32
Wk 9 DE–8024 DISTANCE EDUCATION B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014. CLASSICAL ALGEBRA Time : Three hours Maximum : 100 marks Answer any FIVE questions. Each questions carries equal marks. (5 × 20 = 100) 1. (a) Explain the first five terms of the series = - + 1 2 2 1 2 1 2 n n n . (10) (b) Prove that the harmonic series p n 1 convergence if 1 > p and diverges if 1 p . (10) 2. (a) Test the convergence of the series n x n n 1 + Σ , where x is any positive real number. (10) (b) State and explain ‘Leibnitz's test’ in details. (10) 13

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Page 1: DE–8024 13 DISTANCE EDUCATION B.Sc ... - IndCareer

Wk 9

DE–8024

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

CLASSICAL ALGEBRA

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each questions carries equal marks.

(5 × 20 = 100)

1. (a) Explain the first five terms of the series

∑∞

=

−+

12

2

12

12

n n

n. (10)

(b) Prove that the harmonic series ∑ pn

1 convergence

if 1>p and diverges if 1≤p . (10)

2. (a) Test the convergence of the series nx

n

n

1+Σ ,

where x is any positive real number. (10)

(b) State and explain ‘Leibnitz's test’ in details. (10)

13

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DE–8024

2

Wk 9

3. (a) Find the co-efficient of 2x in the expansion of

xe

xz 2321 −+. (10)

(b) (i) Examine the Binomial series

++−++ L2

!2

)1(

!11 x

mmx

m

LL ++−− nxn

nmmm

!

)1()1(.

(10)

(ii) Examine the logarithmic series

LL +−+−+− −

n

xxxx

nn 1

32

)1(32

.

4. (a) Sum the series

++++++++++!4

3331

!3

331

!2

311

322

to ∞ .

(10)

(b) Let n

n xaΣ be the given power series. Let

α = n

na/1

suplim and α1=R . (If 0=α , ∞=R

and if ∞=α , 0=R ). Then n

n xaΣ converges

absolutely if Rx < . If Rx > the series is not

convergent. (10)

5. (a) Solve the equation 02836 234 =+−+− xxxx given that it has a pair of roots whose sum is zero. (10)

(b) If γβα ,, are the roots of 03 =++ rqxx , find the

value of βααγγβ +

++

++

111. (10)

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DE–8024

3

Wk 9

6. (a) Solve 0841528316 234 =++++ xxxx by

removing the second term. (10)

(b) Solve the equation

06113337116 2345 =++−−+ xxxxx . (10)

7. (a) If 222 zyx += , show that

nnn zyx +≥ whenever

2≥n . (10)

(b) If kcba L,,, are positive numbers, then

n

kcba

n

kcba mmmmm )( ++++>++++ LL

except, when m lies between 0 and 1. (10)

8. (a) Diagonalise the matrix

262

222

644

. (10)

(b) Solve the following simultaneous equations. (10)

9414212

52826

461042

=++=++=++

zyx

zyx

zyx

——————

Page 4: DE–8024 13 DISTANCE EDUCATION B.Sc ... - IndCareer

Wk 16

DE–8025

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION,

MAY 2014.

CALCULUS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

All questions carry equal marks.

(5 × 20 = 100)

1. (a) If )sinsin( 1 xmy −= , prove that

122 )12()1( ++ +−− nn xynyx 0)( 22 =−+ nynm . Hence

show that )0()()0( 222 nn ymny −=+

(b) If )3log( 333 xyzzyxu −++= , show that

(i) zyxz

u

y

u

x

u

++=

∂∂+

∂∂+

∂∂ 3

(ii) 2

2

)(

9)(

zyxu

zyx ++−=

∂∂+

∂∂+

∂∂

.

2. (a) Prove that the rp − equation of the cardiod

)cos1( θ−= ar is a

rp

2

32 = .

(b) Find the radius of curvature of the curve 223223 765432 yxyxyxyyxx +−+−++ – 08 =y

at (0,0)

3. (a) (i) Evaluate ∫ − 22 ax

dx

(ii) Evaluate ∫ − 22 xa

dx

(b) Evaluate ∫ dxex x23 .

