PAPER : STATISTICS-I

Reg. No. : PART – A

Answer ALL the Questions: (20x1=20) Choose the correct Answer: Answer

1. The arithmetic mean and geometric mean of 4 numbers 2,4,6,27 are respectively___________. a) 6;9.75 b) 9.75;6 c) 8;9 d) 9;8 2. The coefficient of variation of a frequency distribution is

a) x

b) 100

x

c) 100x

d)

x

3. The range of 20,22,27,30,40,48,45,32,31,35 is _____________. a) 28 b) 32 c) 30 d) 20 4. The mode of the set of number 63,65,66,65,64,65,65,61,67,68 is a) 64 b) 65 c) 66 d) 68 5. For a frequency distribution given that its mean is 120, mode is 123 and karl pearson's coefficient of skewness is 0.3 then the coefficient of variation

is_______________.

a) 112

b) 112

c) 8.33 d) 8.33

6. 12

=____________

a) 12 1

b) 2

c) 2

12 1

d) 11

7. 1 =_____________

a) 1 b)

1

c) 2

112

d) 11

8. 2 __________

a) 422

b)

2332

c) 2233

d) none of these

9. If 1bxy then bxy

a) >1 b) =1 c) <1 d) <2

ST. XAVIER'S COLLEGE (AUTONOMOUS), PALAYAMKOTTAI - 627 002.

SEMESTER EXAMINATIONS – NOVEMBER – 2016 15UMTA31

B.Sc. MATHS SEMESTER - III

PAPER : STATISTICS-I Time : 3 Hrs. MAX. MARKS: 100

10. If the variables x and y are uncorrelated then cov( , )x y is

a) 1 b) 0 c) -1 d) 2 11. For rank correlation coefficient the correct statement is

a) 1 1 b) 1 1

c) 0 1 d)

12. If 1 in a set of bivariation data then there exists

a) Perfect and positive correlation b) Perfect and negative correlation c) No correlation d) Perfect correlation 13. For the two attributes A and B the 2x2 contingency table is given

B Non B

A a b

Non A c d

Then the expected frequency of the attribute AB is

a) a b b d

a b c d

b)

a b a c

a b c d

c) c d a c

a b c d

d)

c d b d

a b c d

14. Given n attributes ,then total number of positive class frequencies is

a) 2n b) 3n

c) 54 d) 45

15. From the 2x2 contingency table for the two attributes sex and nature of work given below the expected frequency of females among unskilled is

skilled unskilled

males 40 20

females 10 30

a) 30 b) 20 c) 40 d) 60 16. If 90, 50 , then A

______________.

a) 20 b) 30 c) 40 d) 50 17. Two coins are tossed simultaneously. The probability of getting a head and a tail is

a) 12 b) 1

6

c) 14

d) 13

18. A discrete random variable can take integer values from 1 to k each with 1k

probability. Then the mean value is

a) k b) 1

2

k

c) 2

k d)

1

2

k k

19. If ( ) 0.4, ( ) 0.3, ( ) 0.2,P A P B P A B ( )then P A B _____________.

a) 0.2 b) 0.3 c) 0.5 d) 0.4

20. If ,A B then________.

a) ( ) ( )P A P B b) ( ) ( )P A P B

c) ( ) ( )P A P B d) ( ) ( )P A P B

Reg. No. : PART – B

Answer ALL the Questions: (5x3=15)

21. Find the G.M & H.M of the four numbers 2,4,6,27. 22. The first three moments about the origin are

1 1 1 21 1 11 ; 1 2 1 , 1

1 2 32 6 4n n n n n . Examine the skewness of the

distribution. 23. The two variables x and y have the regression lines 3 2 26 0 6 31 0x y and x y . Find the mean values of x and y.

24. Find whether the following data are consistent: N=600,(A)=300, (B)=400 (AB)=50. 25. If A and B are any two events of a sample space S, then prove that ( ) ( ) ( )P A B P A P B P A B .

PART – C


26. a) Find the median and quartile marks of 10 students in statistics test whose marks are 40,90,61,68,72,43,50,84,75,33.

(OR) b) Give that the mode of the following frequency distribution of 70 students is 58.75 . Find the missing frequencies &1 2f f .

Class 52-55 55-58 58-61 61-64

Frequency 15 1f 25 2f

27. a) The mean and standard deviation of 200 items are found to be 60 and 20.If at the time of calculation two items are wrongly taken as 3 and 67 instead of 13 and 17. Find the correct mean and standard deviation.

(OR) b) Mean and S.D of the marks of two classes of sizes 25 and 75 are given below

Class A Class B

Mean 80 85

S.D 15 20

Combined mean and S.D of the marks of the student of the two classes, which class is performing a consistent progress?

28. a) P.T the angle between the two regression lines is 2 11tan

2 2

x y

x y

(OR) b) Find the correlation co-efficient between x and y .

x 51 63 63 49 50 60 65 63 46 50 y 49 72 75 50 48 60 70 48 60 56

29. a) Investigate from the following data between inoculation against small pox and presentation from attack

Attacked Not attacked Total

Inoculated 25 220 245

Not inoculated 90 160 250

total 115 380 495

(OR)



B.Sc. MATHS SEMESTER - III

PAPER : STATISTICS-I Time : 3 Hrs. MAX. MARKS: 100

b) Given N 1200, ABC 600, 50 , 270 , ( ) 36,A 204,B ( ) ( ) 192,A

( ) ( ) 620B Find the remaining ultimate class frequencies?

30. a) State and prove Baye's theorem.

(OR)

b) The Chances that 4 students A,B,C,D solve a problem are 1 1 1 1, , ,2 3 4 4

respectively. It all of them try to solve the problem, What is the probability

that the problem is solved. PART – D

Answer ANY THREE Questions: (3x10=30) 31. An incomplete distribution is given below.

class 0-10 10-20 20-30 30-40 40-50 50-60 60-70

frequency 10 20 ? 40 ? 25 15

32. Calculate the first four central moments from the following data to find

1 2and and discuss the nature of the distribution.

X: 0 1 2 3 4 5 6

F: 5 15 17 25 19 14 5

33. From the following data of marks obtained by 10 students in physics & chemistry, calculate the rank correction coefficient.

