# BSC MATHS

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```					 DE–1327                                                             13

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

CLASSICAL ALGEBRA

Time : Three hours                                Maximum : 100 marks
All questions carry equal marks.

(5  20 = 100)

1.    (a)    Discuss the convergence of

np
(i)
n 1  n

xn
(ii)    n  1    n
,x  0

(b)   Examine the convergence of the series

1 4    9 2           n2
(i)      x    x  ...         x n 1  ... , x  0
2 9   28           1  n3

1 1.2 1.2.3
(ii)              ... .
3 3.5 3.5.7

2.    (a)    Test the convergence of

3n n!
(i)    
n 1 n
n


1
(ii)    4n
n 1
2
1

 n 1 
(b)   (i)    Show that the sequence          is convergent.
 2n  7 

n3  1
(ii)   Test for convergency the series             n
n 1 2  1
.

3.    (a)    Find the sum to infinity :

1            DE–288
15 15.21 15.21.27
               ...
16 16.24 16.24.32

3          1 1 3 1 32 
(b)   Show that       log 10  7   14   21   log 2 .
10         2   2 2    3 2 

4   1   1   1   1
4.    (a)   Show that log                   ...
e 1.2 2.3 3.4 4.5
23    33 2
(b)   Sum the series : 1       x    x  ...   .
1!    2!

5.    (a)   Form the equation whose roots are the squares of the differences of the
roots of x 3  px  q  0 where p, q are real. Hence deduce the condition
that all the roots of the cubic shall be real.
(b)   Increase by two the roots of x 4  x 3  10x 2  4x  24  0 and hence solve the
equation.

1            1          1
6.    (a)   If  ,  ,  are the roots of the equation x 3  7x  7  0 , find       4
       4
            .
                      4

(b)   If a, b, c are the roots of x 3  px 2  qx  r  0 , form the equation whose roots

are bc  a 2 , ca  b2 and ab  c 2 . Hence find the value of ab  c 2            bc  a       2

ca  b  .
2

3 2         2  3
7.    (a)                                             7 5  , B   1 3  and verify
Find the inverse of the two matrices A                   
                 
the relation  AB 
1
 B 1 A 1 .

(b)   Solve : x  y  2z  4 ; 2x  y  3z  9 ; 3x  y  z  2 by Cramers rule.

3  4 4
8.    (a)   Find the eigen values and eigen vectors of A  1  2 4 .
       
1  1 3
       
(b)   Solve, by using matrices, the equations
x  2 y  z  2 ; 3x  8 y  2z  10 ; 4x  9 y  z  12 .

—————————
DE–1328                                                            14

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

2       DE–288
CALCULUS

Time : Three hours                                            Maximum : 100 marks

All questions carry equal marks.

1. (a)      Differentiate sin(cos x ) and log(cos x ) .

dy y( y  x log y )
(b)   If x y  y x , prove that                                 .
dx x( x  y log x )

2.    (a)   Show that the envelope of the polar of the points on the ellipse
x 2 y2                          x2  y2        a 2 x 2 b2 y2
 2  1 w.r.t. to the ellipse 2  2  1 is         4 1.
a2 b                            A   B          A4      B

(b)    Find the radius of curvature of the curve x  3t2 , y  3t  t3 at t  1 .

dx
3.    (a)   Evaluate

  2   x   3
x2  9
.

(b)   Evaluate      log sin x dx .
0

 2

 cos
7
4.    (a)   Evaluate                         x dx
0

 2

 sin
2
(b)   Evaluate                          cos 7  d .
      2

dy
5.    (a)   Solve x( x  2)              2( x  1) y  x 3 ( x  2) given that y  9 when x  3 .
dx
(b)   Solve ( y  px )( p  1)  p .

d2 y      dy
6.    (a)   Solve x 2      2
 2x     6x 2  2x  1 .
dx        dx

(b)   Solve ( D 2  a 2 ) y  e nx  e ax .

7.    (a)   (i)    Find L(cos t cos 2t) .

3        DE–288
 s 1 
(ii)    Find L1  2      .
 s  2s 

(b)    Find L(t e t cos t ) .

8.    (a)    Solve       p q y.

(b)    Solve ( y  z  x ) p  ( z  x  y)q  x  y  z .

—————————
DE–1329                                                                  15

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

ANALYTICAL GEOMETRY AND VECTOR CALCULUS

Time : Three hours                                 Maximum : 100 marks

All questions carry equal marks.