14

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DE–8025

2

Wk 16

4. (a) Establish a reduction formula for

∫= xdxI nn sin where Nn∈ and hence find

∫2

0

sin

π

dxxn .

(b) If ),( nmf ∫=2

0

sincos

π

dxnxxm , prove that

)1,1(1

),( −−+

++

= nmfnm

m

nmnmf . Hence deduce

that 12

1),( +=

mnmf

++++m

m2...

3

2

2

2

1

2 32

.

5. (a) Solve xxxydx

dysin2cot =−

(b) Solve 2pppxy −+= .

6. (a) Solve xx eeyD 422 )4( −+=− .

(b) Solve xexyDDD 223 )133( =−+− .

7. (a) Find

−x

xL

cos1.

(b) Find

++−

)2)(1(

11

SSSL .

8. (a) Solve zyqxP cotcotcot =+ .

(b) Find the complete integral of pqqypx =+ .

–––––––––––––––

Page 6: DE–8024 13 DISTANCE EDUCATION B.Sc ... - IndCareer

Sp 2

DE–8026

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

ANALYTICAL GEOMETRY AND VECTOR CALCULUS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each question carries equal marks.

(5 × 20 = 100)

1. (a) Show that two of the straight lines

03223 =+++ dycxyybxax will be perpendiculars to

each other if 022 =+++ acbdda . (10)

(b) A variable circle is drawn to touch the y-axis at the

origin. Find the locus of the pole of the straight line

0=++ nmylx with respect to it. (10)

2. (a) Find the circles which cut orthogonally each of the

following circles : (10)

056

034

0142

22

22

22

=+++

=+−+

=++++

yyx

xyx

yxyx

(b) Prove that the tangents at the extremities of any

focal chord of a conic interest on the corresponding

directrix. (10)

15

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DE–8026

2

Sp 2

3. (a) If the direction cosines of two lines satisfy

0=++ nml , and 022 =−+ nlmnlm , then find the

direction cosines. (10)

(b) Prove : Length of the perpendicular from a point

( )111 ,, zyxA to the plane 0=+++ kczbyax . Is

( )

++

+++±222

111

cba

dczbxax. (10)

4. (a) Find the equation of the plane passing through the

points ( )2,8,3 − and ( )1,1,2 −− and perpendicular to

the plane 01 =+++ zyx . (10)

(b) Find the image of the point ( )4,3,2 under the

reflection in the plane 652 =+− zyx . (10)

5. (a) Prove that 4

3

3

2

2

1 −=−=− zyx and

5

4

4

3

3

2 −=−=− zyx are coplanar and find the

equation of the plane containing them. (10)

(b) Prove : The lengths of two opposite edges of a

tetrahedron are a, b; their shortest distance is equal

to d and the angle between them is θ . Prove that

its volume is 6

sinθabd. (10)

6. (a) Prove that a plane section of a sphere is a circle. (10)

(b) Find the equation of tangent plane at a point

111 ,, zyx to the sphere

0222222 +++++++ dwzvyuxzyx . (10)

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DE–8026

3

Sp 2

7. (a) Show that ( )36 zxyF += ( )zxi −+ 23 ( )kyxzj −+ 23

is irrotational and find φ such that ∇=F φ . (10)

(b) Prove that ( ) BcurlAAcurlBBAdiv .. −=× . (10)

8. (a) Use Gauss divergence theorem find ∫∫s dsnF .

where hzjyixF 333 ++= and S is the surface of

the sphere 2222 azyx =++ . (10)

(b) Verify Stoke’s theorem for

( ) kzyjzyiyxf 222 −−−= where S in upper half

surface of the sphere 1222 =++ zyx and C is its

boundary. (10)

–––––––––––––––

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Sp 3

DE–8027

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

MECHANICS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

All questions carry equal marks.

