CODE: GR22Axxxxx GR 22 SET - 1
GR11D5104
II B.Tech I Semester Regular Examinations, January/February 2024
GR11D5104
Discrete Mathematics
GR11D5104
GR11D5104
(Common to CSE, IT, AIML & DS)
Time:GR11D5104
3 hours Max Marks: 60
Instructions:
1. Question paper comprises of Part-A and Part-B
2. Part-A (for 10 marks) must be answered at one place in the answer book.
3. Part-B (for 50 marks) consists of five questions with internal choice, answer all questions.
4. CO means Course Outcomes. BL means Blooms Taxonomy Levels.
PART – A
(Answer ALL questions. All questions carry equal marks)
10 * 1 = 10 Marks
< Note: Type the questions in the given format only, Times New Roman font , size 12 >
1. a) State which of the following sentences are propositions: i) A triangle 1M CO1 Analyze
contains three lines ii) x+2 is a positive integer
b) Write down the following prepositions in symbolic form and find its 1M CO1 Understand
negation: “All integers are rational numbers and some rational numbers are
not integers”
c) Let f & g be functions from R to R defined by f(x)=ax+b and g(x)=1-x+x2 . 1M CO2 Understand
If (gof) (x)=9x2 -9x+3 , determine a.b
d) 5.Consider the sets A={a,b,c} and B={1,2,3} and the relations. R={(a,1), 1M CO2 Remember
(b,1),(c,2),(c,3)} and S={(a,1),(a,2),(b,1),(b,2)} from A to B.Determine
R͞ ,S͞ ,RUS,R∩S,Rc ,Sc
e) In how many ways can we distribute 10 identical marbles among 6 distinct 1M CO3 Analyze
containers?
f) Find the number of arrangements of the letter TALLAHASSEE which have 1M CO3 Evaluate
no adjacent A’s
g) Find the sequences generated by the following functions : i) (3+x)3 1M CO4 Evaluate
ii) 3x3+ e2x
h) Find the generating functions for the sequences 12 ,22 ,32…….. 1M CO4 Evaluate
i) Draw a diagram of the graph G=(V,E) where V={A,B,C,D} , E={(A,B), 1M CO5 Apply
(A,C),(A,D),(C,D)}
j) Define complete bipartite graph and kuratowski’s second graph. 1M CO5 Understand
PART – B
(Answer ALL questions. All questions carry equal marks)
5 * 10 = 50 Marks
< Note: Marks can be distributed evenly/unevenly between a) & b) >
2. a) Obtain principal conjuction normal form for the following: 5M CO1 Evaluate
Page1 of 4
CODE: GR22Axxxxx GR 22 SET - 1
GR11D5104
GR11D5104
GR11D5104
GR11D5104
b) Prove using rules inference or disprove. 5M CO1 Evaluate
GR11D5104
(a) Duke is a Labrador retriever All Labrador like to swim. Therefore Duke
likes to swim.
(b) All even numbers that are also greater than 2 are prime 2 is an even
number 2 is a prime Therefore some even numbers are prime UNIVERSE =
numbers (c) if it is hot today or raining today then it is no fun to snow ski
today Therefore it is hot today UNIVERSE=Days
OR
3. a) a)Use De Morgan’s Law to write the negation of each statement : 5M CO1 Evaluate
i.I want a car and a worth cycle
ii. My cat stay outside or it makes a mess
iii.I’ve faller and I can’t get up iv. You study or you don’t get a good grade.
b) Are (p→q) →r and p→ (q→r) logically equivalent? Justify your answer
by using the rules of logic to simply both expression and also by using truth
tables
b) Express the formula P→Q in terms of {↑} only (A^ (~AvB))v~(A^B))= B 5M CO1 Apply
Which of the following formula is not a tautology i)(P→Q) →(Q→R) ii) )
(P→Q) ^(Q→P)
4. a) Let A={1,2,3,4,6,8,12} on An define the partial ordering relation R by aRb 5M CO2 Analyze
iff a/b. a) Draw the Hasse diagram for R. b) Write down the relation matrix
for R.
b) Let A={a,b,c,d} R be a relation on A that has the matrix 5M CO2 Apply
Mr = [ 1 0 0 0
0100
1110
0 1 0 1 ]. Construct the digraph of R and list the indegrees and out-
degrees of all vertices.
OR
5. a) Let A ={1,2,3,4,5} and R={(1,1),(2,2),(3,3),(1,3),(3,4),(3,5),(1,4),(4,4),(1,5), 5M CO2 Analyze
(2,3),(2,4), (2,5),(5,5)}. Draw the Hasse diagram for R.
b) Let A={1,2,3,4,6} and R be a relation on A defined by aRb if and only if”a” 5M CO2 Evaluate
is a multiple of “b”. Represent the relation R as a matrix and draw its
digraph.
6. a) Find the coefficient of 5M CO3 Analyze
(i) x9 y 3 in the expansion of (x+2y)12
Page2 of 4
CODE: GR22Axxxxx GR 22 SET - 1
GR11D5104
(ii) (ii) x5 y 2 in the expansion of (2x-3y)7
GR11D5104
b) a) Find the value of n so that 2P(n,2)+50=P(2n,2) b) 5M CO3 Evaluate
Prove that ,for all the integers n, r≥ 0,if n+1>r, j then P(n+1,r)=( 𝑛+1/
GR11D5104
𝑛+1−𝑟 ) P(n,r).
GR11D5104
GR11D5104
OR
7. a) Find the number of integer solutions of the equation x1+x2+x3+x4+x5 = 30 5M CO3 Analyze
under the constraints x1 >=0 for i=1,2,3,4,5 and further x2 is even and x3 is
odd
b) In how many ways can the letters of English alphabet be arraged so that 5M CO3 Analuze
there are exactly 6 letters between the letetrs b and c?
How many different outcomes are possible by tossing 15 similar coins?
8. a) Find the sequence generated by the following functions: (i) (3+x)3 (ii) 2x2 5M CO4 Evaluate
(1-x)-1 (iii) (1-x)-1+2x3 (iv) (1+3x)-1/3 (v) 3x3+e2x
b) Determine the coefficient of (i) x12 in x3 (1-2x)10 (ii) x10 in (x3 - 5x)/ (1- 5M CO4 Analyze
x)3 (iii) x5 in(1-2x)-7
OR
9. a) Solve the recurrence relation an+3-3an+2+3an+1-an =3+5n for n>=0 5M CO4 Analyze
b) Find a generating function for recurrence relation an+1-an=n2 ,n>=0 and a0 5M CO4 Analyze
=1.Hence solve it.
10. a) Let D be the digraph whose vertex set is V={ v1 ,v2,v3, v4, v5 }and the 5M CO5 Apply
directed edge set is E={(v1,v4),( v2,v3),( v3,v5),( v4,v2), ( v4,v4),( v4,v5),
( v5,v1) } write down a diagram of D and indicate the outdegrees and
indegrees of all vertices.
b) How many vertices will the following graphs have if they contain (i) 16 5M CO5 Analyze
edges and all vertices of degree 4? (ii) 21 edges,3 vertices of degree 4 and
other vertices of degree 3 (iii) 12 edges , 6 vertices of degree 3, and other
vertices of degree
OR
11. a) 5M CO5 Apply
Page3 of 4
CODE: GR22Axxxxx GR 22 SET - 1
GR11D5104
b) 5M CO5 Apply
GR11D5104
GR11D5104
GR11D5104
GR11D5104
*****
Page4 of 4