1. (a)     Show that the equation of the pair of straight lines each inclined at an
angle 45 to one of other of the lines given by the equation ax 2  2hxy  by 2  0 is
a  2h  b x 2  2a  bxy  a  2h  b y 2    0.

(b)   Show that if the circle x 2  y 2  a 2 cuts off from the line y  mx  c ,
       
chord of length 2b , then c 2  1  m 2 a 2  b2 .        
2.    (a)    Show         that          the    circles          x 2  y 2  6 x  9 y  13  0   and
2     2
x  y  2x  16 y  0 touch each other and find the co-ordinates of the
point of contact.
(b)   Find the equation of the circle passing through the intersection of
x 2  y 2  6  0 and x 2  y 2  4 y  1  0 and through the point  1,1 .

3.    (a)    Show that the straight lines whose direction cosine are given by
2l  m  2n  0 and lm  mn  nl  0 are at right angles.

4               DE–288
(b)   A moving plane passes through a fixed point  ,  ,   and intersects the
coordinates axes at A, B, C . Show that the locus of the centroid of the
               
 ABC IS                        3.
x       y       z

4.   (a)   Find in symmetrical forms the equation of the line given by
x  5 y  z  7 ; 2x  5 y  3z  1  0 .

(b)   Find the perpendicular distance of the point P 1, 1, 1 from the line
x 2 y3    z
        . Also find the foot of the perpendicular.
3   2    1

5.   (a)   Find the equation in the symmetrical form of the orthogonal projection of
x 1 y 1 z  3
the line                  in the plane x  2 y  z  12 .
2    1       4
(b)   Find the shortest distance and the equation of the line of shortest
x 3 y 6 z
distance  between    the   straight  line                    and
4      6    2
x 2 y z 7
        .
4     1   1

6.   (a)   Show that the plane 2x  2 y  z  12  0 touches the sphere

x 2  y 2  z 2  2x  4 y  2z  3  0 . Also find the point of contact.

(b)   Find the equations of the spheres which pass through the circle

x 2  y 2  z 2  2x  2 y  4 z  3  0 ;    2x  y  z  4  0   and   the   plane
3x  4 y  14  0 .

7.   (a)   If r is the position vector of any point                     P x, y, z  , prove that
grad r n  nr n  2 r .

(b)   Prove that curl r  a   2a and div r  a   0 , where a is constant vector.

8.   (a)   Evaluate     f  dr                          
where f  x 2  y 2 i  2xyj and the curve C is the
C
rectangle in the x  y plane bounded by y  0 , y  b , x  0 , x  a .

(b)   Verify Stoke’s theorem for the vector function f  y 2i  yj  xz k and S
is the upper half of the sphere x 2  y 2  z 2  a 2 and z  0 .

5          DE–288
—————————
DE–1330                                                     23

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

MECHANICS

Time : Three hours                        Maximum : 100 marks

(5  20 = 100)

1. (a)     State and prove the converse of triangle of forces.
(b)   A weight is supported by a string inclined to the horizon at angle  on a
inclined plane of inclination  . If the slope of the plane is increased to 
and slope of the string unaltered the tension of the string is doubled.
Prove that cot   2 cot   tan  .

2.   (a)   E is the middle point of the side CD of a square ABCD . Forces 16, 20,
4 5 , 12 2 kgwt act along AB , AD , EA , CA . Show they are in
equilibrium.
(b)   Prove : Any system of forces acting in one plane on a rigid body can be
reduced to a single force or a couple.

3.   (a)   State and prove Varignon’s theorem.
(b)   Discuss the equilibrium of a body on a rough inclined plane under a force
parallel to the plane.

4.   (a)   A heavy uniform rod of length 2a rests partly within and partly without
a smooth spherical bowl of raidus r , fixed with its rim horizontal. If  is
the inclination of the rod to the horizon show that 2r cos2  a cos .
(b)   A uniform ladder rests in limiting equilibrium with its lower end on a
rough horizontal plane and its upper end against an equally rough
vertical wall. If  be the inclination of the ladder to the vertical prove
2
that tan          where  is the coefficient of friction of wall, ground.
1  2

5.   (a)   Determine when the horizontal range of a projectile is maximum, given
the magnitude u of the velocity of projection.
(b)   Derive the equation of the enveloping parabola.

6          DE–288
6.   If two spheres of given masses with given velocities impinge directly then show
that there is a loss of kinetic energy and find the amount of loss.