Physics(P) 35 56 50 65 44 38 44 50 15 26

Chemistry(Q) 50 35 70 25 35 58 75 60 55 35

34. Use the method of least squares and fit a straight line trend to the following data given from 82 to 92. Hence estimate the trend value for 1993.

Year: 82 83 84 85 86 87 88 89 90 91 92

Production: 45 46 44 47 42 41 39 42 45 40 48

35. The contents of 3 urns are urn I: 1white 3 red 2 black balls; urn II: 3 white 1 red 1 black balls; urn III : 3 white 3 red 3 black balls. Two balls are

chosen from a randomly selected urn. If the balls are 1 white and 1 red ball, what is the probability that they come from urn II.

**************

Reg. No. :

PART – A Answer ALL the Questions: (20x1=20) Choose the correct Answer: Answer 1. Which is countable? a) Q b) R c) C d) (0,1] 2. The intersection of the open intervals (0, 1/n) is

a) {0} b) c) {1} d) none

3. If 1 1 – :S n Nn

then inf S =___________.

a) 0 b) 1/2 c) 1 d) none 4. ………….is an unbounded subset of R a) (0,∞ ) b) [0,1] c) (-1,1) d) none 5. In a discrete metric space M, Int A=____________.

a) A b) c) M d) none

6. In R with usual metric, which is an open set a) {0} b) Q c) Z d) (0,1) 7. In any metric space, the intersection of an infinite number of open sets a) is open b) is closed c) need not be open d) both open and closed 8. The set of integers Z in R is ____________. a) closed b) open c) both open and d) dense 9. The interior of Q is _______________. a) R b) Q c) [0,1] d)

10. Which of the following is not complete?

a) l2 b) C with usual metric c) any discrete metric space d) Q 11. In a discrete metric space, if A is compact, then a) countable b) uncountable

c) finite d) none 12. Which is false?

a) BABA b) BABA

c) BABA d) none


SEMESTER EXAMINATIONS – NOVEMBER – 2016 12UMT52

B.Sc. MATHS SEMESTER - V

PAPER : REAL ANALYSIS Time : 3 Hrs. MAX. MARKS: 100

13. Which of the following subsets of R is compact? a) (0,1) b) [0,1) c) (0,1] d) [0,1] 14. If f is a continuous real valued function defined on M, then {xM / f(x) = 0} is a) closed b) open c) not open d) none

15. Define f: R→ R by f(x)= irrational

rational

is

is

x

x

if

if

x

0

. Then f is

a) continuous b) not continuous c) continuous at 0 and discontinuous at all other points d) discontinuous at 0 and continuous at all other points

16. Define f:[0,1]→[0,2] by f(x) = 2x. Then 1f is

a) not defined b) discontinuous c) uniformly continuous d) none of these 17. Which of the following subsets of R is connected? a) (0,1] [1,3) b) (0,1) (1,3) c) [0,1) (1,3] d) (0,2) (2,3) 18. The open interval (0, 1) is a) not connected b) totallly bounded c) connected d) none 19. Any totally bounded metric space is a) complete b) separable c) compact d) connected 20. A contraction mapping defined on a metric space

a) has a fixed point b) need not be continuous c) uniformly continuous d) none.

PART – B Answer ALL the Questions: (5 x 3 = 15) 21. State and prove the density theorem. 22. Prove that in any metric space, every closed ball is a closed set. 23. Prove that any discrete metric space is complete.

24. If A and B are connected subsets of a metric space M and if A B=. Prove

that A B is connected. 25. Prove that continuous image of a compact metric space is compact.

PART – C Answer ALL the Questions: (5 x 7 = 35) 26. (a). State and prove Archimedean property and the nested intervals property.

(OR) (b). State and prove Bolzano – Weierstrass theorem. 27. (a). Let (M, d) be a metric space. Let ., MBA Then prove that Int A = Union

of all open sets contained in A. (OR)

(b). Prove that in any metric space, every closed ball is a closed set. 28. (a). Prove that a subset A of a complete metric space M is complete iff A is

closed. (OR)

(b). Let 𝑀1 ,𝑑1 and (𝑀2 ,𝑑2) be two metric spaces. Then prove that 𝑓:𝑀1 → 𝑀2

is continuous iff 𝑓−1(𝐺) is open in 𝑀1 whenever G is open in 𝑀2. 29. (a). Prove that a closed subspace of a compact metric space is compact.

(OR) (b). Prove that a subspace of R is connected if it is an interval. 30. (a). Prove that a metric space M is totally bounded iff every sequence in M has

a Cauchy Subsequence. (OR)

(b). Prove that any continuous mapping defined on a compact metric space into any other metric space is uniformly continuous.

PART – D

Answer ANY THREE Questions: (3 x 10 = 30) 31. Prove that there exists a positive real number x such that x2 = 2. 32. Prove that in any metric space the intersection of a finite number of open sets is open. Is the result true for infinite collection? Justify your answer. 33. State and Prove Baire’s category theorem. 34. Prove that a subspace of R is connected iff it is an interval and deduce the intermediate value theorem. 35. Define a contraction mapping. State and prove Contraction mapping

theorem.




PAPER : REAL ANALYSIS Time : 3 Hrs. MAX. MARKS: 100

Reg. No. :

PART – A


Choose the correct Answer: Answer

1. The characteristic function of the differential equation 2 2( 7 12) xD D y e is

a) 4 3x xAe Be b) 4 3x xAe Be

c) 4 3x xAe Be d) None of these

2. The particular integral of the differential equation 2 3( 4 13) xD D y e is

a) 3

34

xe b)

3

34

xe

c) 3

10

xe d)

