### 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**

*Answer all questions.*

**Instruction:**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=

*c*

^{2}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

*x*

^{3}+ x

^{2}+

*y*

^{2}- x - 4y + 3 = 0.

12) Find the envelope of the family of circles (

*x-a*)

^{2}+

*y*

^{2}=

*a*

^{2}, where 'a' is a parameter.

13)Find the asymptotes parallel to the coordinate axes for the curve

*x*

^{2}

*y*

^{2}-

*a*

^{2}

*x*

^{2}=

*a*

^{2}

*y*

^{2}.

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 : (

*x*

^{2}- ay) dx + (

*y*

^{2}- 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+p

^{2}.

20) Find theothogonal trajectories of the curve

*x*

^{2}+

*y*

^{2}

*=a*

^{2}.

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 a

^{p}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 (

*C*

_{3}+

_{3}). Write all the generators.

**5**) State and prove Langrange's theorem.

Please upload Business Mathematics repeater question paper of 2015,Please help me.

ReplyDelete