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# Bangalore University 2008-5th Sem B.A I /BSc-mathematics-e - Question Paper

Thursday, 21 March 2013 10:25Web

SM - 223

IV Semester B.A./B.Sc. Degree Examination, June 2008

(Semester Scheme)

MATHEMATICS (Paper-IV)

Time : 3 Hours    Max. Marks : 90

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.

00 v    3

9)    Prove that f x'2e~x dx = Jn .

o    4

10)    Prove that r (n + 1) = n F(n).

Vi

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.

Ililllllllllll

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.

OR

 a 6

OR

 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.

I t J i

i s

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

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