Bangalore University 2008-5th Sem B.A I /BSc-mathematics-e - Question Paper
IV Semester B.A./B.Sc. Degree Examination, June 2008
(Semester Scheme)
MATHEMATICS (Paper-IV)
Time : 3 Hours Max. Marks : 90
Instructions :1) Answer all questions.
2) Answers should be written completely either in English or in Kannada.
I. Answer any fifteen of the following : (15x2=30)
1) Show that the subgroup H = { 1,-1 } of the multiplicative group G = { 1, -1, i, -i } is normal in G.
2) If G is a group and H is a subgroup of index 2 in a group G, show that H is normal subgroup of G.
3) Show that every quotient group of an abelian group is an abelian group.
4) Define a homomorphism of groups.
5) Show that f : G > G defined by f (x)=2x is not a homomorphism, where G is the multiplicative group of non-zero real numbers.
6) Prove that the function f (x, y) =
Xy for(x,y)*(0,0) .
x + y
0 for(x,y) = (0,0)
at (0, 0).
7) State Taylors theorem for a function of two variables.
8) Prove that there is a minimum value at (0, 0) for the function x3 + y3 - 3xy.
9) Prove that f x'2e~x dx = Jn .
o 4
10) Prove that r (n + 1) = n F(n).
11) Evaluate J yjcot 0 d0.
dv dy
12) Solve y-6+ 8y = 0. dx2 dx
13) Find the particular integral of (D2 - 6D + 13) y = 5e2x.
14) Show that the equation (1 + x2)y' + 3xy' + y = 1 + 3x2 is exact.
15) Verify the integrability condition of the equation yz dx - 2xz dy + (xy - zy3)
dz = 0.
16) Evaluate : L {e 1 Sin 2t }.
17) Evaluate : L '
S
(S + 2)2
18) Define Heavyside unit function.
19) Find all the basic solutions of the system of linear equations 3x + 2y + z = 22, x + y + 2z = 9.
20) Solve graphically the following system of inequalities.
2x + y > 3, x - 2y < -1, y < 3.
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 cosets of H in G is also a right coset of H in G.
2) Prove that a subgroup H of a group G is normal if and only if ghg-1 e H, for all g e G.he H.
3) If f : G > G' is a homomorphism and H is a subgroup of G, then prove that f (H) is a subgroup of G'.
4) Prove that every finite group is isomorphic to a permutation group.
SM - 223
-3-
III. Answer any three of the following :
(3x5=15)
1) Expand f (x, y) = ex cos y at the point (1, 4) using the Taylors theorem
upto that second degree term.
2) Find the extreme values of the function f (x, y) = x3 + y3 - 3x - 12y + 20.
3) Find the extreme values of x2 + y2 subject to the condition 2x2 + 3xy + 2y2 = 1.
|
m-l . n-1 |
 |
where m and n are both positive.
|
a 6 |
 |
 |
|
0 |
Evaluate
IV. Answer any three of the following :
(3x5=15)
1) Solve (D2 + 3D + 2) y = x2 + cos x.
2) Solve x2 4--x-2y = xlogx.
dx dx
3) Solve xV-x(x + 2)y' + (x + 2)y=xV. using e~x as a part of complimentary function.
4) Solve (x2 + l)y"- 2xy' + 2y = 6(x2 +1)2by the method of variation of parameters.
dx dy dz
5) Solve -=-=-.
mz - ny nx - Iz ly - mx
1) If L { f (t) } = F (s), then prove that L J } = f F(S) ds.
2) i) Evaluate : I/1
(S -13-
CO
, ii) Prove that f te~l Sint dt =
I . i 50
o 50
d~ d
J 3) Solve : + = &iven y(0) - 1- y'CO) - 0 by using Laplace
transform method.
VI. Answer any two of the following : (2x5=10)
1) Maximize Z = 2x + 3x2 + 4x3 + 7x4 subject to the constraints 2Xj + 3x2 - 4x3 + 4x4 = 8, xx - 2 + 6x3 - 7x4 = -3
Xj, x2, X3, x4 > 0.
2) The manager of an oil refinery must decide on the optimal mix of two possible blending process of which the inputs and outputs per production run as follows.
|
Process |
Input Units |
Output Units | |
|
Crude A Crude B |
Gasoline X |
Gasoline Y | |
|
1 |
5 3 |
5 |
8 |
|
2 |
4 5 |
4 |
4 |
The maximum amounts available of Crude A and B are 200 units and 150 units respectively. Market requirements show that atleast 100 units of gasoline X and 80 units of gasoline Y must be produced. The profit per production run from process 1 and process 2 are Rs. 300 and Rs. 400 respectively. Solve LPP by graphical method.
3) Solve the following L.P.P. by simplex method. Maximize : f = x - y + 3z. subject to the contraints x + y + z < 10, 2x - z < 2, 2x - 2y + 3z < 0, x, y, z > 0.
r
|
Attachment: |
| Earning: Approval pending. |
