university, bhopal assignment question paper - Madhya Pradesh ...

M P BHOJ (OPEN) UNIVERSITY, BHOPAL 

ASSIGNMENT QUESTION PAPER 

2009-10 

CLASS : M.Sc. Final Year SUBJECT: Mathematics 

Paper - I - Integration theorey and Functional Analysis 

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First Assignment Max Marks - 30 

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. 

Second Assignment Max Marks - 30 

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.



2009-10 

CLASS : M.Sc. Final SUBJECT: Mathematics 

Paper - II - Partial differential Equations & Mechanics 

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ds vkSlr vad l=kar ijh{kk ifj.kke esa tksM+s tk,axsA lHkh iz'uksa ds vad leku gSaA 

Q1. Find the solution of ∂ 2 u/∂x 2 + 3∂ 2 u/∂x∂y + 2∂ 2 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) = X0 , Y′ (0) = V0. 

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 = p0 cos pt (p0 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 2 + ∂ 2 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



2009-10 


Paper - III - Operation Research 

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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 

W1 21 16 25 13 8 

W2 17 18 14 23 10 

W3 32 27 18 41 12 

DEMAND 20 15 15 30 20 

Q3. Use Branch & Bound technique to find an solution 

Max. Z = x 1 + 4 x2 

Sub. to 2x1 + 4x2 ≤ 7 

5x1 + 3x2 ≤ 15, x1 , x2 ≥ 0. 

Q4. Solve the following pay-off matrix , determine the optimal strategies and value of game 

A = 5 1 

3 4 

Q5. Solve the following by Simplex method to 

Minimize z = x1 – 3x2 + 2x3 

Sub. to 3x1 – x2 + 3x3 ≤ 7 

-2x1 + 4x2 ≤ 12 & x1, x2 ≥ 0 

Second Assignment Max Marks – 30 

P.T.O.

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



2009-10 


Paper - IV - Integral transform with applications 

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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



2009-10 


Paper - V - Programming in C (Theory & Practical) 

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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++ + (b35; 

(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).

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