(5 × 20 = 100)

1. (a) The resultant of two forces P,Q acting at a certain

angle is X and that P,R acting at the same angle is

also X. The resultant of Q,R again acting at the

same angle is Y. Prove that

( ) ( )222

21

2

yRQ

RQQRQRXP

−++=+=

Prove also that if 0=++ RQP , XY = . (10)

(b) ABC is a given triangle, Forces P,Q,R acting along

the lines OA, OB, OC are in equilibrium. Prove that

P : Q : R = ( )2222 acba −+ : ( )2222 bacb −+ :

( )2222 cbac −+ if 0 is the circumcentre of the

triangle. (10)

2. (a) State and prove Lami's theorem. (10)

(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 ON

R

OM

Q

OL

P =+ . (10)

23

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DE–8027

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Sp 3

3. (a) Find the resultant of two unlilke and unequal

parallel forces acting on a rigid body. (10)

(b) If D is any point on the base BC of triangle ABC

such that n

m

DC

BD = and θ=∠ADC , α=∠BAD and

β=∠DAC then prove the following

(i) ( ) βαθ cotcotcot nmnm −=+

(ii) ( ) CmBnnm cotcotcot −=+ θ . (10)

4. (a) A particle of weight 30 kgs resting on a rough

horizontal plane is just on the point of motion when

acted on by horizontal forces of 6 kgwt and 8 kgwt

at right angles to each other. Find the coefficient of

friction between the particle and the plane and the

direction in which the friction acts. (10)

(b) A beam of weight W hinged at one end is supported

at the other end by a string so that the beam and

the string are in a vertical plane and make the

same angle θ with the horizon. Show that the

reaction at the hinge is θ2cos84

ecW + . (10)

5. (a) Show that the greatest height which a particle with

initial velocity ν can reach on a vertical wall at a

distance 'a' from the point of projection is 2

22

22 v

ga

g

v − .

(10)

(b) The range of a rifle bullet is 1000 m, when α is the

angle of projection. show that if the bullet is fired with

the same elevation from a car travelling

36 km/h towards the target, the range will be increased

by 7

tan1000 αm. (10)

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Sp 3

6. (a) A shot of mass m penetrates a thickness t of a fixed

plate of mass M. If M were free to move and the

resistance supposed to be uniform. Show that the

thickness penetrated is mM

Mt

+. (10)

(b) A particle is projected from a point on an inclined

plane and at the rth impact it strikes the plane

perpendicularly and at the nth impact is at the point

of projection, show that 012 =+− rn ee .

(10)

7. (a) A particle is moving with S.H.M and while making

an oscillation from one extreme position to the

other, its distances from the centre of oscillation at

3 consecutive seconds are 321 ,, xxx . Prove that the

period of oscillation is

+−

2

311

2

2

x

xxCos

π. (10)

(b) If the displacement of a moving point at any time be

given by an equation of the form

wtbwtax sincos += , show that the motion is a

simple harmonic motion. If 3=a , 4=b , 2=w

determine the period, amplitude, maximum velocity

and maximum acceleration of the motion.

(10)

8. (a) Discuss the composition of two simple harmonic

motions of the same period and in the same straight

lines. (10)

(b) Show that the path of a point p which possesses two

constant velocities u and v, the first of which is in a

fixed direction and the second of which is

perpendicular to the radius OP drawn from a fixed

point O, is a conic whose focus is O and whose

eccentricity is v

u. (10)

–––––––––––––––

Page 12: DE–8024 13 DISTANCE EDUCATION B.Sc ... - IndCareer

WK 6

DE–8028

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

ANALYSIS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

All questions carry equal marks.

(5 × 20 = 100)

1. (a) Let 2, Ryx ∈ . Then ),( 21 xxx = and ),( 21 yyy =

where Ryyxx ∈2121 ,,, . We define

2211),( yxyxyxd −+−= then prove that d is a

metric on 2R .

(b) Prove that in any metric space ),( dM , each open

ball is an open set.

(c) Let ),( dM be a metric space. Let Mx∈ . Show that

cx }{ is open. (8 + 8 + 4 = 20)

2. (a) Let M be a metric space and 1M a subspace of M.