3

9

xe

3. The complementary function of 2( 6 9) xD D y e is

a) 3 ( )xe ax b b) 3 ( )xe ax b

c) 2 3x xae be d) 2 3x xae be

4. The auxiliary equation of the differential equation '' cosy y ecx

a) m 1 b) 2m 1

c) m-1 d) 2m -1

5. Which one of the following is linear differential equation

a) 2'' '' 0y yy x y b) '' (sin ) 0y x y

c) 2'' sin 0y x y d) 2'' siny y x

6. The solution of the equation dx dy dz

x y z is

a) 21 2x c y and z c x b) 1 2x c y and x c z

c) sin cos1 2y c x and z c x d) None

7. The genetal solution of 2p q is

a) ( 2 )( 3 ) 0y x c y x c b) ( 3 )( 3 3 ) 0y x c y x c

c) ( 3 )( 3 ) 0y x y x d) none of those

8. The general solution of 2( 4) 0D y is

a) 2 2x xy Ae Be b) 4 4x xy Ae Be

c) 3x xy Ae Be d) 4xy Ae B

9. The partial differential equation by eliminating the arbitrary constants a and b

from z axy b is

a) 0px qy b) 0py qx

c) 0px qy d) 0py qx




PAPER : DIFFERENTIAL EQUATIONS & FOURIER SERIES Time : 3 Hrs. MAX. MARKS: 100

10. The partial differential equation by eliminating the arbitrary constants a and b

from z ax by ab is

a) z p q pq b) z px qy pq

c) z px qy pq d) z px qy

11. Which one of the following equation has a solution yz f

x ?

a) 0px qy b) 0px qy

c) pq x d) p

yq

12. The complete integral of 0pq p q is

a) 1

az ax y c

a

b) z axy x y c

c) ( )z axy b x y c d) ( ) ( )z f y g x c

13. (1)L

a) 1 b) 1

s

c) 1

2s d)

1

3s

14. 1( )nL t =

a) !n

ns b)

!

1

n

ns

c) n

ns d)

1

n

ns

15.

11

2 2L

s s a

=

a) 1 cos

2

at

a

b)

1 sin

2

at

a

c) 1 cos

2

at

a

d)

1 sin

2

at

a

16. 12( 2)

sL

s

=

a) 2(1 2 ) xx e b) 2(1 2 ) xx e

c) 2 sinxe x d) 2 cosxe x

17. 2

cosnxdx

when n is integer

a) 0 b) 1

c) 2 d) 3

18. Which one of the following functions is even?

a) 4 23x x b) 2 2x x

c) 3x x d) 3 2x x

19. The period of tanx is

a) 0 b)

c) 2 d) 2

20. Even functions are

a) symmetric b) antisymmetric

c) periodic d) none of these



21. Solve 2 5 6 0D D y .

22. Solve dx dy dz

x y z .

23. Solve 2 2 2x p y q z .

24. Find sinhL ax .

25. Find a sine series for ( )f x c in the range 0 to .

PART – C


26. a) Solve 2 1 sin2D D y x .

(OR)

b) Solve 2'' (2 1) ' ( 1) xxy x y x y x e .

27. a) Solve 2 2 22 2 0z dx z yz dy y yz xz dz .

(OR)

b) Solve '' (2tan ) ' 5 0y x y y by removing the first derivative.

28. a) Solve 2 2 2y zp x zq xy .

(OR)

b) Find the complete integral of px qy pq .

29. a) Find sinat

Lt

.

(OR)

b) Find

s1

22 1

L

s

.

30. a) Express 1

( )2

f x x as a fourier series with period 2 , to be valid in the

interval 0 2to .

(OR)

b) Find a cosine series in the range 0 to for 0

2( )

2

x x

f xx x

.

PART – D

Answer ANY THREE Questions: (3x10=30)

31. Find the particular integral of 2 4 3 cos2xD D y e x .

32. Verify the Condition of integrability of 2 22 2 0xz yz dz yz zx dy x zx y dz

and solve.

33. Solve 2'' cot 4 cos 0y y x y ec x by changing the independent variable x to z .

34. Solve the equation 2

2 3 sin2

d y dyy t

dtdt given that 0

dyy

dt when t=0.

35. Show that 2 cos2 4 ( 1)

231

nxnxnn

in the interval x .

************




PAPER : DIFFERENTIAL EQUATIONS & FOURIER SERIES Time : 3 Hrs. MAX. MARKS: 100

Reg. No. :

PART – A Answer ALL the Questions: (20x1=20) Choose the correct Answer: Answer

1. The resultant of two forces P,P which make an angle is

a) 2P b) 2 sin2

P

c) 2 cos2

P

d) None of these

2. If the forces P and Q act along the same line but opposite directions, then their resultant is__________.

a) P Q b) P+Q c) 2P+Q d) P+2Q 3. Two forces of given magnitudes P and Q act at a point at an angle . Then

the maximum value of resultant is___________.

a) P+Q b) P Q c) 2P d) P+2Q 4. The resolved part of a force F in its own direction is______. a) F,F b) 2F c) F d) None of these

5. Two parallel forces are said to be like when they act in__________. a) Different direction b) Same direction c) Opposite parallel direction d) none of these 6. The moment of a force about a point is a ______ quantity. a) Vector b) Scalar c) negative d) none of these 7. If a system of coplanar forces is in equilibrium, then the algebraic sum of their moment about any point in their plane is___________. a) Zero b) not equal to zero c) Product of moments d) None of these 8. The magnitude of resultant of two like parallel forces is a) their difference b) Their sum c) 0 d) none of these 9. If three forces acting on a rigid body are in equilibrium, then they must be a) Coplanar b) Parallel c) perpendicular d) none of these 10. With usual notation, when R=G=0 , then the system is in a) Not equilibrium b) Equilibrium c) Couple d) None of these

11. When R=0 but 0G , then the system reduce to a

a) equilibrium b) Not equilibrium c) single force d) couble




PAPER : STATICS Time : 3 Hrs. MAX. MARKS: 100

12. A uniform rod cannot rest entirely within a smooth hemisphere bowl except in a _____ position. a) Vertical b) Perpendicular c) Horizontal d) none of these 13. When motion ensues by one body sliding over another, then the friction exerted is called____________.

a) Dynamical friction b) Statical friction c) Limiting friction d) None of these 14. The limiting friction is independent of the a) Extent b) Shape of the surface c) extend and shape of the surface d) None of these 15. If F is the friction and R is the normal reaction between two bodies when equilibrium then F=_____.

a) b) R

c) R

d) R

16. With usual rotation =________________.

a) sin b) cos

c) cot d) tan

17. Intrinsic equation of catanary is______________. a) sins c b) tans c

c) coss c d) none of these

18. Cartesion equation to the catenary is____________.

a) coshx

y cc

b) sinh

xy

c

c) tanhy x d) None of these

19. If two points A and B from where the string is suspended are in horizontal line then distance AB is called_____________. a) Sag b) Span c) radius d) none of these 20. Tension T=________________.

a) 2w y b) wy

c) 2wy d) w

y

Reg. No. :

PART – B Answer ALL the Questions: (5x3=15) 21. State and prove parallelogram of forces theorem. 22. Find the conditions of equilibrium of three coplanar parallel forces. 23. If three coplanar forces acting on a rigid body keep it in equilibrium, then prove that they must either be concurrent or be all parallel. 24. Write the law of dynamical friction. 25. Explain about geometrical properties of the common catenary.

