Some sets of questions Third Semester B.E. IA Test, 2011
USN
PES SCHOOL OF ENGINEERING (Hosur Road, 1KM before Electronic City, Bangalore-560 100) Date: 02/09/2011 MaxMarks: 50 Subject & code: 10CS34 Mins. Name of faculty: Dr. Snehanshu Saha
Time: 90
Note: 1. Answer one question from Part A and FOUR questions from Part B with a total 5 out of 8 including both units. 2. Your work must be neat in order to receive full credit. 3. DO NOT ANSWER MORE THAN FIVE QUESTONS!! PART-A 1 a) by the principle of mathematical induction, prove that (5) 12+22+32+…………+n2 =n(n+1)(2n+1)/6 b) Give a recursive definition of each of the following integer sequence (5) ii) bn=2-(-1)n Z+ i) an= 7n 4 a) A sequence {an} is defined recursively by a1=4, an=a n-1+ n for n>=2. Find an in explicit form. (4) b) Chebyshev polynomials are defined as follows (6) P0(X) =1, P1(X) =X Pn+1(X)= XPn(X)-Pn-1(X), n>0 Show by mathematical induction, Pn(2cosθ)= sin(n+1)θ/sinθ
Part B 5 a) For any three non empty sets A, B, C.Prove that A X (B-C) = (AXB)-(AXC) (5) A X (B∩ C) = (AXB)∩ (AXC) (5)
6 a) let f, g: RR defined by f(x) =2x+5, g(x) =1/2(x-5) (4) Show that f and g are invertible b) Let A=B=C=R and f: AB, g: BC be defined by f (a) =2a+1, g (b) =1/3b (6) , compute gof and show that gof is invertible. What is (gof)-1 7 a) let A= {1, 2, 3, 4, 5, 6} and R be a relation on A defined by aRb iff
,
‘a’ is a multiple of ‘b’
(7) i) Write down R as a set of ordered pairs ii) Represent R using a 6X6 matrix(rows are 1 to 6, columns are 1-6) [if there exists an ordered pair (1,3) then the intersection of 1 and 3 from row and column respectively will be 1, otherwise 0] For example consider R={(1,1),(1,3),(1,1),(2,2),(3,2),(3,3)} b) Define 1) function 2) Cartesian product (3) 8 a) let f: ZZ be defined by f (a) = a+1 for a € Z. find whether f is 1-1 or onto (or both or neither) (6) b) Let A & B be finite sets with |B|=3. If there are 4096 relations between A & B what is |A| (4)
B.E. CS & E- Semester III IA Test III, 2013 USN PES Institute of Technology-South Campus (Hosur Road, 1 kmbefore Electronic City, Bangalore-560 100) Date: 02/09/2011 MaxMarks: 50 Subject & code: 10CS34 Time: 90 Mins. Name of faculty: Dr. Snehanshu Saha Note: 1. Answer ANY FIVE questions ( at least one from PART-A 2. Your work must be neat in order to receive full credit.
PART-A
1. a) by the principle of mathematical induction, prove that (5) 1²+2²+…………+n² =n(n+1)(2n+1)/6 b) Give a recursive definition of each of the following integer sequence (5) i) a_{n}= 7n ii) b_{n}=2-(-1)ⁿ n is any positive integer 2. a) A sequence {an} is defined recursively by a1=4, a_{n}=a_{n-1}+ n for n >=2. Find an in explicit form. (4) b) Chebyshev polynomials are defined as follows (6) P0(X) =1, P1(X) =X P_{n+1}(X)= X P_{n}(X) - P_{n-1}(X), n ≥0 Show by mathematical induction, Pn(2cosθ)= sin(n+1)θ/sinθ Part –B 3. a) For any group G, G is abelian iff (ab)²=a²b² b) State and prove Lagrange's theorem
(5) (5)
4. a) Define a cyclic group. Prove that every subgoup of a cyclic group is cyclic b) Prove that every cyclic group is abelian but the converse may not be true
(5) (5)
5. a) Prove that every field is an integral domain b) Check if (Z, +, .) is a ring with the binary operations x+y=x+y-7; x.y=x+y-3xy
(5)
6. a) A={1,2,3,4}; R= {(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3),(4,4)} Construct a valid partition of A. b) A={1,2,3,4}; R= {(1,1),(1,2),(2,2),(2,4),(1,3),(3,4),(1,4),(3,3),(4,4)} Is R a partial order? Draw a Hasse diagram of R.
(5) (5) (5)
7. a) Prove that a function f:A→B is invertible iff f is one-to-one and onto. (7) b) A= {1, 2, 3, 4, 5}. R on A×A is defined by (x1, y1) R (x2, y2) (3) iff x1+y1=x2+y2 Determine the equivalence classes of [(1, 3)] and [(2, 4)] 8. Prove that every group of prime order is cyclic.
Third Semester B.E. IA Test, 2012(diploma)
USN
PES Institute of Technology South Campus
(10)
(Hosur Road, 1KM before Electronic City, Bangalore-560 100) Date: 17/10/2012 MaxMarks: 25 Subject & code: 10CS34 Mins. Name of faculty: Dr. Snehanshu Saha
Time: 45
Note: 1. Answer the following question 2. Your work must be neat in order to receive full credit.
1 a) By the principle of mathematical induction, prove that (7) 12+22+32+…………+n2 =n(n+1)(2n+1)/6 b) Give a recursive definition of each of the following integer sequence (6) i) an= 7n ii) bn=2-(-1)n Z+ c) A sequence {an} is defined recursively by a1=4, an=a n-1+ n for n>=2. Find an in explicit form. (6) d) Chebyshev polynomials are defined as follows (6) P0(X) =1, P1(X) =X Pn+1(X) = XPn(X)-Pn-1(X), n>0 Show by mathematical induction, Pn(2cosθ)= sin(n+1)θ/sinθ
Third Semester B.E. IA Test-II(Diploma), 2012
USN
PESIT South Campus (Hosur Road, 1KM before Electronic City, Bangalore-560 100) Date: 16/11/2012 MaxMarks: 25 Subject & code: 10CS34 Mins. Name of faculty: Dr. Snehanshu Saha Answer the following questions logically, clearly and neatly!
Time: 45
Part B 1 a) For any three non empty sets A, B, C.Prove that i) A X (B-C) = (AXB)-(AXC) (3) ii) A X (B∩ C) = (AXB)∩ (AXC) (3) b) let f, g: RR defined by f(x) =2x+5, g(x) =1/2(x-5) (4) Show that f and g are invertible. c) Let A=B=C=R and f: AB, g: BC be defined by f (a) =2a+1, g (b) =1/3b (7) , compute gof and show that gof is invertible. What is (gof)-1 d) Let A= {1, 2, 3, 4, 5, 6} and R be a relation on A defined by aRb iff
‘a’ is a multiple of ‘b’
Write down R as a set of ordered pairs (4) e) Let A & B be finite sets with |B|=3. If there are 4096 relations between A & B, what is |A| ? (4)
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