7.   (a)   Find the composition of two SHM (Simple Harmonic Motion) of same
period and in two perpendicular directions.
(b)   A particle is suspended from a fixed point by a spiral string of length a
and modulus  . If it is slightly displaced in the vertical direction,
discuss the subsequent motion.

8.   (a)   Derive the differential equation of a central orbit in polar coordinates.
(b)   Find the law of force towards pole under which the curve r n  a n cos n
can be described.

–––––––––––––––
DE–1331                                                         24

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

ANALYSIS

Time : Three hours                          Maximum : 100 marks

All questions carry equal marks.
(5  20 = 100)

d x , y 
1. (a)     If   M, d  is a metric space then prove that        d1 x, y                    is a
1  d x , y 
metric on M .     (10)
(b)   Prove that in any metric space, every closed ball is a closed set.
(10)

2.   (a)   If A is any non-empty subset of a metric space M, d  then prove that A
is open if and only if A can be expressed as the union of a family of open
balls.                                           (10)

(b)   If M1 , d1 , M 2 , d2  are two metric spaces and if a M1 , then prove that
f : M1  M 2 is continuous at ‘ a ’ if and only if xn   a  f xn   f a  .
(10)

7           DE–288
3.   Prove that a subspace of R is connected if and only if it is an interval.
(20)

4.   State and prove Cantor’s intersection theorem.           (20)

5.   (a)   Prove that any compact subset of a metric space is closed.
(10)
(b)   Prove that any closed interval a, b is a compact subset of R .
(10)

6.   (a)   Prove that continuous image of a compact metric space is compact.
(10)
(b)   Prove that any continuous mapping f defined on a compact metric space
M1 ,d1  into any other metric space M2 ,d2  is uniformly continuous on
M1 . (10)

7.   (a)   Define pointwise convergence of a sequence of functions and give one
example with illustration.                   (10)
(b)   State and prove Cauchy criterion for uniform convergence.
(10)

8.   State and prove Picard’s existence theorem.              (20)

————————
DE–1332                                                     25

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

PROBABILITY AND STATISTICS

Time : Three hours                        Maximum : 100 marks

Each question carries 20 marks.

1. (a)      If A and B are any 2 events of a sample space S then prove that
P  A  B   P  A   P B   P  A  B  .

8          DE–288
1 1 1 1
(b)   The chances that 4 students A , B, C, D solve a problem are, , ,
2 3 4 4
respectively. If all of them try to solve the problem, what is the
probability that the problem is solved?

2.   (a)   A random variable X has the following probability functions.
xi :     –2     –1          0           1       2   3

P xi  : 0.1          k       0.2 2k 0.3 k

Find :
(i)    k
(ii)   mean
(iii) variance
(iv)     p x  2, v, p x  2 .

(b)   Find :
(i)    the mean deviation from the mean
(ii)   variance of the arithmetic progression a, a  d , a  2d,...,a  2nd .

3.   (a)   Fit a second degree parabola by taking x i as independent variable.
x: 0 1           2           3       4
y : 1 5 10 22 38

(b)   Prove that the correlation coefficient is independent of the change of
origin and scale.

4.   (a)   10 competitors in a beauty contest were ranked by 3 judges in the
following order.
J1 : 1 6 5 10 3                       2       4       9   7 8
J2 : 3 5 8            4       7 10 2                  1   6 9
J3 : 6 4 9            8       1       2       3 10 5 7

Use rank correlation coefficient to discuss which pair of judges have the
nearest approach to common tastes in beauty.
(b)   The two variables x and y have the regression lines 3x  2 y  26  0 and
6x  y  31  0 .

Find :

9                   DE–288
(i)    Mean values of x and y
(ii)   Correlation coefficient between x and y.
(iii) The variance of y if the variance of x = 25.

5.   (a)   Find  and  coefficients for the Binomial distribution and discuss the
results with reference to skewness and kurtosis.

(b)   Fit a Poisson distribution for the following data :
x:      0       1       2        3 4 Total
f:   123 59 14 3 1                      200

6.   (a)   The marks of 1000 students in a university are found to be normally
distributed with mean 70 and S.D. 5. Estimate the number of students
whose marks will be (i) between 60 and 75 (ii) less then 60.

(b)   Assuming that 1/80 births is a case of twins calculate the probability of 2
of more births of twins on a day when 30 births occur using (i) Binomial
distribution (ii) Poisson distribution.

