BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2015

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1 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 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.

2 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 V n N. 4 9 n2 3 Find the solution of the recurrence relation an = 4 an _ + 5 an _ 2, ao = 0, al = (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

3 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 (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.

4 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) MTE-13 4

5 tuiltict) 441 ki chl t (Att.*. TrEff trftati til-elt, ;19441 xibrstoi :751W 11.t.-13:fkilW lard FITRI : 2 Wu affiiw-dv 3 i : 50 (()0 : 70%) 3/?-1 R. / qww..9ravref * JR7 # " 4 ow WIT 3R.1 *AV / 47c-35dej TR-27/ arylift q-e g/ 1. PHrcirtgcl 4144 war 1:rm atur? 3Tcr4 371k ftz AT-4R I 5x2=10 () (p A q) = p V q (1.4) c1;11?. aript 7, 7, 7, 5, 5, 3, 3 etic, wo 1T9 W. t (TI) an = an/2 + n, ai = 0,,46 n, 2 lefta. t, 7T11 WEIATt I (V) fkigilft t I (3) T'T K4, 5 aikeft t 2. () x5 W'Wet * Trql eeitil AT4R a* tits; m o, 1, 2, 3, 4, 5 S5 +1,t,t1ictoi *1177 I 5 MTE-13 5 P.T.O.

6 ZEFA:VXE ( ) 1f t I.4'zi-r. (ii) 11% 717 i4-0)1:1 W-A-R I 3 ( (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 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 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.

7 5. ( ) 5. era 800 All de.'tb-rft 7rr t n lc4i l tik 4 cr)r. - quo ki,t9eit an * I-Z-R Tui%Triwiz trr-a7 I R* UZI ki441 zff "RiTql *UT -0-r-*-grTht 800 Q. clr1 *'t trft I 1TF #0.R atrek xrruc (*A *:1 titizll arftrrral Tr* ucnik tt 2 4)14 art t eirc m, n 2 * rc..r Km, n t, m 3 n rchti 514)1( ITIAHNU t? 3ltr4 3- 'Cr-4R I 3 turf eirc 50 Tri&O. 4Z cia 7 ii T 1.4) t -5F* tt 14 *4z Rhell t, 8 til*rchril irf.)111 I 3 "cr OM At 4 Tkc1 t Iw ootil rllrl t 1411, TErtterr Nireichoi? 3 I*01 ill NTT G 7rtli tv-ciovict) ti -coei xritvrem W-Au PHiCiRgo tii-swim atik -qftti ti wrff*if-a-l : 4 MTE-13 7 P.T.O.

8 (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 _ ZA 11?_ 1 3* ao = Tik a5 1-ft Tff Alf4R (w) -47ETT 7 TET 4-"ftegrff T9 01? ` ~ld i -ktrr TiT -4r-4.R zrr ` cit 1-R R (TT) iii(unmog -1:1 1:1ftITTREfffc * DNF VtzT oeiact, lid*tr-4 : x y z f(x, y, z) MTE ,000

BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination. June, 2018 ELECTIVE COURSE : MATHEMATICS MTE-02 : LINEAR ALGEBRA

No. of Printed Pages : 8 1MTE-021 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination 046OB June, 2018 ELECTIVE COURSE : MATHEMATICS MTE-02 : LINEAR ALGEBRA Time : 2 hours Maximum Marks : 50 (Weightage