M P BHOJ (OPEN) UNIVERSITY, BHOPAL ASSIGNMENT QUESTION PAPER

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1 CLASS : M.Sc. Final Year SUBJECT: Mathematics Paper - I - Integration theorey and Functional Analysis 1- lhkh iz- Lo;a dh glrfyfi esa gy djuk vfuok;z gsa 2- nksuksa l=h; iz-i= gy djuk vfuok;z gsa ds vkslr vad l=kar ijh{kk ifj.kke esa tksm+s tk,axsa lhkh iz-ksa ds vad leku gsaa Q.1 State and prove Hahn decompodition theorem. OR State and prove Radon-Nikodym theorem. Q.2 If a real valued continuous function f on x is st. the set N(f) ={ x:f(x) 0} is σ -bounded then f is Baire measurable. OR State and prove Riesz- Markoff theorem. Q.3 Show that the real linear space R & the complex linear space c are Banach space under the norm x = x x ε c or R OR Define Dual spaces and give an example. Q.4 State and prove Boundedness theorem. OR State and prove Riesz lemma. Q.5 If x is inner product space then x,y (x,x) (y,y) x,y ε x i.e. x,y x. y OR An operator T on Hilbert space H is said to be self adjoint it T*=T. Q.1 State and prove Fubini s theorem. OR State and prove Extension theorem. Q.2 Let µ* be a topologically regular outer measure on x then suc Barel set is µ* measur able. OR A finite disjoint union of inner regular set of finite measure is inner regular. Q.3 Let x 1 and x 2 be two normed linear space of same finite dimention n with same scalor field then x 1 and x 2 are topological isomorphism. OR For a bounded linear transformation T the follouing are equivalent - i) T = { T(x) : x 1} T = sup{ T(x) : x = 1} Q.4 State and prove uniform bounded theorem. OR State and prove Hahn Banech theorem for complex linear space. Q.5 If x is an inner product space tan (x,x) has the property of a norm. OR Show that the unitory operators on a Hilbert space H form a group.

2 Paper - II - Partial differential Equations & Mechanics Q1. Find the solution of 2 u/ x u/ x y u/ y 2 = 0. Q2. Solve y - 3y + 3y - y = t 2 e t, where y(0) = 1, y (o) = 0 & y (0) = -2, by using Laplace transform, where y = y/ x. Q3. A particle of mass m moves along the x axis & is attracted towards origin O with a force numerically equal to kx, k > 0. A damping force given by β dx/ dt, β > 0 also acts. Discus the motion treating all cases, assuming that X(0) = X 0, Y (0) = V 0. Q4. A tightly streached flexible string has its end fixed at x = 0 & x = l. At time t = 0 the string has given a shape defined by F(x) = µx ( l x), where µ is a constant and then released. Find the displacement of any point x of the string at any time t > 0. Q5. Find the solution of the wave equation 2 y/ t 2 = c 2 2 y/ x 2 Such that y = p 0 cos pt (p 0 is a constant) when x = l and y = 0 when x = 0. Q1. Solve u/ t = 2 2 u/ x 2, u(0, t) = 0, u(5,t) = 6, u(x, 0) = 10sin4πx. Q2. Solve 2 z/ x z/ y 2 = cosmx cosny Q3 A string is stretched between the fixed points (0, 0) & (1, 0) & released from rest from the position y(x,0) = Asin2πx.. Find the displacement y(x, t). Q4. Find the temperature u(x, t) in a bar of length l, which is perfectly insulated whose ends are already kept at temp. zero & initial temp. is x, 0 < x < l/2 F(x) = l-x, l/2 < x <l Q5. State and prove Lee Hwa Chung s theorem.

