## Sunday, April 29, 2018

2015
(May)
MATHEMATICS
(Major)
Course: 603
(Algebra II and Partial Differential Equations)
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions.
A: Algebra II
(Marks: 40)
1. (a) Choose the correct answer for the following: 1
Isomorphism of a group onto itself is called

1. Automorphisms.
2. Homomorphism.
3. Endomorphism.
4. Ideal.
(b) Suppose a group G is an internal direct product of its subgroups H and K, then write the element which is common to both H. and K. 1
(c) Let G be a group and and automorphisms of G. If , then prove that . 3
(d) Prove that if G is a group, their Aut (G), the set of automorphisms of G is also a group under composition of functions. 4
(e) Let , be normal subgroups of a group . Then prove that is an inner direct product of and if an only if (i) and (ii) 4
Or
If such that is an automorphisms, where is some fixed integer, then show that for all , is the centre of group 2. (a) Write the binary compositions needed to form a ring in a non-empty set. 1
(b) Write an example of a commutative ring without unity. 1
(c) If R is a ring such that  then prove that . 3
(d) Show that a Boolean ring is commutative. 4
(e) Prove that a commutative ring R is an integral domain if and only if for all 4 Or
Show that intersection of any two left ideals of a ring is again a left ideal of the ring.
3. (a) Write the another name of the ring of all equivalent classes. 1
(b) Let be an onto homomorphism, where is a ring with unity. Show that is unity of 3
(c) If be an onto homomorphism, then prove that is isomorphic to a quotient ring of .     5
Or
Prove that any ring can be imbedded into a ring with unity.
(d) Let be a commutative ring with unity. Prove that an ideal of is maximal ideal of if is a field.    5
Or
Show that in a Boolean ring , every prime ideal is maximal.

B: Partial Differential Equations
(Marks: 40)
4. (a) Write the number of arbitrary constants that the solution of the partial differential equation , where possesses. 1
(b) Write the auxiliary equations of the equation . 1
(c) Solve any two equations: 4x2=8
1. 2. 3. (d) Solve any one equation: 5
1. 2. (e) Find the integral surface of the linear partial differential equation. 5 which contains the straight line? 5. (a) In Charpit’s method for solving a partial differential equation, a second equation is introduced. Write the order of the equation. 1
(b) Write the complete integral of the equation 1
(c) Write the Charpit’s auxiliary equations for the equation . 2
(d) Find complete integrals of any three of the following: 4x3=12
1. 2. 3. 4. (e) Find a complete integral of using Jacobi’s method. 4