7.   (a)   A group of 10 rats fed on a diet A and another group of 8 rats fed on a
different diet B recorded the following increase in weight in grams.
Diet A : 5 6 8 1 12                      4    3 9 6 10
Diet B : 2 3 6 8                    1    10 2 8 –                 –

Test whether diet A is superior to diet B.
(b)   Five coins are tossed 320 times. The number of heads observed is given
below. Examine whether the coin is unbiased.
No. of Heads :       0       1        2    3     4        5    Total

Frequency :          15 45 85 95 60 20                          320

8.   (a)   Analyse the variance in the following :
Latin square A8 C18                           B9
C9        B18 A16
B11 A10 C20

(b)   Verify time reversal test and factor reversal test from the following data.
Commodities Base Year Current Year
Price Qty             Price          Qty

10             DE–288
A          10          25   12    30
B             8        21   9     25
C          4.5         28   6.5   35
D          3.5         16   4     20

————————
DE–1333                                                             31

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

ALGEBRA

Time : Three hours                                  Maximum : 100 marks

(5  20 = 100)

1. (a)     Show that if A and B are two sets then A  B , B  A and A  B are
pairwise disjoint.   (6)
(b)   Give an example of a subset of R R which can not be expressed as a
Cartesian product of two subsets of R .
(6)

(c)   Show that the union of two equivalence relations need not be an equivalence
relation.  (8)

2.    (a)   Out of 200 persons living in a street 120 drink coffee, 71 drink tea. 60
persons drink both coffee and tea. Find the number of persons who drink
neither coffee nor tea.                         (6)

(b)   State and prove De Morgans’ laws.                              (6)

(c)   Show that a function f : A  B is a bijection iff there exists a unique function
g : B  A such that g  f  i A and f  g  iB .                   (8)

3.    (a)   Show that        c       is a group under usual multiplication given by
a  ibc  id   ac  bd   iad  bc .            (6)

Let G be a group in which ab   a mbm for three consecutive integers and for
m
(b)
all a, b  G . Prove that G is abelian.                           (6)

11        DE–288
(c)   Let A and B be two subgroups of a group G . Show that AB is a subgroup of
G iff AB  BA .    (8)

x        x
4.    (a)   Show that the set of all matrices of the form 
x          where x  R  is a
         x

group under matrix multiplication.                                    (10)
(b)   State and prove Lagrange’s theorem.                      (10)

5.    (a)   Prove that the set of all real numbers of the form a  b 3 where a, b  Q
under usual addition and multiplication is a ring.         (6)
(b)   Show that the ring of quarternions is a skew field but not a field.
(8)
(c)   Show that a finite commutative ring R without zerodivisors is a field.
(6)

6.    (a)   Show that the ring of Gaussian integers R  a  bi / a, b  Z  is an
Euclidean domain where we define d a  ib  a 2  b2 .        (6)
(b)   Show that the field of quotients F of an integral domain D is the smallest
field containing D . (6)
(c)   State and prove division algorithm.                         (8)

7.    (a)   Let A and B be subspaces of a vector space V . Show that A  B  0
iff every vector V  A  B can be uniquely expressed in the form
V  a  b where a  A and b B .            (10)
(b)   If T : V  W be an isomorphism, prove that T maps a basis of V onto a basis
of W .   (10)

8.    (a)   Let V be a vector space over a field F . Let A and B be subspaces of V .
AB       B
Prove that               .
B     AB
(10)
(b)   Prove that every finite dimensional inner product space has an orthonormal
basis.    (10)

–––––––––––––––
DE–1334                                                     32

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

OPERATIONS RESEARCH

Time : Three hours                          Maximum : 100 marks

12         DE–288

All questions carry equal marks.
(5  20 = 100)

1. (a)      A manufactures produces two types of models M1 and M2. Each M1 model
requires 4 hours of grinding and 2 hours of polishing; whereas each M2 model
requires 2 hours of grinding and 5 hours of polishing. The manufacturer has 2
grinders                                                                       and
3 polishers. Each grinder works for 40 hours a week and each polisher was for 60
hours a week. Profit on an M1 model is RS. 3.00 and on an M2 model is 4.00.
Whatever is produced in a week is sold in the market. How should the manufactures
allocate his production capacity to the two types of models so that he may make the
maximum profit in a week?

(b)   Solve the following LPP graphically.