Let 11 MA ⊆ . Then prove that 1A is open in 1M if

and only if there exists an open set A in M such that

11 MAA ∩= .

(b) Let ),( dM be a metric space. Let MA ⊆ . Then

prove that x is a limit point of A iff each open ball

with centre x contains an infinite number of points

of A. (10 + 10 = 20)

24

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DE–8028

2

WK 6

3. (a) Prove that C with usual metric is complete.

(b) State and prove Cantor’s Intersection theorem.

(10 + 10 = 20)

4. (a) Prove that a subspace of R is connected if and only

if it is an interval.

(b) Prove that the closure of a totally bounded set is

totally bounded. (12 + 8 = 20)

5. (a) Prove that any compact subset A of a metric space

M is bounded.

(b) Prove that a metric space M is compact if and only if

any family of closed sets with finite intersection

property has non empty intersection. (8 + 12 = 20)

6. (a) Let M be a metric space let MA ⊆ . Prove that A is

compact iff given a family of open sets }{ αG in M

such that AUG ⊇α there exists a subfamily

nGGG ααα L,,

21 such that U

n

i

AGi

1=

⊇α .

(b) Prove that any continuous real valued function f

defined on a compact metric space is bounded and

attains its bounds. (12 + 8 = 20)

7. (a) Consider RRfn →: defined by 221

)(xn

nxxfn +

= .

Determine whether the convergence is uniform or

not.

(b) Prove that the uniform limit of a sequence of

continuous functions is continuous.

(c) Let MMT →: be a contraction mapping. Then

prove that T is continuous on M. (8 + 8 + 4 = 20)

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DE–8028

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WK 6

8. (a) State and prove Piccard’s theorem.

(b) Let Rfn →]1,0[: be defined by n

xxfn =)( . Prove that

the sequence )( nf converge pointwise to

Rf →]1,0[: defined by 0)( =xf . (12 + 8 = 20)

————————

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WK 5

DE–8029

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

PROBABILITY AND STATISTICS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each question carries equal marks.

(5 × 20 = 100)

1. (a) Prove : If A and B are any two events of a random

experiment with sample space S then

)()()()( BAPBPAPBAP ∩−+=∪ . (10)

(b) The probability that India wins a cricket test match

against West Indies is known to be 2/5. If India and

West Indies play three test matches what is the

probability that :

(i) India will loose all the three test matches.

(ii) India will win atleast one test match.

(iii) India will win all tests.

(iv) India will win atmost one match. (10)

2. (a) Obtain the

(i) Mean

(ii) Median and

(iii) Mode for the following distribution :

<<−=

elsewhere0

10if)(6)(

2 xxxxf (10)

25

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DE–8029

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WK 5

(b) Find the correlation coefficient from the following

data : (10)

X : 10 12 18 24 23 27

Y : 13 18 12 25 30 10

3. (a) If X , Y and Z are uncorrelated random variables

each having same standard deviation obtain the

coefficient of correlation between the r.v’s YX +

and ZY + . (10)

(b) Let 1x , 2x , ...., nx be the ranks of n individuals

according to a character A and 1y , 2y , ...., ny the

ranks of the same individuals according to another

character B . It is given that nyx ii +=+ 1 for

ni ...,,2,1= . Show that the value of the rank

correlation ρ between the character A and B

is –1. (10)

4. (a) The random variables X and Y are connected by

the equation 0=++ cbyax . Show that −=XYγ 1

or 1 according as a and b are of the same sign or of

opposite sign. (10)

(b) Find the correlation coefficient between X and Y

from the following table : (10)

x

y

5 10 15 20

4 2 4 5 4

6 5 3 6 2

8 3 8 2 3

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DE–8029

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WK 5

5. (a) Find β and γ coefficients for the Binomial

distribution and discuss the results with special

reference to skewness and Kurtosis. (10)

(b) Assuming that one in 80 births is a case of twins

calculate the probability of 2 or more births of twins

on a day when 30 births occur using

(i) Binomial distribution and

(ii) Poisson distribution. (10)

