### II Sem B.Sc/B.A. - Mathematics - Question Paper 2012

II Sem B.Sc/B.A. Examination, April/May 2012
(New Syllabus Scheme) (2011 - 12 Onwards)
MATHEMATICS (Paper - II)
(For Fresh Students of 2011-2012)

Time : 3 Hours                                                                             Max. Marks : 100

Instruction: Answer  all questions.

1. Answer any fifteen questions :                                                         (15x2=30)

1) Define an order of an element of a group and give an example.

2) If a is a generator of a cyclic group G, then prove that a-1 is also a generator.

3) Find the right cosets of the subgroup H={0,3} in Z6.

4) Find the index of the subgroup {0,3} of the group (Z6,+6).

5) If G is a finite group and a G, then prove that 0(a) divides 0(G).

6) Find the angle between the radius vector and the tangent for the curve r=a(1+sin0) at θ=π/6.

7) Show that for the curve r=a0, the polar sub tangent, varies as the square of the radius vector.

8) Find ds/dt for the curve x=a cos t, y = b sin t.

9) Find the radius of curvature for the curve xy=c2 at (x1, y1).

10) Write the formula for the radius of curvature at any point on acurve in polar form for the curve r=f(θ).

11) Find the singular point on the curve x3+ x2 + y2 - x - 4y + 3 = 0.

12) Find the envelope of the family of circles (x-a)2 + y2 = a2, where 'a' is a parameter.

13)Find the asymptotes parallel to the coordinate axes for the curve x2y2- a2x2 = a2y2.

14) Find the length of the curve y= log sec x from x = 0 to x =π/3.

15) Write the formula for the volume obtained by revolving the arc of the curve y=f(x) and the x-axis.

16) Solve : dy/dx + y=e-x.

17) Verify the exactness : (x2 - ay) dx + (y2 - ax) dy = 0.

18) Find the integrating factor of the equation dy/dx + (2/x)y = x log x.

19) Find the singular solution of y=px+p2.

20) Find theothogonal trajectories of the curve x2+y2=a2.

II. Answer any three questions :                                                         (3x5=15)

1) If the order of an element 'a' of a group G is 'n' and p is an integer prime to 'n' then prove that ap is also of order n.
2) Prove that every subgroup of a cyclic group is cyclic.
3) Prove that there is a one to one correspondence between the set of all distinct right cosets and the set a of all distinct left cosets of a subgroup of a group.
4) Find the number of generators of the cyclic group (C3+3). Write all the generators.
5) State and prove Langrange's theorem.

1. 