Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics

Internal Examination

Session 2020-21

M.Sc. (Mathematics) Semester - I

Paper IV (Advanced Complex Analysis)

Note: All questions are compulsory and carry equal marks. (Max Marks- 40)

1. State and prove Cauchy-Goursat theorem.

2. State and prove Schwarz’s lemma.

3. Apply calculus of residues to prove that

, <1

4. Find the bilinear transformation which maps the points z1 = 0, z2 = 1, z3= ∞

into the point w1 = 1, w2 = i, w3 = -1

5. State and prove Hurwitz’s theorem.

Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics

Internal Examination

Session 2020-21

M.Sc. (Mathematics) Semester - I

Paper -I (Advanced Abstract Algebra)

Note: All questions are compulsory and carry equal marks. (Max Marks- 40)

1. Prove that any two composition series of a finite group are equivalent.

2. Let H be a normal subgroup of a group G. If both H and G/H are solvable, then

G is also solvable.

3. Let G be a nilpotent group. Then every subgroup of G and every homomorphic

image of G are nilpotent.

4. If E is a finite extension of a field F, then E is an algebraic extension of F.

5. Show that every field F has an algebraic closure F.

Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics

Internal Examination

Session 2020-21

M.Sc. (Mathematics) Semester - I

Paper -II (Real Analysis)

Note: All questions are compulsory and carry equal marks. (Max Marks- 40)

1. State and prove Weierstrass M- test for uniform convergence.

2. State and prove Weierstrass approximation theorem.

3. State and prove Abel’s theorem.

4. State and prove Tauber’s theorem.

5. State and prove Riemann’s theorem.

Govt. Pt. J.L.N.PG College, Bemetara

Department of Mathematics

Internal Examination

Session 2020-21

M.Sc. (Mathematics) Semester- III

Paper III (Fuzzy theory & its application)

Note: All questions are compulsory and carry equal marks. (Max Marks- 40)

1. A fuzzy set A on R is convex iff

A( ) min [A( ) , A( ) ]

for all , R and all [0 , 1], where min denotes the minimum

operator.

2. Let R be a reflexive fuzzy relation on X2, where |X| = n Then RT(i)

= R(n-1).

3. Let a function c : [0 ,1] [0 ,1] satisfy axioms c2 and c4. Then, c also

satisfies axioms c1 and c3.Moreover, c must be a bijective function.

4. Explain max-min compositions with example.

5. For every A ∈P(X), any necessity measure, Nec, on P(X) and the

associated possibility measure, Pos, satisfy the following

implications:

A Nec A > 0 Pos A =

B Pos A < Nec A =0