V Semester B.Sc./B.A Examination, October/November 2011
(Semester Scheme)(NS)
MATHEMATICS (Paper - V)
Time: 3 Hours Max.Marks: 60
Instructions : Answer all questions.
I. Answer any fifteen of the following. Each sub-question carries 2 marks. (15x2=30)
1) In a Ring (R, +,,), prove that (-a)(-b)=ab for a,b ER.
2) Give an example of
a) a ring with zero divisions.
b) a ring without zero divisions.
3) Define left ideal and right ideal of a Ring R.
4) Define a Field. Give an example.
5) Prove that the intersections of two subrings of a ring is subring.
6) If R is a ring and aER such that r(a)={xeR|ax=0 } is a right ideal.
7) If F=ti=t2j+sin t.k find dr/dt and d2r/dt2 at t=0.
8) If r=(coswt)i+(sinwt)j show that d2r/dt2=-w2r.
9) Show that a necessary condition for a curve to lie on a plane is that r=0 at all points.
10) For a curve x=t, y=t2 and z=2t3/3 find the unit tangent vector at t=1.
11) Find spherical co-ordinates of a point whose Catesian co-ordinates are (3/2, 3/3,-1)
12) If 0 (x,y,z)=xy2+yz3. Find v- at (2,-1,1).
13) Find the constant a so that F=(x+3y)i+(y+2z)+(x-az)k.
14) Prove that div (Curl F)=0.
15) Show that V2(1/r)=0 where r>xi+yk+zk.
16) Show that F=(siny+z)i+(xcosy-z)j+(x-y)k is irrotational.
17) Show that 2x-3x3=1/5P1(x)-6/5P3(x).
18) Show that Pn(1)=n(n+1)/2.
19) Show that J-a(x)=(-1)aja(x).
20) Show that J1/2(x)=2/rx x sinx.
II. Answer any four questions. Each question carries five marks : (4x5=20)
1) Prove that R = {0, 1, 2, 3, 4} is a ring under addition modulo 5 and multiplication modulo 5
2) Define the centre of a ring and prove that centre of a ring is a sub-ring of R.
3) Prove that every finite integral domain is a field.
4) Find all principal ideals of the ring R = {0, 1, 2, 3, 4, 5} with respect to addition modulo 6 and multiplication modulo 6.
(Semester Scheme)(NS)
MATHEMATICS (Paper - V)
Time: 3 Hours Max.Marks: 60
Instructions : Answer all questions.
I. Answer any fifteen of the following. Each sub-question carries 2 marks. (15x2=30)
1) In a Ring (R, +,,), prove that (-a)(-b)=ab for a,b ER.
2) Give an example of
a) a ring with zero divisions.
b) a ring without zero divisions.
3) Define left ideal and right ideal of a Ring R.
4) Define a Field. Give an example.
5) Prove that the intersections of two subrings of a ring is subring.
6) If R is a ring and aER such that r(a)={xeR|ax=0 } is a right ideal.
7) If F=ti=t2j+sin t.k find dr/dt and d2r/dt2 at t=0.
8) If r=(coswt)i+(sinwt)j show that d2r/dt2=-w2r.
9) Show that a necessary condition for a curve to lie on a plane is that r=0 at all points.
10) For a curve x=t, y=t2 and z=2t3/3 find the unit tangent vector at t=1.
11) Find spherical co-ordinates of a point whose Catesian co-ordinates are (3/2, 3/3,-1)
12) If 0 (x,y,z)=xy2+yz3. Find v- at (2,-1,1).
13) Find the constant a so that F=(x+3y)i+(y+2z)+(x-az)k.
14) Prove that div (Curl F)=0.
15) Show that V2(1/r)=0 where r>xi+yk+zk.
16) Show that F=(siny+z)i+(xcosy-z)j+(x-y)k is irrotational.
17) Show that 2x-3x3=1/5P1(x)-6/5P3(x).
18) Show that Pn(1)=n(n+1)/2.
19) Show that J-a(x)=(-1)aja(x).
20) Show that J1/2(x)=2/rx x sinx.
II. Answer any four questions. Each question carries five marks : (4x5=20)
1) Prove that R = {0, 1, 2, 3, 4} is a ring under addition modulo 5 and multiplication modulo 5
2) Define the centre of a ring and prove that centre of a ring is a sub-ring of R.
3) Prove that every finite integral domain is a field.
4) Find all principal ideals of the ring R = {0, 1, 2, 3, 4, 5} with respect to addition modulo 6 and multiplication modulo 6.
Question paper requested by: Ambika
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