PART – C

Answer ALL the Questions: (5x7=35) 26. a) State and prove Lami's theorem.

(OR) b) OA, OB,OC are the lines of action of two forces P and Q and their resultant R respectively. Any transversal meets the lines in L,M and N

respectively. Prove that P O R

OL OM ON .

27. Two like parallel forces P and Q act on a rigid Rod at A and B respectively.

If Q be Changed to 2P

Q, then show that the line of action of the resultant is

the same as it would be if the forces were simple interchanged. (OR)

b) If Two couples, whose moments are equal and opposite act in the same plane upon a rigid body, then prove that they balance one another.

28. a) If D is any point on the base BC of ABC such that BP m

DC n and ADC ,

BAD , DAC .Then prove that ( )cot cot cotm n m n .

(OR) b) Find the conditions for a system of forces to reduce to a single force or to a couple. 29. a) Explain about angle of friction.

(OR) b) A weight can be supported on a rough inclined plane by a force P acting along the plane or by a force Q acting horizontally. Show that the weight is

2 2 2sec

PQ

Q P

,where is the angle of friction.

30. a) A uniform chain of length l is to be suspented from two points in the

same horizontal line so that either terminal tension is n times that at the

lowest point. Show that the span must be 2log 12 1

ln n

n

.

(OR) b) Prove that if a uniform inextensible chain hangs freely under gravity, the difference of the tension at two points varies as the difference of their weights.




PAPER : STATICS Time : 3 Hrs. MAX. MARKS: 100

PART – D Answer ANY THREE Questions: (3x10=30) 31. Two beads of weights W and W' can slide on a smooth circular wire in a vertical plane. They are connected by a light string which subtends an angle 2 at the centre of the circle whether beads are in equilibrium on

the upper half of the wire. Prove that the inclination of the string to the

horizontal is given by

'tan tan

'

w w

w w

.

32. Find the resultant of two like parallel forces acting on a rigid body. 33. A uniform rod, of length a hangs against a smooth vertical wall being supported by means of a string of length l tied to one end of the string

being attached to a point in the wall; Show that the rod can reset inclined

to the wall at the angle given by 2 2

223

l aCos

a

.

34. A uniform rod reset in limiting equilibrium within a rough hollow sphere. If the rod subtends an angle 2 at the centre of the sphere and if be the

angle of friction, show that the inclination of the rod to the horizontal is

sin21tan

cos2 cos2

.

35. Show that the length of an endless chain which will hang over a circular pully of radius "a" so as to be in contact with two-thirds of the

circumference of the pully if

3 4

3log 2 3a

.

*************

Reg. No. :

PART – A


1. Which of the following is not correct about LPP? a) All constraints must be linear relationships b) Objective function must be linear c) All the constraints and decision variables must be of either or types

d) All decision variables must be linear 2. A constraint in an LPP restricts a) Value of objective function b) Value of a decision variable c) Use of variable resource d) Uncertainty of optimum value 3. The set of all feasible solutions to an LPP is a ______ set. a) Concave b) convex c) Connected d) Compact 4. The coefficient of slack or surplus variables in the objective function are always assumed to be a) -1 b) -M c) 0 d) M 5. For maximization LPP, the simplex method is terminated when all the net- evaluations are a) Negative b) Non -negative c) zero d) none-positive 6. For maximization LPP, the simplex method is terminated when all the net- evaluations are a) +M b) -M c) +1 d) 0 7. The role of an artificial variable in simplex method is a) To aid in finding initial basic feasible solution b) to start phases of simplex method c) to find shadow prices from the final simplex table d) none of the above 8. If the LPP is in canonical form the primal dual pair is said to be a) symmetric b) asymmetric c) equal d) Unequal 9. The solution to a T.P with m-sources and n-destinations is feasible, if the number of allocations are a) m+n-1 b) m+n+1 c) m+n d) m x n

10. The initial solution of a T.P can be obtained by applying any known method. However the only condition is a) The solution must be optimum b) The solution must be non-degenated c) The rim conditions are satisfied d) all of the above


SEMESTER EXAMINATIONS – NOVEMBER – 2016 12UMTE51


PAPER : LINEAR PROGRAMMING & GAME THEORY Time : 3 Hrs. MAX. MARKS: 100

11. If we were to use opportunity cost value for non-basic call to test optimatily, it should be a) Most negative number b) Most positive number c) Equal to zero d) Any value 12. A necessary and sufficient condition for the existence of a feasible solution to the transportation problem is that a) ai bj b) ai bj

c) ai bj d) none

13. If there are n workers and n jobs,there would be a) n solutions b) n! solutions c) (n-1)! Solutions d) (n!)n solutions 14. The method used for solving an A.P is called a) Modi method b) reduced matrix method c) Hungarian method d) None of the above 15. For a salesman, who has to visit n cities, following are the ways of his tour plan. a) n b) n! c) (n+1)! d) (n-1)! 16. In an A.P, an optimum assignment has been obtained if the minimum number of lines______ the order of the cost matrix a) Equal to b) Greater than c) Less than d) None of these 17. A two person game is said to be zero-sum, if a) Gain of one player is exactly matched by a loss to the other so that their sum is equal to zero. b) Gain of one player does not match the loss to the other c) both the players must have an equal number of strategies

d) diagonal entries of the pay-off matrix are zero 18. A game is said to be fair,if a) Upper value is more than lower value of the game b) Upper and lower values of the game are not equal c) Upper and lower values of the game are same and zero d) none of the above 19. The pay-off value for which each player in a game always selects the same strategy is called the a) Equillibrium point b) Saddle point c) Both a) and b) d) none of the above 20. A mixed strategy game can be solved by a) Matrix method b) Algebraic method c) Graphical method d) All of the above

