.(a) A plank of mass is initially at rest along a line of greatest slope of a smooth plane inclined at an angle alpha to the horizon, and a mass of mass starting from the upper end walks down the plank so that it does not move. Show that he gets to the other end in time <img src='./qimages/1144-8a.jpg'> where a is the length of the plank.
State the conditions under which Euler's equations of motion can be integrated. Show that
<img src='./qimages/1144-8b.jpg'>
where the symbols have their usual meaning.