6. (a) A random variable X has the probability function

2

1)( =xp , ...,3,2,1=x find its moment generating

function and variance. (10)

(b) In a normal distribution 31% of the items are under

45 and 8% are over 64. Find the mean )(µ and

standard deviation )(σ . (10)

7. (a) The theory predicts that the proportion of an object

available in four groups A , B , C , D should be

9 : 3 : 3 : 1. In an experiment among 1600 items of

this object the numbers in the numbers in the four

groups were 882, 33, 287 and 118. Use 2χ – test to

verify whether the experimental results supports

the theory. (10)

(b) The table gives the biological values of protein from

6 cow’s milk and 6 buffalo’s milk. Examine whether

the differences are significant. (10)

Cow’s milk Buffalo’s milk

1.8 20

2.0 1.8

1.9 1.8

1.6 2.0

1.8 2.1

1.5 1.9

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DE–8029

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WK 5

8. (a) The following is the statistics showing the life times

in hours of four batches of electric bulbs sold in

different shops. Perform an analysis of variance and

state your conclusion. (10)

Batches S1 S2 S3 S4 S5 S6 S7 S8

A 1600 1610 1650 1680 1700 1720 1800 –

B 1580 1640 1640 1700 1750 – – –

C 1460 1550 1600 1620 1640 1660 1740 1820

D 1510 1520 1530 1570 1600 1680 – –

(b) Calculate :

(i) Laspeyre’s

(ii) Paasches’

(iii) Fishers’ index numbers for the following data.

Hence or otherwise find Edgeworth and

Bowley’s index numbers. (10)

Base Year 1990 Current year 1992

Commodities Price Quantity Price Quantity

A 2 10 3 12

B 5 16 6.5 11

C 3.5 18 4 16

D 7 21 9 25

E 3 11 3.5 20

–––––––––––––––

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wk ser

DE–8030

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

ALGEBRA

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

All questions carry equal marks.

(5 × 20 = 100)

1. (a) If A, B and C are any three finite sets prove that

−∩−∩−++=∪∪ CBBACBACBA

CBAAC ∩∩+∩ . (10)

(b) If A, B and C are any three sets prove that

(i) ( ) ( ) ( )CBCACBA ×∩×=×∩

(ii) ( ) ( ) ( )CABACBA ×−×=−× . (10)

2. (a) If ρ and σ are equivalence relations defined on a set S, prove that σρ ∩ is also an equivalence

relation. (10)

(b) Let YXf →: be a function. If XA ⊆ and YB ⊆

show that :

(i) ( )[ ]AffA 1−⊆ and

(ii) ( )[ ] BBff ⊆−1 .

(iii) Give an example to show that equality need

not hold in (i) and (ii).

(iv) In each case state when will the equality hold?

(10)

31

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wk ser

3. (a) Prove : Let G be a group and Gba ∈, , then

(i) order of a = order of 1−a .

(ii) order of a = order of abb 1− .

(iii) order of ab = order of ba . (10)

(b) Let H and K be two subgroups of G of finite index

in G. Prove that KH ∩ is a subgroup of finite

index in G. (10)

4. (a) State and prove ‘Lagrange’s theorem’ on group. (10)

(b) State and prove ‘Fundamental theorem’ of

homomorphism. (10)

5. (a) Let Z be the ring of integers and R any ring. Define

( ){ }RxZmxmRZ ∈∈=× and/, , ⊕ and . on RZ × as follows ( ) ( ) ( );,.. yxnmynxm ++=⊕

( ) ( ) ( )xynxmymnynxm ++= .... where nx and

my denotes respectively the concerned multiples of

the elements x and y in R . Prove that RZ × is a

ring under ⊕ and . . Also prove that RZ × is

commutative iff R is commutative. (10)

(b) Prove : The field of quotients F of an integral

domain D is the smallest field containing D. (ie) If F’

is any other field containing D then F’ contains a

subfield isomorphic to F. (10)

