# 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)=2^{x} 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 x^{3} + y^{3} - 3xy.

9) Prove that f x'^{2}e~^{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. dx^{2} dx

13) Find the particular integral of (D^{2} - 6D + 13) y = 5e^{2x}.

14) Show that the equation (1 + x^{2})y' + 3xy' + y = 1 + 3x^{2} is exact.

15) Verify the integrability condition of the equation yz dx - 2xz dy + (xy - zy^{3})

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) = e^{x} 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) = x^{3} + y^{3} - 3x - 12y + 20.

3) Find the extreme values of x^{2} + y^{2} subject to the condition 2x^{2} + 3xy + 2y^{2} = 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 (D^{2} + 3D + 2) y = x^{2} + cos x.

2) Solve x^{2} 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 (x^{2} + l)y"- 2xy' + 2y = 6(x^{2} +1)^{2}by 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 : ^{+ =} &i^{ven} y(0) - 1- y'CO) - 0 by using Laplace

transform method.

VI. Answer any two of the following : (2x5=10)

1) Maximize Z = 2x + 3x_{2} + 4x_{3} + 7x_{4} subject to the constraints 2Xj + 3x_{2} - 4x_{3} + 4x_{4} = 8, x_{x} - 2 + 6x_{3} - 7x_{4} = -3

Xj, x_{2}, X_{3}, x_{4} > 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. |