Reg. No. :

PART – B Answer ALL the Questions: (5x3=15) 21. Three grades of coal A,B and C contains ash and Phosphorous as impurities. In a particular industrial process a fuel obtained by blending the above grades containing not more than 25% ash and 0.03% phosphorous is required. The maximum demand of the fuel is 100 tons. Percentage impurities and cost of various grades of coal are shown below. Assumming that there is an unlimited supply of each grade of coal and there is no loss in blending , formulate the blending problem to minimize the cost. Coad grade %ash %phosphorus cost per ton (Rs) A 30 0.02 240 B 20 0.04 300 C 35 0.03 280 22. Formulate the dual of the following LPP Maximize 5 31 2z x x

Subject to 3 5 151 2x x

5 2 101 2x x

, 01 2x x

23. Write the steps involved in matrix minima method. 24. Given below is an A.P , write it as T.P

A1 A2 A3

R1 1 2 3 R2 4 5 1

R3 2 1 4

25. For the game with the following pay off matrix determine the optimum strategies and the value of the game

2

1

5 1

3 4

P

P

PART – C Answer ALL the Questions: (5x7=35) 26. a) Use the graphical method to solve the following LPP

min 21 2

3 101 2

61 2

21 2

, 01 2

imize z x x

subject to x x

x x

x x

x x

(OR) b) Rewrite in standard form the following LPP.

min 2 41 2 3

2 4 41 2

2 51 2 3

2 3 21 2

, 01 2 3

imize z x x x

subject to x x

x x x

x x

x x and x unrestricted

27. a) Use simplex method to solve the following LPP

min 4 101 2

2 501 2

2 5 1001 2

2 3 901 2

, 01 2

imize z x x

subject to x x

x x

x x

x x


SEMESTER EXAMINATIONS – NOVEMBER – 2016 12UMTE51


PAPER : LINEAR PROGRAMMING & GAME THEORY Time : 3 Hrs. MAX. MARKS: 100

(OR) b) Write the dual of the following LPP

min 3 41 2 3

2 2 141 2 3

2 4 131 2

4 7 101 2 3

, 01 2 3

imize z x x x

subject to x x x

x x

x x x

x x and x unrestricted

28. a) Obtain an initial basic feasible solution to the following T.P using north west corner rule.

(OR) b) Find the initial basic feasible solution to the following T.P using VAM

supply

29. a) A department head has 4 tasks and 3 subordinates. The estimates of the time each subordinate would take to perform is given below. How should he allocate the task me to each man so as to minimize the total man hours?

TASK 1 2 3

I 9 26 15 II 13 27 6 III 35 20 15 IV 18 30 20

b) A Salesman is planning to tour eities B,C, D and E from A. The inter city distances are shown below.

A B C D E

A 103 188 136 38

B 103 262 176 52

C 188 262 85 275

D 136 176 85 162

E 38 52 275 162 Solve? 30. a) Solve the following game.

3 2 4 0

3 4 2 4

4 2 4 0

0 4 0 8

Player B

I II III IV

I

IIPlayer A

III

IV

(OR) b) Consider payoff matrix with respect to player A and solve it optimally.

A

B 1 2

1 6 9 2 8 4

PART – D Answer ANY THREE Questions: (3x10=30) 31. The advertising agency wishes to reach two types of audiences. Customers with annual incomes greater than Rs.40,000 and customers with annual incomes of less than Rs.40,000 . The total advertising budget is Rs.200000.

D E F G

A 11 13 17 14 250

B 16 18 14 10 300 Available

C 21 24 13 10 400

Requirement 200 225 275 250

D1 D2 D3 D4

S1 20 25 28 31 200 S2 32 28 32 41 180 S3 18 35 24 32 110

Demand 150 40 180 170

One programme of T.V advertising costs Rs.50,000; One programme of radio advertising costs Rs.20,000. For contract reasons atleast 3 programmes ought to be on TV and the number of radio programmes must be limited to 5. Surveys indicate that a single TV programme reaches 7,50,000 customers in target audience A and 1,50,000 in target audience B. One radio programme reaches 40,000 in target audience A and 2,60,000 in target audience B. Formulate this as a LPP and determine the media mix to maximize the total reach by graphical method.

32. Use simplex method to

min 3 22 3 5

3 2 72 3 5

2 4 122 3

4 3 8 102 3 5

, , 02 3 5

imize z x x x

subject to x x x

x x

x x x

x x x

33. Find the starting solution in the following T.P by VAM. Also obtain optimum solution.

Demand

D1 D2 D2 D4 supply

S1 3 7 6 4 5 S2 2 4 3 2 2 S3 4 3 8 5 3

3 3 2 2 34. A manufacturing company has 4 zones A,B,C,D and four sales engineers P,Q,R,S respectively for assignment. Since the zones are not equally rich in sales potential, it is estimated that a particular engineer operating in a particular zone will bring the following sales. Zone A: 4,20,000 Zone B: 3,36,000 Zone C: 2,94,000 Zone D: 4,62,000 The engineers are having different sales ability working under the same conditions their yearly sales are proportional to 14,9,11 and 8 respectively. The criteria of maximum expected total sales is to be met by assigning the best engineer to the richest zone, the next best to the second richest zone and so on. Find the optimum assignment and the maximum sales.

35. Solve the following 3x5 game using dominance property.

Player B Player A

1 2 3 4 5

1 2 5 10 7 2 2 3 3 6 6 4 3 4 4 8 12 1

******************

Reg. No. : PART – A



1. If 2xy then 4y =_________.

a) 32 (log 2)x b) 4(log 2) 2x

c) 2 log 2x d) 22 (log 2)x

2. If log(2 3)y x then yn =________.

a)

11 1 !2

2 3

n nn

nx

b)

11 2

2 3

n n

nx

c)

11 1 !