Maximize :       z 100x1  40x 2

Subject to :      5x1  2x 2 1000
3x1  2x 2  900
x1  2x 2  500
and x1 , x 2  0

2.   (a)   Prove that the dual of dual is primal.

(b)   Solve the following LPP by Simplex method

Maximize : z  3x1  2x 2  5x 3
Subject to : x1  4 x 2  420
3x1  2x 3  460
x1  2x 2  x 3  430
and x1 , x 2 , x 3  0

3.   (a)   Explain Big M-method Algorithm.

(b)   Use two phase simplex method to

Minimize z  2x1  x 2

13   DE–288
x1  x 2  2
Subject to the constraints :
x1  x 2  4 and x1 , x 2  0

4.   (a)   Explain cutting plane method for pure integer programming problem.

(b)   Find the optimum integer solution to the all integer programming
problem.

Maximize : z  x1  x 2
Subject to the constraints : 3x1  2x 2  5, x 2  2 & x1 , x 2  0

5.   (a)   Explain V.A.M. to find an initial basic feasible solution to L.P.P.

(b)   Find the initial basic feasible solution for the following transformation
problem by matrix minima method.
To                   Supply

1        2       1       4       30

From         3        3       2       1       50

4        2       5       9       20

Demand 20 40                 30 10

6.   Solve the following assignment problem
Machines

1        2       3       4    5

1       9        22 58 11 19

Jobs     2       43       78 72 50 63

3       41       28 91 37 45

4       74       42 27 49 39

5       36       11 57 22 25

7.   (a)   Define zero-sum game.

(b)   Solve the following 2  2 game
B

14                  DE–288
A 5 1
3 4

8.   (a)   Explain the following terms :
(i)    Activity
(ii)   Dummy activity
(iii) Dangling.
(b)   Calculate the earliest start, earliest finish, latest start and latest finish of
each activity given below and determine the critical path of the project.
Activity :               1-2 1-3 1-5 2-3 2-4
Duration (in weeks) : 8          7    12    4     10
Activity :               3-4 3-5 3-6 4-6 5-6
Duration (in weeks) : 3          5    10    7     4

——————
DE–1335                                                      33

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

NUMERICAL METHODS

Time : Three hours                         Maximum : 100 marks

(5 × 20 = 100)

1. Find a root of the equation x 3  4x  9  0 correct to three decimal places using
bisection method.

2.   Solve the equations

2x  y  4 z  12
8x  3 y  2z  20
4 x  11 y  z  33

by (a) Gauss-elimination method and (b) Gauss-Jordon method.

3.   Derive the Gregory-Newton forward interpolation formula.

15          DE–288
4.   Derive the Newton’s interpolation formula for unequal intervals and using this
formula, find the value of f 8  given

x:      4     5    7    10   11     13

f x  : 48 100 294 900 1210 2028

5.   (a)   Obtain Newton’s forward difference formula to compute the derivatives.
dy
(b)   From the following table of values of x and y, find         at x  1.10 .
dx
x:   1.00     1.05        1.10   1.15      1.20    1.25      1.30
y : 1.00000 1.02470 1.04881 1.07238 1.09544 1.11803         1.14017

6.   Dividing the range into 10 equal parts, find the approximate value of


 sin x dx , using
0

(a)   Trapezoidal rule and
(b)   Simpson’s rule.

7.   Find the first six terms of the power series solution of y  xy  2 y  0 if
y1  0 , y1  1 .

dy
8.   Solve the equation         1  y with the initial condition x  0, y  0 , using
dx
Euler’s algorithm and tabulate the solutions at x  0.1 , 0.2, 0.3 and 0.4 . Get
the solutions by Euler’s improved method and Euler’s modified method.

———————
DE–1336                                                       34

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

COMPLEX ANALYSIS

Time : Three hours                           Maximum : 100 marks

All questions carry equal marks.

16           DE–288
1. (a)     Prove that the function                                           (12)
x 2 y 5 x iy  if z  0
f z   
0               if z  0

satisfies C-R equations at z  0 , but it is not analytic at z  0 .