6. (a) Prove that the ring of Gaussian integers

{ }ZbabiaR ∈+= ,/ is an Eucledian domain, where

( ) 22 babiad +=+ . (10)

(b) State and prove ‘Division algorithm’. (10)

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wk ser

7. (a) Let V be a vector space over a field F. Let S, VT ⊆ ,

prove that

(i) ( ) ( )TLSLTS ⊆⇒⊆

(ii) ( ) ( ) ( )TLSLTSL +=∪

(iii) ( ) SSL = iff S is a subspace of V. (10)

(b) Prove : Let V be a vector space over a field F. Let

{ }nVVVS ,......,, 21= spans V, and { }nWWWS .....,,, 21=

be a linearly independent set of vectors in V, then

nm ≤ . (10)

8. (a) Prove that every finite dimensional inner product

space has orthonormal basis. (10)

(b) Prove : Let V be a finite dimensional inner product

space, and W be a subspace of V, then V is the direct

sum of W and ⊥W (ie) ⊥⊕= WWV . (10)

__________________

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Ws 18

DE–8031

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014

OPERATIONS RESEARCH

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each question carries equal marks.

(5 × 20 = 100)

1. (a) Solve the graphically the LPP;

minimize yxZ 32 += ,

Subject to :

120,200

,0

,0

,30

≤≤≤≤≥

≥−≤+

yx

y

yx

yx

(10)

(b) Explain Simplex Method to find an optimal basic feasible solution to LPP. (10)

2. (a) A farmer has 100 acres of land. He cultivates tomato, brinjal and green chilly in this land. He can sell all vegetable in the market and earn a profit of Rs. 2per kg on tomatoes and brinjal and Rs .3 per kg on green chillies . The average yield per acre in 2000 kgs of tomatoes, 3000 kgs of brinjals and 500 kgs of green chilly. Fertilizers are available at Rs. 5 per kg and the amount required per acre is 100 kgs each for tomatoes, brinjals and 50 kgs for green chillies. Labour required for sowing, cultivating and harvesting per acre is 5 man days for tomatoes, 4 man days for brinjal and 6 man days of labour are available at Rs. 60 per man-day. Formulate this problem as an LPP to maximize the farmer’s total profit. (10)

32

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DE–8031

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Ws 18

(b) Use Simplex method to solve the following LPP :

Maximize 21 32 xxz += ,

Subject to the constrains :

ed.unrestrict,,93

6

42

2121

21

21

xxxx

xx

xx

≤+≤+

≤+−

(10)

3. (a) Use the penalty method to solve the following LPP.

Maximize 21 32 xxz +=

Subject to the constrains :

0and0,3

and42

2121

21

≥≥=+≤+

xxxx

xx

(10)

(b) Solve the following LPP using Two-phase Simplex method :

Minimize 21 8060 xxZ += ,

Subject to :

0,,12003040

9003020

2121

21

≥≥+≥+

xxxx

xx

(10)

4. (a) Explain Two-phase Simplex method algorithm. (10)

(b) Find the integer solution to the L.P.P. :

Maximize 21 22 xxz +=

Subject to the constrains :

0,,42

,835

2121

21

≥≤+≤+

xxxx

xx and are integers. (10)

5. (a) Using Row Minima method find a basic feasible solution to the following transportation problem : (10)

1w 2w 3w ai

1F 8 10 12 900

2F 12 13 12 1000

3F 14 10 11 1200

bj 1200 1000 900 3100

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(b) A company produces a small component for an

industrial product and distributes it to five

wholesale at a fixed delivered price of Rs. 2.50 per

unit. Sales forecast indicate that monthly deliveries

will be 3000, 3000,10000, 5000 and 4000 units to

wholesalers 1,2,3,4 and 5 respectively. The monthly

production capacities are Re. 1.00, Re 0.90 and

Re. 0.80 at plants 1, 2 and 3 respectively. The

transportation costs of shipping a unit from a plant

to a wholesaler are given below :

Wholesaler

1 2 3 4 5

1 .05 .07 .10 .15 .15

Plant 2 .08 .06 .09 .12 .14

3 .10 .09 .08 .10 .15

Find how many components each plant supplies to

each wholesaler in order to maximize its profit. (10)

6. A departmental head has four subordinates, and four

tasks to be performed. The subordinates differ in

efficiency. And the tasks differ in their intrinsic difficulty.