2 3

nn

nx

d)

1 2

2 3

n n

nx

3. If xy

ux y

then u u

x yx y

=_________.

a) 0 b) u

c) 1 d) 2u

4. If 3 3 3( , , ) 3u x y z x y z xyz then 2u

x y

=________.

a) 3z b) 3

c) 23 3x yz d) 6x

5. The radius of curvature in p-r coordinates for the curve which is in polar form is

a) dr

rdp

b) dprdr

c) 1 dr

r dp d)

1 dp

r dr

6. The abscissa of the centre of curvature of the curve 2y x at the origin is____.

a) 1 b) -1

c) 0 d) 2

7. The evolutes of the parabola 2 4y ax is________.

a) a semi cubical parabola b) a circle

c) a straight line d) an ellipse

8. The curvature of the straight line y mx c is_______.

a) m b) 0

c) d) c

9. If (a,b) denotes the solution of the equations 0f

x

and 0

f

y

then ( , )f x y has a

maximum at ( , )a b if

a) 2 0AC B and 0( 0)A orB

b) 2 0AC B and 0( 0)A orB

c) 2 0AC B d) 2 0AC B

10. If cosx r and siny r then ( , )

( , )

x y

r

=________.

a) 2r b) r

c) cos d) cosr



B.Sc. MATHS SEMESTER - I

PAPER : DIFFERENTIAL AND INTEGRAL CALCULUS Time : 3 Hrs. MAX. MARKS: 100

11. A point (x,y) on the curve ( , )f x y =0 is called a multiple point if and only if .

a) 0fx b) 0fy

c) 0 0f and fx y d) 0fx

12. An asymptote of an algebraic curve of degree n cuts the curve in

a) n points b) n-1 points

c) n+1 points d) n-2 points

13. If (2 ) ( )f a x f x then 2

( )

0

a

f x dx

a) ( )

0

a

f x dx b) 2 ( )

0

a

f x dx

c) ( )

a

f x dx

a d) 0

14. 2

sin cos

sin cos0

a x b xdx

x x

a) 0 b) ( )3a b

c) ( )4

a b

d) ( )2

a b

15. xxe dx

a) ( 1)xe x b) ( 1)xe x

c) ( 1)xx e d) ( 1)xx e

16. 2

5 5sin cos

0

x xdx

a) 1

30 b)

1

60

c) 5

18 d)

1

40

17. 12

2

00

xy dydz

a) 3

2 b)

3

4

c) 4

3 d)

2

3

18. 2 3

0

xydxdy

x

a) 1 b) 2

c) 3 d) 4

19. 4 =

a) 4 b) 6

c) 8 d) 16

20. 2

0

xe dx

a) 2

b)

c) 3

d)

3

Reg. No. :

PART – B


21. If 1tany x find yn .

22. Find the pedal equation of r a .

23. Find the asymptotes of the curve sinxy e x x .

24. Evaluate 1sin xdx .

25. Prove that ( 1) ( )n n n .

PART – C


26. a) Find the nth derivation of

23 1

2( 1) 2 1

xy

x x

.

(OR)

b) If 21m

y x x

prove that 2 2 21 2 1 ( ) 02 1x y n xy n m yn n n .

27. a) Find the centre of curvature of the curve (cos sin ), (sin cos )&x a t t t y a t t t

P.T its evolutes is a circle.

(OR)

b) From any point of the ellipse 2 2

12 2

x y

a b ,perpendiculars are drawn to the

coordinate axes. Prove that the envelope of the straight lines joining the feet of

these perpendiculars is the curve

2 2

3 3 1x y

a b

.

28. a) Show that the curve 3 2( ) (2 )y x a x a has a single cusp of first species at

(a,0).

(OR)

b) Find a point within a triangle such that the sum of the squares of its

distances from three vertices is minimum.

29. a) Express

1

2 2 2lim 1 21 1 ........ 1

2 2 2

nit n

n n n n

as a definite integral and hence

evaluate.

(OR)

b) If 2

cos

0

nI x xdxn

prove that ( 1) 22

nI n n In n

.

30. a) Change the order of integration in 2

0 0

a ax

x dydx and hence evaluate.

(OR)

b) Evaluate 2 2

xydxdy

x yD by transforming to polar coordinates where D is the

region enclosed by the circles 2 2 2x y a and 2 2 24x y a in the first

quadrant.



B.Sc. MATHS SEMESTER - I

PAPER : DIFFERENTIAL AND INTEGRAL CALCULUS Time : 3 Hrs. MAX. MARKS: 100

PART – D


31. a) Find yn if 3 axy x e .

b) If 3 3

1tanx y

ux y

find 2 22x u u y uxx xy xy yy .

32. Find the evolute of the curve given by 3 3cos , sinx a y a .

33. Show that the asymptotes of the curve 2 2 2 0x y xy xy y x y cut the curve

again in three points which lie on the line x y 0.

34. a) Evaluate 4

log(1 tan )

0

d

. b) Evaluate 2

sin cos

sin cos0

a x b xdx

x x

. (6+4)

35. Evaluate xyzdxdydz

D where D is the positive octant of the ellipsoid

2 2 2

12 2 2

x y z

a b c .