2    2                 2 2    2
(b)   If              , prove that      4       .                     (8)
x y y x              x 2 y  z z

2.   (a)   Prove that an analytic function in a region with constant modulus is
constant. (6)

(b)   Prove that u  2x  x 3  3 xy 2 is harmonic and find its harmonic conjugate.
Also find the corresponding analytic function.                                       (8)

u u
(c)   If ux, y  is a harmonic function in a region D, prove that f z                 i  is
x y
analytic in D.             (6)

3.   (a)   Show that the transformation w  z 2 transforms the families of lines x  h
and y k into confocal parabolas, having w 0 as the common focus.
(12)

(b)   Find the bilinear transformation which maps 1, 0,1 of the z-plane anto
1, i,1 of the w-plane. Show that under this transformation the upper
half of the z-plane maps anto the interior of the unit circle w  1 .
(8)

1
4.   (a)   Show that by means of the inversion w                       the circle given by z  3  5 is
z
3   5
mapped into the circle w                        .                                  (8)
16 16

(b)   Show that the transformation w  z1 / 2 maps the upper half of the inside
of the parabola                            
y 2  4c 2 c 2  x      into the infinite strip bounded by
0  u  , 0  v  c where w uiv .                              (12)

x            
2
5.   (a)   Evaluate the integral                       i y2 dz where C is the parabola y  2x 2 from
C

1, 2   to 2, 8 .       (8)

17         DE–288
ez
(b)     Evaluate      z  2z 1
C
2
dz where C is z  3 .   (12)

6.   Let f be a function which is analytic at all points inside and on a simple closed

curve C. Then prove that                f z  dz  0 .
C

(20)

7.   (a)     State and prove Cauchy’s residue theorem.                      (14)

dz
(b)     Evaluate      2z  3
C
where C is z  2 , using residue theorem.

(6)

8.   (a)     State and prove Rouche’s theorem.                              (12)
2
d
(b)     Evaluate      5  4 sin
0
.                            (8)

——————————–
DE–1337                                                                     35

DISTANCE EDUCATION

B.Sc. (Mathematics) DEGREE EXAMINATION, MAY 2011.

DISCRETE MATHEMATICS

Time : Three hours                                        Maximum : 100 marks

All questions carry equal marks.

1. (a)     Define a well formed formula. Also check whether the following are
formulae or not.
(i)     (( p  q)  q)
(ii)    (( p  q)  ( q))

(iii)     pq

(iv)    (( p  q)  (r  q))

(v)      ( p  q )  (r  ( s))

18         DE–288
(b)   Write down the truth table for the following statements and state which of
them are tautologies.
(i)     (q  r )  ( p  r )

(ii)    ( p  ( p  q))  q

(iii)   ( p  q)  ( p  q)

2.    (a)     Show       that      SR         is               a     tautologically   implied   by
( P  Q)  ( P  R)  (Q  S ) .
(b)   By using truth table, obtain the principal disjunctive normal form for
( P  Q )  ( P  R)  (Q  R) and also find the same for P  Q without
using the truth table.

3.    (a)     Show that R  S can be derived from the premises P  (Q  S ) ,
R  P and Q.
(b)   Verify the validity of the following arguments :
(i)     Lions are dangerous animals. There are lions. Therefore there are
dangerous animals.
(ii)    All integers are rational numbers. Some integers are powers of 2.
Therefore, some rational numbers are powers of 2.

4.    (a)     Show that
( x ) ( P ( x )  Q( x ))  ( x ) P ( x )  ( x ) Q( x ) .

(b)   Show that R  S is a valid conclusion from the premises C  D , C  D                     H,
H  ( A  B ) and ( A  B )  ( R  S ) .

5.    (a)     Define the following terms with an example each with respect to graph
theory :
(i)     Simple graph
(ii)    Graph isomorphism
(iii) Complete graph
(iv) Bipartite graph
(v)     Geodesic
(b)   Prove that a simple graph with n vertices must be connected if it has more
(n  1) (n  2)
than                     edges. Also give an example for a graph with
2
(n  1) (n  2)
edges which is not connected.
2

6.    (a)     State and prove Cayley’s formula.

19           DE–288
(b)   Prove that the number of vertices n in a full binary tree is always odd. Also
(n  1)
prove that the number of leaves is         .
2

7.    (a)   State and prove Euler’s formula. Show that if G is a simple planar graph
with atleast three vertices, e(G )  3 n(G )  6 .
(b)   Prove that a given connected graph G is Eulerian if and only if all vertices of G
are of even degree.

8.    (a)   For a graph G, show that vertex connectivity  edge connectivity 
minimum degree  .
(b)   State and prove the max-flow min-cut theorem. Also write down the algorithm
to find a maximum flow.

———————

20         DE–288

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