His estimate, of the time each man would take to perform

each tasks, is given in the matrix below :

Men Tasks

E F G H

A 18 26 17 11

B 13 28 14 26

C 38 19 18 15

D 19 26 24 10

How the tasks should be allocated one to a man, so as to

minimize the total man-hours? (20)

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7. (a) Use graphical method in solving the following game

(10)

Player A

Player B 2 2 3 –2

4 3 2 6

(b) (i) Define zero-sum game. (5)

(ii) Define payoff matrix. (5)

8. (a) Discuss about PERT algorithm. (10)

(b) For the following directed network

(i) Determine a directed path from node 1 to node

6. Identify three undirected paths connecting

node 1 and node 6.

(ii) Find three directed cycles.

(iii) Grow a tree one link at a time until a

spanning tree has been obtained.

(iv) Repeat the process of (c) to obtain another

spanning tree. (10)

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DE–8032

DISTANCE EDUCATION

B.Sc. (Mathematics). DEGREE EXAMINATION, MAY 2014

NUMERICAL METHODS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

All questions carry equal marks.

(5 × 20 = 100)

1. (a) Find a real roots of the equation 0113 =−− xx by

using bisection method. (10)

(b) (i) Explain the method of false position is solving

an algebraic equations. (5)

(ii) Evaluate 12 to four places of decimals by

Newton – Rapshon method. (5)

2. (a) Find the inverse of the matrix

=941

323

112

A using

Gaussian method. (10)

(b) Solve the system of equations by Gauss elimination

method : (10)

42

332

323

=++−=−−

=++

zyx

zyx

zyx

33

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Ws 1

3. (a) Using Newton’s forward interpolation find ( )2.0f

from the following data : (10)

x : 0 1 2 3 4 5 6

( )xfY = : 176 185 194 203 212 220 229

(b) Define Langrange’s intepolation formula. (10)

4. (a) Using Bessel’s formula find ( )25Y from the given

data : (10)

X : 20 24 28 32

Y : 2854 3162 3544 3992

(b) Using Hermite’s Interpolation to find sin (1.05) from

the following data : (10)

X : 1.0 1.1

:sin xY = 0.84147 0.89121

xY cos' = : 0.5403 0.45360

5. (a) Derive Trapezoidal rule for integration. (10)

(b) Evaluate ∫ +

1

0 21 x

dx using Simpson’s percentage

rule. (10)

6. (a) Derive Weddle’s rule for integral. (10)

(b) Evaluate ∫ +1

0 1 x

dx, using Weddle’s rule. (10)

7. Using Runge–Kutta method to find ( )1.0y , ( )2.0y and

( )3.0y from the differential equation

,1 xydx

dy += ( ) 20 =y . (20)

8. Find an approximate solution of the initial value problem

( ) 00,1' 2 =+= yyY by Picards methods and compare it

with the exact solution. (20)

————————

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DE–8033

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

COMPLEX ANALYSIS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each questions carries equal marks.

(5 × 20 = 100)

1. (a) Prove that ( ) zzf = lm z is differentiable only at

0=z and find ( )0'f . (10)

(b) Prove that the functions ( )zf and ( )zf are

simultaneously analytic. (10)

2. (a) Show that

( ) xyyxyxyxyxu 4sinhcos2coshsin, 22 +−++= is

harmonic. Find an analytic function ( )zf in terms of

zwith the given u for its real part. (10)

(b) Prove :

If u and v are harmonic functions satisfying the

Cauchy-Riemann equations in a region D then

ivuf += is analytic in D. (10)

3. (a) Show that the region in the z - plane given by 0>x

and 20 << y is mapped to the region in the w -

plane given by 11 <<− u and 0>v under the

transformation 1+= izw . (10)

(b) Prove that any bilinear transformation can be

expressed as a product of translation, rotation,

magnification or contraction and inversion. (10)

34

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2

4. (a) Find the bilinear transformation which maps

the points ∞−= ,1,1z respectively on iiw ,1,−−= .