****************

Reg. No. :

PART – A


1. If a, b are positive and a>b then

a) 1 1

a b b)

1 1

a b

c) 1

ba d)

1b

a

2. For any 'n' positive real numbers not all of them being equal then

a) A>G>H b) A<G<H

c) A<G d) A<H

3. If 'x' is any positive real number then

a) 1 1n

x nx b) 1 1n

x nx

c) 1 1n

x x d) 1 1n

x nx

4. If x, y,z are positive real numbers such that 1x y z then

2 2 2

x y z

x y z

is

a) 3

5 b)

5

3

c) 5

3 d)

3

5

5. The g.l.b of the sequence 1 1 1

1, ,1, ,1, ...2 3 4

is

a) 1 b) 0

c) 1

2 d)

1

3

6. The sequence 1n

n

converges to

a) 1 b) 0

c) -1 d) 2

7. The sequence 1 1 1

1, ,2, ,3,.... , ,....2 3

nn

is

a) Divergent b) Convergent

c) Finitely oscillating d) Infinitely oscillating

8. If an and bn then a bn n

a) Converges to 0 b) Diverges to

c) Diverges to - d) Oscillates

9. The sequence 0nnr if

a) 1r b) 1r

c) 0r d) 0r



B.Sc. MATHS SEMESTER -III

PAPER : SEQUENCES AND SERIES Time : 3 Hrs. MAX. MARKS: 100

10. The peak point of the sequence 1,2,3…… is

a) 1 b) 2

c) 3 d) has no peak point

11. The limit point of the sequence 1,1,1….. is

a) 0 b) 1

c) d)-1

12. For the sequence 1,2,3,1,2,3 … liman ______.

a) 1 b) 2

c) 3 d) 0

13. The series 1 1 1

1 .... ..1! 2! !n

a) Converges to C b) Converges to 1

c) Diverges to d) Converges to 2

14. The series 1

pn

n n converges if

a) p 1 b) p 1

c) p 2 d) 1p2

15. The series 2ne

a) non converges b) Converges

c) Neither converges nor diverges

d) Absolutely convergent

16. If 0an then 1

nan

a) Diverges to b) Diverges to -

c) Converges to 0 d) Converges to 1

17. The series ( 1)

2 1

nn

n

a) Oscillates b) Diverges

c) Converges d Converges to 0

18. The series ( 1)n

n

a) Convergent b) Conditionally Convergent

c) Diverges d) Oscillates

19. For the geometric series nx , the radius of convergence=______.

a) 1 b) 0

c) d) -

20. The series formed by the set of positive terms of an absolutely convergent

series is

a) Convergent b) Divergent

c) Conditionally Convergent d) Absolutely convergent


Answer ALL the Questions: (5x3=15) 21. Prove that if a,b,c are three positive real numbers then 9a b c ab bc ca abc .

22. Prove that any sequence an diverging to is bounded below but not bounded

above.

23. Prove that lim

0!

nx

n n

.

24. Test the convergence of 1

logn n .

25. Show that the series

1

1 !

nx

n

converges absolutely for all values of x.

PART – C Answer ALL the Questions: (5x7=35)

26. a) If a,b,c are positive real numbers such that 2 2 2 27a b c then show that

3 3 3 81a b c .

(OR)

b) If m>n, prove that n m

m m n nx y x y .

27. a) Show that if an is a monotonic sequence then ...1 2a a an

n

is also a

monotonic sequence.

(OR)

b) Prove that any convergent sequence is a bounded sequence.

28. a) Show that limlim 1 1 111 1 ....

1! 2! !

ne

n nn n

.

(OR)

b) State and prove Cauchy's general principle of convergence.

29. a) Discuss the convergence of the series 2 31 2 3

1 .....2 3 42 3 4

(OR)

b) Test the convergence of 3

2

n a

n a

.

30. a) State and prove Leibnitz's test.

(OR)

b) Discuss the convergence of the series 1 1 sin

1 ....2

n

n n

.

PART – D Answer ANY THREE Questions: (3x10=30) 31. For any two real numbers x and y, prove that i) x y x y

ii) x y x y

32. i) It a a and b bn n then prove that a b a bn n .

ii) If a a and b bn n then prove that a b abn n .

33. State and prove Cauchy's first limit theorem.

34. State and prove Cauchy's integral Test.

35. State and prove Merton's theorem.

***********



B.Sc. MATHS SEMESTER -III

PAPER : SEQUENCES AND SERIES Time : 3 Hrs. MAX. MARKS: 100

Reg. No. :

PART – A



1. The coefficient of nx in the expansion of 22 4 1 2x x

is

a) 2n b) 12n

c) 12n d) 22n

2. log2

a) 1 1

1 ....2 3

b) 1 1 1

1 ....2 3 4

c) 1 1 1

1 ....2 3 4

d) 1 1

1 ....3 5

3. 1 1

1 ....2! 3!

a) 1

ee

b) 1

ee

c) 1 1

2e

e

d)

1 1

2e

e

4. 2 3(log 2) (log 2)

log 2 ...2! 3!

=

a) 32

b) 1

2

c) 1 d) none

5. One real root of the equation 3 24 24 23 18 0x x x is

a) 1 b) 2

c) 1 d) 2

6. The real root of the equation 3 22 9 18 0x x x is

a) 0 b) 1

c) 2 d) 3

7. The number of complex roots of 4 3 1 0x x is

a) 0 b) 2

c) 3 d) 1

8. If ( )f x is a polynomial of degree n then the equation '( ) 0f x has

a) n roots b) n 1 roots

c) n+1 roots d) n 2 roots

9. A square matrix A is singular if A =

a) 0 b) 1

c) 2 d) 3

10. sin( )ix

a) sinhi x b) sinhi x

c) sinhx d) sinhx



B.Sc. CHEMISTRY & PHYSICS SEMESTER - I

PAPER : MATHS-I Time : 3 Hrs. MAX. MARKS: 100

11. The value of Logi is

a) 2i

b) 22

i n

c) 4 12

i n

d) 2

12. The value of 1 11 1tan tan2 3

is

a) 2

b)

3

c) 6

d)

4

13. The characteristic polynomial of 0 1

0 0A

is

a) 2x b) 2 1x

c) 21 x d) 1x

14. The product of the eigen values of 3 3

2 4

is

a) -6 b) 6

c) 0 d) none

15. The sum of the roots of 4 26 13 6 0x x x is

a) 136

b) 136

c) 0 d) 613

16. The rank of the matrix

1 1 1 3

0 1 3 4

0 0 4 4

is

a) 1 b) 2

c) 3 d)4

17. xxe dx

a) ( 1) xx e b) ( 1) xx e

c) ( 1) xx e d) ( 1) xx e

18. 2

6 5sin cos

0

x xdx

=

a) 693

8 b)

8

693

c) 1

60 d) 60

19. An example of a function which is neither even nor odd is

a) 3 sin2x x b) cos2x

c) sin2 cos3x x d) sin2 cos3x x

20. An example of an even function is

a) x b) x

c) 3x x d) 2x x



21. What is the coefficient of nx in the expansion of 1(1 ) xx e in ascending powers

of x.

