IGNOU MCA MCS-033 Advanced Discrete Mathematics SOLVED ASSIGNMENTS 2014-2015

IGNOU SOLVED ASSIGNMENTS FOR 3rd SEMESTER MCS-033 2014-2015  Advanced Discrete Mathematics

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Q.1. Define each of the following concepts from graph theory and give one suitable example for the concept:
 i) Complete graph                  ii) Path
 iii) Cycle                               iv) Subgraph
 v) Complement of a graph    vi) Connected components of a graph
vii) Bipartite                         viii) Spanning
 ix) Vertex cut-set                   x) Eulerian curcit
xi) Eulerian graph                 xii) Hamiltonian graph
xiii) Open trail                     xiv) Edge traceable graph
xv) Biapartite graph

Q.2. A person deposits Rs. 250, 000/-in a bank in a saving bank account at a rate of 8 % per annum. Let Pn be the amount payable after n years, set up a recurrence relation to model the problem. Also using the recurrence relation, find amount payable after 9 years.

Q.3. For each of the following recurrences, find its order and degree and also tell whether it is homogeneous or non-homogeneous

i) an = an-1 + an-2 + … + a0 ii) an = nan-2 + 2n iii) an = √an –1 + (a n –2) 2  9 iv) an = (an–1) 2+ an-2 an-3 an-4 v) an = sin an-1 + cos an-2 + sin an-3 + …+ an vi) bn = bn – 1 + (n + 3) vii) an = an –1     a1 + an – 2 a2 + ……+ a1 an – 1 (for n  2)                       
                                                                    

Q.4. The following recurrence equation represents the Tower of Hanoi problem: Cn = 2 Cn– 1 + 1 (for n  2) and C1 =1 Verify, using Principle of Mathematical Induction that Cn = 2n – 1.

Q.5. Find generating function for each of the following sequences: i) (4, 12,36, 108, 384,…….) ii) (1, 5 k(k+1)/2, 25 k(k+1)(k+2)/6, 125k(k+1)(k+2) (k+3)/24, …… )

Q.6. Find the sequence with each of the following functions as its exponential generating function: i) f (x) = 5x 3x ii) f (x) = (2 – x ) + e 3x

Q.7. What is the solution of the recurrence relation an = 2an – 1 + 3an–2 with a0 = 5 and a1 = 8?

Q.8. Find all solutions of the recurrence relation an = 5 an – 1 + 3n. What is the solution with a1 = 9?
Q.9. Find all solutions of the recurrence relation an = 5an– 1 – 6an –2 + 7n