School of Science

Term End Examination- Nov 2009

Subject: Higher Mathematics (CSE 501)

Common to M.Tech-ITNet &CSE

Time : 3 Hrs Max.Marks: 100

PART – A

Answer any 5 of the following Questions 5 x 8 = 40 Marks 1. Obtain the principal disjunctive normal form of ( ) ( )( )P P Q Q P⎯⎯→ → ∧¬ ¬ ∨¬ .

2. Find and draw all fundamental cycles of the given graph with respect to the spanning tree T, ( ) { }1 2 3 12 7 6 5 e ,e ,e ,e ,e ,e ,e=E T

3. Which of the following posets forms a lattice? Justify briefly your answer (i) (ii) (iii) (iv)

4. a). Let G be a given simple connected graph on ‘n’ vertices such that each edge of G lies on atmost one cycle. If G has ‘r’ cycles, find the number of edges of G.

1e

2e

7e

5e4e 6e8e

9e 10e

12e 11e

3e

a

b c d

e

k

l

m

n o

f

g

h

I

j

kk

l

o

n

m

b).If G represents a graph, whose incidence matrix is given by

I(G) =

0 1 z 1 y 0 11 w y 0 0 0 00 x 0 0 0 0 0x 0 z 1 0 y x zy z x y 0 w w w

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥+⎣ ⎦

Compute the values of x, y, z, w and the adjacency matrix of G.

5. Find the least positive integer x such that x 5(mod 7)≡ and x 7(mod11)≡ and x 3(mod13)≡ 6. Show that R (P Q)∧ ∨ is a valid conclusion from the premises P Q∨ , Q R→ , P M→ , M¬

7. a) Let G be a simple graph with n vertices and q edges, such that each vertex of G is of degree either 3 or 4 . Find the number of vertices of degree 4 in G.

b) Let G be a simple connected graph with all its vertices are of degree 3 or 4. For all edges e {u, v} E(G) d(u) d(v) 7= ∈ ⇔ + = . Does this graph contains any cycle of ODD length? Justify your answer. Also prove that

u X v Yd(u) d(v)

∈ ∈

=∑ ∑ ,

where X = {u V(G) | d(u) 3}∈ = , Y {v V(G) | d(v) 4}= ∈ =

8. At what average rate must a clerk in a supermarket work in order to ensure a probability of 0.90 that the customer will not wait longer than minutes? It is assumed that there is

only one counter at which customers arrive in a Poisson fashion at an average rate of 15 per hour and that the length of the service by the clerk has an exponential distribution.

PART – B

Answer any 3 of the following Questions 3 x 20 = 60 Marks

9. Prove that for any prime p , all the coefficients of the polynomial p 1f (x) (x 1)(x 2) (x p 1) x 1−= − − ⋅⋅⋅ − + − + are divisible by p.

10. i. Let L be a Lattice. Prove that the following statements are true.

1. ∨ ∧a a = a , a a = a 2. ∨ ∨ ∧ ∧a b = b a , a b = b a 3. ( ) ( ) ( ) ( )∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧a b c = a b c , a b c = a b c 4. ( ) ( )∨ ∧ ∧ ∨a a b = a , a a b = a

ii. Compute ( ) ( ) ( ) ( )a c e f a b c e∨ ∧ ∨ ∨ ∨ ∧ ∨⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

11. i. If T1 and T2 are given two spanning trees of a given connected graph G, then prove the following :

( ) ( )E( ) E( ) E( ) E( ) E( ) E( )1 2 1 2 G T T T G T =− −∩ ∩ ii. Find the Minimum and Maximum Spanning Trees of the following connected graph.

12. a. Show that ( )x M(x)∃ follows logically from the premises

( )( )x H(x) M(x)⎯⎯→ and ( )x H(x)∃

0

a b

c

e f

I

0v

1 2

9

2

93

61

56

2

1

8

2

9 1

34 1 4

7

7

0u

b. Prove that the total number of vertices of degree 2 of a given tree ‘T’ on ‘q’ edges ( 2)≥q is equal to [ ]

v V(T)(v) 3

( 1) (v) 1∈

≥

− − −∑qd

d .

Also prove that the expression j

j

jw V(T)

) 3

2 d(w ) 2 d(w

∈≥

⎡ ⎤+ −⎣ ⎦∑ represents the total number of

pendent vertices of T

13. Customers arrive at a one-man barber shop according to a Poisson

process with a mean inter arrival time of 12 min. Customers spend an average of 10 min in the barber’s chair.

(a). What is the expected number of customers in the barber shop and in the queue

(b). Calculate the percentage of time an arrival can walk straight into the barber’s chair without having to wait.

(c). How much time can a customer expect to spend in the barber’s shop? (d). Management will provide another chair and hire another barber , when a

Customer’s waiting time in the shop exceeds 1.25 h. How much must the average rate of arrivals increase to warrant a second barber? (e). What is the average time customers spend in the queue? (f). What is the probability that the waiting time in the system is greater than 30 min

(g) Calculate the percentage of customers who have to wait prior to getting into the barber’s chair.

(h) What is the probability that more than 3 customers are in the system?