22. Diminish the roots of the equation 3 2 100x x x by 4.

23. Prove that 1 tanhcosh2 sinh2

1 tanh

xx x

x

.

24. If 2 4

1 1A

find 3A .

25. Evaluate log xdx .

PART – C


26. a) Sum to the series 1 1.4 1.4.71 ...

5 5.10 5.10.15

(OR)

b) Prove that 1 2 1 2 3 3

1 ..2! 3! 2

eS

.

27. a) If one root of the equation 3 22 11 38 39 0x x x is 2 3i solve the equation.

(OR)

b) Increase the roots of the equation 5 34 2 7 3 0x x x by 2.

28. a) Prove that 1 2cosh log 1x x x

.

(OR)

b) If sinx iy A iB , Prove that 2 2

12 2sin cos

x y

A A .

29. a) Find the rank of the matrix

1 2 2 3

2 5 4 6

1 3 2 2

2 4 1 6

A

.

(OR)

b) Find the inverse of the matrix3 3 4

2 3 4

0 1 1

Using Cayley-Hamilton Theorem.

30. a) Evaluate 4

log 1 tan

0

I d

.

(OR)

b) Evaluate cosaxe bxdx .

PART – D


31. Sum to infinity the series 1 1 1 1 1 1 11 ...

22 3 4 9 5 6 9

32. Show that the roots of the equation 3 2 0px qx rx s are in G.P iff 3 3r p q s .

33. Prove that log tan24

ue

iff secCoshu

34. Find the eigen values and eigen vectors of the matrix 6 2 2

2 3 1

2 1 3

A

.

35. If 2

sin

0

nI dn

and 1n .Prove that 1 1

2 2

nI In n

n n

.Hence deduce that 1495

225I .

**********



B.Sc. CHEMISTRY & PHYSICS SEMESTER - I

PAPER : MATHS-I Time : 3 Hrs. MAX. MARKS: 100

Reg. No. :

PART – A Answer ALL the Questions: (20x1=20) Choose the correct Answer: Answer

1. G {1, 1} is an abelian group of order.

a) 1 b) 2 c) -1 d) 0 2. The set of all n n non-singular matrices with rational entries under

multiplication is a) Abelian b) non-abelian c) Finite d) none

3. The generators of the multiplicative group, 2{1, , }G are,

a) only b) 2and

c) 1 d) -1 4. Every proper subgroup of an infinite cyclic group is, a) Simple b) Infinite c) finite d) not cyclic 5. A group having no proper normal subgroups is called______ group.

a) abelian b) simple c) cyclic d) Quotient 6. Every cyclic group is____________. a) normal b) abelian c) finite d) none 7. Any two finite cyclic groups of same order are, a) simple b) isomorphic c) non-isomorphic d) none 8. A homomorphism of a group onto itself is called______. a) Epimorphism b) Endomorphism c) Isomorphism d) none 9. A ring R is called a Boolean ring if x R ,

a) 1x b) 2x x

c) 2x x d) 2 1x

10. An isomorphism of a group onto itself is called_____. a) epimorphism b) automorphism c) endomorphism d) none 11. A transposition is a cycle of length_________. a) 0 b) 1

c) 2 d) 3




PAPER : ABSTRACT ALGEBRA Time : 3 Hrs. MAX. MARKS: 100

12. The identity permutation is an_______ permutation. a) Single b) Odd c) Even d) None 13. R is a subring of________. a) Q b) W

c) C d) Z 14. The improper ideals of a ring R are a) R only b) {0} only C) {0} and R d) None 15. A homomorphism among rings is said to be an Isomorphism if it is a) not onto b) onto c) 1-1 d) none 16. A homomorphism among rings is 1-1 if, a) Kerf={1} b) kerf {0} c) Kerf={0} d) Kerf {1} 17. The ring of rationals is of characteristic__________. a) 3 b) 1 c) 0 d) 2 18. If a+ib is not a unit in z(i), then,

a) 2 2 0a b b) 2 2 0a b

c) 2 2 1a b d) none

19. 1+i is irreducible in____________. a) C b) Q

c) Z(i) d) R 20. The characateristic of Zp (p-prime) is__________.

a) 2 b) 2p

c) p d) 1

Reg. No. :

PART – B Answer ALL the Questions: (5x3=15) 21. If every element of a group has its own inverse then show that the group is abelian. 22. Show that every subgroup of an abelian group is normal. 23. Show that any permutation can be expressed as a product of transpositions. 24. Prove that the intersection of two subrings of a ring R is a subring of R. 25. Find all units in Z(i).

PART – C Answer ALL the Questions: (5x7=35) 26. a) Prove that the union of two subgroups of a group G is a subgroup of G if and only if one is contained in the other.

(OR) b) Prove that a group of prime order is cyclic. 27. a) Show that any infinite cyclic group isomorphic to a group of integers under addition.

(OR) b) Define the centre ( )Z G of a group G and Prove that ( )Z G is a normal

sub group of G. 28. a) If G is a group, then prove that the set of all automorphisms of G is a group.

(OR) b) Prove that a finite integral domain is a field. 29 a) If f is a homomorphism of a ring R into ring R', then prove that the kernel of f is an ideal of R.

(OR) b) Let R be a commutative ring with unity whose only ideals are (0) and R itself. Then prove that R is a field. 30. a) Prove that the characteristics of an Integral domain is either zero or a prime number.

(OR) b) Prove that Z(i) is a Euclidean ring.

PART – D

Answer ANY THREE Questions: (3x10=30) 31. State and prove Lagrange's Theorem. 32. State and prove the fundamental theorem of homomorphism among groups. 33. State and prove the Cayley's theorem on group of permutations. 34. Let R be a commutative ring with unity and P an ideal of R. Then P is a prime ideal of R if and only if R/P is an Integral domain.

35. Find a greatest common divisor of 14 3a i and 4 7b i and

represent it in the form a b in ( )Z i

*************




PAPER : ABSTRACT ALGEBRA Time : 3 Hrs. MAX. MARKS: 100

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