No. of Printed Pages : 8 I MTE-13 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2015 ELECTIVE COURSE : MATHEMATICS MTE-13 : DISCRETE MATHEMATICS Time : 2 hours Maximum Marks : 50 (Weightage : 70%) Note : Question no. 1 is compulsory. Answer any four questions from questions no. 2 to 7. Use of calculators is not allowed. 1. Which of the following statements are true and which are false? Justify your answer. 5x2=10 (a) -(p q)=-p v -q There exists a simple graph with degree sequence 7, 7, 7, 5, 5, 3, 3. an = an/2 + n, al = 0, where n is a power of 2, is a linear recurrence relation. (d) (e) The partition 4 + 3 + 3 + 1 is self-conjugate. The graph 11/4 5 is Eulerian. 2. (a) Express x5 in terms of falling factorials and hence evaluate S 5m for m = 0, 1, 2, 3, 4, 5. 5 MTE-13 1 P.T.O.
Prove that every tree is bipartite. Is the converse true? Justify. How many five digit numbers are composed of only even digits? 3. (a) Form the truth tables of (p v q) A r and p v (q A r) to verify whether (pvq)a-r -7.-pv(qn-r). Using mathematical induction, prove that 1 1 1 1 + - + - + + 2 - - 1 V n N. 4 9 n2 3 Find the solution of the recurrence relation an = 4 an _ + 5 an _ 2, ao = 0, al = 6. 4 4. (a) Solve the following recurrence relation : un = (n! n) un _ 1 for n 1 with up = 1. 3 Let,$ be a two element Boolean algebra with A, V and ' operations defined in it. Prove that x is also a Boolean algebra stating clearly the operations defined in it. 5 Determine if the following is a valid argument. Explain your conclusion. 2 Proposition : V x e R, x 3 > x2 2 > 0 Proof : V x e R, x x2 (x 1) > 0 (x 1) x3 x2 > 0 x 3 > x 2. MTE-13 2
5. (a) Find a recurrence relation for a n, the number of ways to arrange cars in a row with n spaces if we can use Maruti 800 or Tata Safari or Scorpio. A Tata Safari or Scorpio requires two spaces, whereas a Maruti 800 requires just one space. Assume that you have unlimited number of each type of car and we do not distinguish between 2 cars of the same type. 4 If Km, n for m, n?_ 2, is Hamiltonian, how are m and n related? Justify your answer. 3 Show that if 7 colours are used to paint 50 bicycles and each cycle is coloured with a single colour, at least 8 bicycles will have the same colour. 3 6. (a) A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green? 3 Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph : a MTE-13 3 P.T.O.
How many numbers from 0 to 759 are not divisible by either 3 or 7? 3 7. (a) Solve the following recurrence relation : an = 2an + 1 if n > 1 and NI = 0, using generating function technique. Also find a 5 using your answer. Is there a 4-regular graph on 7 vertices? If `yes', construct such a graph. If your answer is 'no', justify your answer. 2 Find the Boolean expression in the DNF form for the function defined in tabular form below : 3 x y z f(x, y, z) 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 0 MTE-13 4
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ZEFA:VXE ( ) 1f t I.4'zi-r. (ii) 11% 717 i4-0)1:1 W-A-R I 3 (-4.9411(r 1 TR e icn? 2 3. (*) (p v q) A -rap v (q A r) zit Hczllrcl chk4 rc-r (p V q) A r p v (q A r)r ii,k111i -4T4 I 3 (IN) Trrurtzr alrtritq gri 1 1 1 1 2--VneN. 4 9 n` (T ) T7* Triqm an.4an _ 1 -F5an _ 2, a0 =0, a 1 =6 vi W77 I 4 4. () (Si Ci77% TIW IT " (1 *AR : Uo = 1 n 1 * un = (n! n)un _ i 3 (1:4) 1TF #11-4 3l-444 Vtzt 41\41.4 -t fti irl A, V 3-1-N TrIbTrr ista. t x trkesid TcrEz cbk. ratg 1-11 T-41.4.41\41.4 I 5 (T) AtTiftff Pii &irtgo oci) -grrr -t 31Er4 Pitchti Tcrit*11-A7 I 2 MTE-13 6 *49. : V XE R,x3 >x2 R, x2 > 0 x2 (x - 1) > 0 (x - 1) x3 - x2 > 0 x3 > x2.
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(TT) o A 759 i+741 (-Itcsii 3 71T 7 A141TTN-d-? 3 7. () Alt) nab-flay 'I cbt 1 1-1(1 ci WAM- C*1 Arr-A7: an =2an _ 1 + 1 ZA 11?_ 1 3* ao = 0. 3374 3Tik a5 1-ft Tff Alf4R (w) -47ETT 7 TET 4-"ftegrff T9 01? ` ~ld i -ktrr TiT -4r-4.R zrr `.0-101 31-0. 3cit 1-R R (TT) iii(unmog -1:1 1:1ftITTREfffc * DNF VtzT oeiact, lid*tr-4 : 5 2 3 x y z f(x, y, z) 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1. 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 MTE-13 8 2,000