govt. pt. j.l.n.pg college, bemetara · govt. pt. j.l.n.pg college, bemetara department of...
TRANSCRIPT
![Page 1: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/1.jpg)
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.
![Page 2: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/2.jpg)
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.
![Page 3: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/3.jpg)
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.
![Page 4: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/4.jpg)
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
![Page 5: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/5.jpg)
![Page 6: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/6.jpg)
![Page 7: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/7.jpg)
![Page 8: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/8.jpg)
![Page 9: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/9.jpg)
![Page 10: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/10.jpg)
![Page 11: Govt. Pt. J.L.N.PG College, Bemetara · Govt. Pt. J.L.N.PG College, Bemetara Department of Mathematics Internal Examination Session 2020-21 M.Sc. (Mathematics) Semester - I Paper](https://reader036.vdocuments.site/reader036/viewer/2022071505/61255c21ce278d27637bfa0f/html5/thumbnails/11.jpg)