## Friday, April 20, 2012

### BANGALORE UNIVERSITY BSc, IV SEMESTER MATHEMATICS MODEL QUESTION PAPER-1

BANGALORE UNIVERSITY
BSc, IV SEMESTER MATHEMATICS
MODEL QUESTION PAPER-1
Time: 3 Hours                                                                                               Max. Marks: 90
I. Answer any fifteen of the following                                                                     2x15=30
1.      Prove that every subgroup of an abelian group is normal.
2.      Prove that intersection of two normal subgroups of a group is also a normal subgroup.
3.      The center Z of a group G is a normal subgroup of G.
4.      Define a homomorphism of groups.
5.      If G={x+y√2|x,yєQ} and f:G→G is defined by f(x+y√2)=x-y√2, show that f is a homomorphism and find its kernel.
6.      Show that f(x,y)= √|xy| is not differentiable at (0,0).
7.      Show that f(x,y)=tan -1[y x) at (1, 1) has limit.
8.      Prove that there is a minimum value at (0,0) for the fuctions x3+y3-3xy.
9.      Show that  πо2cos10 0d0=½β (11  1
2,  2)
10.  Prove that ┌(n+1)=n!
11.  Prove that о∞2 √xe-x2 dx=½|→3/4
12.  Find the particular integral of y11-2y`+4y=ex cos x.
13.  Show that x(2x+3)y11+3(2x+1)y1+2y=(x+1)ex is exact.
14.  Verify the integrability condition for yzlogzdx-zxlog zdy+xydz=0.
15.  Reduce x2 y11-2xy11+3y=x to a differential equation with constant coefficients.
16.  Find ∟[sin3t].
17.  Find {∟-1 (s+2) (s-1)}
18.  Define convolution theorem for the functions f(t) and g(t).
19.  Find all basic solutions of the system of equations: 3x+2y+z=22, x++y+2z+9.
20.  Solve graphically x+y≤3, x-y ≥-3, Y ≥0, x≥-1, x≤2.
II. Answer any two of the following                                                                            2x5=10
1.      Prove that a subgroup H of a group G is a normal subgroup of G if and only if the product of two right coses of H in G is also a right coset of H in G.
2.      Prove that the product of two normal subgroups of a group is a normal subgroup of the group.
3.      If G and G1 are groups and F:G→G1 is a homomorphism with kernel K, prove that K is a normal subgroup of G.
4.      If G=(Z6, +6), G1=(Z2, +2) and the function f: G→G1 is defined by f(x)=r where r is the remainder obtained by dividing x by 2, then verify whether f is homomorphism. If so, find its kernel. Is f an isomorphism?
III. Answer any three of the following                                                                            3x5=15
1.      State and prove Taylor’s theorem for a function of two variables.
2.      Find Maclaurin’s expansion of log (1+x-y).
3.      Find the stationary points of the function f(x,y)=x3y2(12-x-y) satisfying the condition x>0 and examine their nature.
4.      Show that оx4(1+x5) dx =    1
(1+x)15             5005
OR
If n is a positive integer, prove that ┌1(m+1/2)= 1.3.5.......(2n-1) √π
2n
5. Show that оπ/2 √sin0d0. 1 sin0 do  + оπ/2          1  2n
√sino      d0
OR

Evaluate о  dx
01+x4
IV. Answer any three of the following                                                                            3x5=15
1.      Solve y11 1-2y11+4y=ex cosx.
2.      Solve x3y111+2x2y+2y11+10 (x+1/x).
3.      Solve d2y  -(1+4ex) dy   + 3e2xy=e2(x+ex) using changing the independent variable method.
dx2                dx
4.      Solve dx/dt+3x - y, dy/dt=x+y
5.      Solve dx x2+y2+yz = dy/x2+y2-zx = dz/z(x+y)
V. Answer any two of the following                                                                            2x5=10
1.      Find (i) ∟{2sin st sin 5t/t (ii) ∟-1log s2+1 s(s+1)}
2.      Verify convolution theorem for the functions (t)=et and g(t)=cost.
3.      Solve y11+2y1=10sin tsy given y(0)zo, y(0)=1 using Laplace transform method.
VI. Answer any two of the following:                                                                       2x5=10
1.      Find all the basic feasible solutions to the LPP:
Maximize: z=2x+3y+4z+7t
Subject to the constrains: 2x+3y-z+4t=8, x-2y+6z-7t=-3 x,y,z,t >0.
2.      A quality engineer wants to determine the quantity produced per month of products A and B.
 Source Product A Product B Available month Material Working hours Assembly man hours 60 8 3 120 5 4 12,000 600 500 Sale price Rs. 30 Rs. 40 -

Find the product mix that give maximum profit by graphical method.
3.      Using Simplex and method to maximize f=5x+y+4z subject to x+z<8, y+z<3 x+y+z<5.