3 Paper - III - Operation Research Q.1. A Company produces two kinds of leather belts A and B. A is of superior quality and B is of lower quality. The respective profits are Rs. 10 and Rs. 5 per belt. The supply of raw material is sufficient for making 850 belts per day. For belt A special type of buckle is required and 500 are available per day. There are 700 buckles available for belt B per day. Belt A needs twice as much time as that required for belt b. & company can produce 500 belts of all of them were of type A. Formulate LPP & solve it graphically. Q2. Solve following transportation problem & test the optimality by MODI method. F1 F2 F3 F4 SUPPLY W W W DEMAND Q3. Use Branch & Bound technique to find an solution Max. Z = x x 2 Sub. to 2x 1 + 4x 2 7 5x 1 + 3x 2 15, x 1, x 2 0. Q4. Solve the following pay-off matrix, determine the optimal strategies and value of game A = Q5. Solve the following by Simplex method to Minimize z = x 1 3x 2 + 2x 3 Sub. to 3x 1 x 2 + 3x 3 7-2x 1 + 4x 2 12 & x 1, x 2 0 Second Assignment Max Marks 30 P.T.O.

4 From Pre Page Q1. Show that the collection of all feasible solutions to LP problems constitutes a conver set whose extreme pairts correspond to the basic feasible solutions. Q2. State the priiciple of optionality in dynamic programming and give a methematical formalation of a dynamic programming problem. Q3. What is scpoe of O.R. in daily life. Q4. Define convex programming. Q5. Write short notes on the following (a) Network simplex method (b) Game Theory

5 Paper - IV - Integral transform with applications Q1. Find the Laplace transform of t 2, 0 < t < 2 F(t) = t 1, 2 < t < 3 7, t > 3 Q2. Solve ( t D 2 + (1-2t) D 2) y = 0, where y(0) = 1, y (0) = 2. Q3. An alternating EMF esinωt is applied to an inductance L & a capacitance C in series. Show that the current in the circuit is { eω / (n 2 - ω 2 ) L} (cos ωt cosnt), where n 2 = 1/ LC. Q4. Find the fourier series for the periodic function f(x) defined by - π where - π < x < 0 f(x) = x 0 < x < π Q5. A string is stretched between the fixed points (0,0) & (1,0) & released at rest from the position u(x, 0) = A sin 2πx. Find the displacement u(x, t). Q1. Using Laplace transform solve d 2 y/dx 2 + y = 0, under the condition that y = 1, dy/dt = 0, when t = 0. Q2. Find L -1 {log(s 2 +1/s(s+1)) Q3. Find the surface satisfying t = 6x 3 y, containing two lines y = 0 = z, y =1 = z using partial differential equation. Q4. Find the Fourier half range cosine series of the function 2t, 0< t <1 F(t) = 2(2-t), 1 < t <2 Q5. Find thefourier cosine transform of f(x) = e -mx, m> 0. Q6. Use Fourier transform to solve boundary value problem u/ t = k 2 u/ x 2, u(x, 0) = f(x), u(x,t) <M, where - < x <, t > 0.

6 Paper - V - Programming in C (Theory & Practical) Q.1 write a program to print series of prime no. from 1 to n. Where n is a user defined no. Q.2 Write a program to add digits of a no. given by the user. Q.3 write a program to print given no. in reverse order without using array. Q.4 Write a program for reversing an array. Q.5 Give characterstics of good programming? Q.6 write short notes on (i) OOPS; (ii)tokens. Q.7 What will be the value of a & b after the execution of the following statement: a=a a + b++ + (b<8 && a==6); {WHERE:- (a=10;b=5)} Q.8 write a program to check whether a given word is palendrome. Q.1 write a program to print first 10 no.s of a fabonacci series. Q.2 write a program to print the given structure(diamond shape), using loop Q.3 write a program : where an insurance company provides insurance to its imployees according to the following criteria:- (i) If the employee is married. (ii) If the employee is not married & male, age>35; (iii)if the employee is not married & female, age> 30. In all other cases insurance is denied. Age, gender & martial status is given by user & check wheather he or she is eligible for insurance or not. Usintg neste if & logical operators. Q.4 Take two number in two variables and swap them by using functions (by passing address of the wwo no.s in the function).

M P BHOJ (OPEN) UNIVERSITY, BHOPAL ASSIGNMENT QUESTION PAPER

M P BHOJ (OPEN) UNIVERSITY, BHOPAL ASSIGNMENT QUESTION PAPER 2009-10 CLASS : PGDCI SUBJECT :... PAPER : I- Basic of Chemoinformatics funsz'k %& 1- lhkh iz'u Lo;a dh glrfyfi esa gy djuk vfuok;z gsa 2- nksuksa