(10)

(b) Prove let C be a circle or a straight line and 21 ,zz be

inverse points or reflection points with respect to C .

Let 21 ,ww and 1C be the image of 21 ,zz and C under

bilinear transformation. Then 1w and 2w are

inverse points or reflection points will respect to 1C .

(10)

5. (a) Prove : Let ( )zf be a function which is analytic

inside and on a simple. closed curve C and 0z be

any point in the interior of C , then

( ) ( )∫ −

=c

dzzz

zf

izf

0

02

1

π.

(10)

(b) Evaluate dzz

z

c∫ −12 where C is the positively

oriented circle 2=z . (10)

6. (a) State and prove Liouvilie’s theorem. (10)

(b) Evaluate ( )∫ +c

dziz

z

2

3

, where C is the unit circle

2=z . (10)

7. (a) For the function ( )( )izz

zzf

++2 13

, find

(i) a Taylor’s series valid in a neigh-bourhood

of iz = and

(ii) a Laurent’s series valid within an annulus of

which centre is the origin. (10)

(b) Calculate the residue of ( )zz

z

2

12 −

+ at its poles. (10)

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3

8. (a) Show that the function izez −+ 22 has Precisely one

zero in the open upper half plane. (10)

(b) Prove that ( )∫ =

+c

z

e

idz

z

e23

2 4

1

π, where C is

2

3=Z .

(10)

———————

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DE–8034

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2014.

DISCRETE MATHEMATICS

Time : Three hours Maximum : 100 marks

Answer any FIVE questions.

Each questions carries equal marks.

(5 × 20 = 100)

1. (a) Explain: Negation, Conjunction, Disjunction, Conditional statement and Biconditional statement. (10)

(b) Construct a truth table for the following compound propositions:

(i) ( )rpqp →↔∨ )(

(ii) ( )( )srqp →↔→ . (10)

2. (a) Indicate which ones are tautology or contradiction

(i) ( ) ( )rprq ∧→∨

(ii) ( )( ) ( ) ( )( )RPQPRQP →→→⇒→→ . (10)

(b) Obtain the principal disjunctive normal form for

( ) ( )( )PQQPP ∨∧→→ . (10)

3. (a) Show that:

( ) ( ) ( ) ( ) ( ){ }rputustqsrqp →∧→∧→→∧→ ,,,

pq⇒ .

(10)

35

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DE–8034

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2

(b) Determine the validity of the following argument:

My father praises me only if I can be proud of

myself. Either I do well in sports or cannot be proud

of myself. If I study hard, then I cannot do well in

sports. Therefore, if father praises me, then I do not

study well. (10)

4. (a) Show that:

( ) ( ) ( )( ) ( ) ( ) ( )( )yWyMyxSxFx →∀→∧∀ . (10)

(b) Show that ( ) ( )xMx∀ follows logically from the

premises ( ) ( ) ( )( )xMxHx →∀ and ( ) ( )( )xHx∀ . (10)

5. (a) Prove in any graph, the number of vertices of odd

degree is always even. (10)

(b) Define the adjacency matrix and incidence matrix of

a graph with an example. (10)

6. (a) (i) Prove: A connected graph with n vertices and

1−n edges is a tree. (5)

(ii) Define binary tree with an example. (5)

(b) Prove Let T be a full binary tree with n vertices,

then the number of leaves in T is ( ) 2/1+n . (10)

7. (a) Prove: Every cycle has an even number of edges in

common with any cut-set. (10)

(b) Prove that if ( )1≠n is a positive odd integer, then

the complete graph nK contains ( )

!2

1−n edge —

disjoint Hamilton cycles. (10)

8. (a) Explain Ford and Fulkerson algorithm. (10)

(b) State Euler’s formula for a graph ( )EVG ,= . (10)

————————