Math DM

c, d

1) Study the following sequences: an=n(n+1), n is integer, n>0 bn=bn-1 + bn-2 + bn-3, b1=1, b2=-2, b3=3 a. b5 = 6 b. a3 = 15 c. a3 = 12 d. b5 = 3

d, e

10) Let ~p be not p. Select incorrect statements a. X (+) X = 1 b. ~X (+) X = X c. X (+) ~X = 0 d. X (+) ~X = 1 e. X (+) X = 0

a, d

11) Let R be a relation from {1,2,3} to {a,b,c,d} and R = {(1,c), (2,a), (2,d), (3,d)} Let R-1 be the inverse relation of R and ~R be the complementary relation of R Select incorrect statements a. ~R contains (c,2) b. ~R has 3 elements and R-1 has 8 elements c. ~R contains (2,c) and (3,d) d. ~R has 8 elements and R-1 has 4 elements e. R-1 contains (2,c) and (2,b)

b

12) Which of the following propositions are FALSE? (n=1,2,3,...) a. 1.3 + 2.4 + 3.5 + ... + n(n+2) = n(n+1)(2n+7)/6 b. 1+2+3+...+2n-1+2n=(2n+1)n c. 1+3+5+...+2n+1 = (n+1)(n+1) d. 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/3

b

13) A recursive definition the function f(n) = n is a. f(n) = f(n-1) + 1 for all n ≥ 1 b. f(1) = 1, f(n) = n + f(n-1) for n > 1 c. none of the others d. f(1) = 1, f(n) = n for all n > 1

a, b

14) Study the following recursive algorithm: Procedure T (n:integer) If n < 3 then T(n) ≔2 Else T(n) ≔ T(n-2) + T(n-3) Select incorrect statements: a. T(6) = 10 b. T(5) = 8 c. T(6) = 8 d. T(5) = 6

b, e

15) Select correct statements a. If a full binary rooted tree has height 3, then all its subtrees has height 2 b. A single vertex is a full binary rooted tree c. A full binary rooted tree with height 3 has exactly 15 nodes d. A rooted tree cannot have 3 subtrees e. There is a rooted tree with height 0

d

16) Which the following propositions is true for all positive integers n: a. n3 > n2 + 3 b. none of the others c. 2n + 3 ≤ 2n d. 2 | (n2 + 3n)

c, d

17) Select corrects statements a. {~, +} is functional completeness b. There are 7 Boolean functions F(x,y,z) with 3 variables such that F(1,1,1) = 1 c. There are 1024 Boolean functions with 10 variables d. The sum of product expansion of F(x,y,z) has 8 products

b

18) The sum of products expansion of the function F(x,y,z) = xy + ~y has ... minterms a. 2 b. 6 c. 4 d. 5

d

2) A recursive definition the function f(n) = n is: a. None of the others b. f(n) = f(n-1) + 1 for all n≥1 c. f(1) = 1, f(n) = n for all n>1 d. f(1) = 1, f(n) = n + f(n-1) for n>1

c

21) Let R be an equivalence relation on a set A. Suppose that a and b are elements of A and aRb. Choose the correct statement a. [a]~[b] b. a belongs to R and b belongs to R c. a belongs to [b] and b belongs to [a] d. none of the others

c, d

22) Select correct statements a. The "<" relation on the set of positive integers is symmetric b. A relation on the set A is symmetric if it is not antisymmetric c. The relation R = {(1,1), (2,1), (1,2), (3,3)} on {1,2,3} is symmetric d. Let A be the set {1,2,3,4}. The relation {} in A is symmetric e. The relation R = {(a,b) | a,b are integers and ab = 2012} is symmetric

c

23) (1) 1+2+3+...+n = n(n+1) ; (2) 1+2+22+23+24+...+2n-1 = 2n - 1 a. true, false b. true, true c. false, true d. false, false

c

24) Which of the following is a recursive definition of the sequence {an}, n = 1,2,... if an = 3n-2? a. none of the others b. a1 = -2, an = an-1 + 3 if n>1 c. a1 = 1, an = an-1 + 3 if n>1 d. a1 = 1, an = an-1 + 2 if n>1 e. a1 = 1, an = an-1 - 2 if n>1

a, c

25) Suppose X is a varibale such that probability given by P(X=2) = 0.4, P(X=3) = 0.3, P(X=4) = 0.2 and P(X=5) = 0.1. Find E(X) and V(X) a. V(X) = 1 b. V(X) = 0.8 c. E(X) = 3 d. E(X) = 2.6 e. E(X) = 3.5

b, d

26) Flip an unfair coin, where p(heads) = ¾ and p(tails) = ¼, ten times a. p(exatly 7 heads) = C(10,7).(3/4)3.(1/4)7 b. p(exatly 7 heads) = C(10,7).(3/4)7.(1/4)3 c. p(exatly 9 heads) = C(10,9).(3/4).(1/4)9 d. p(exatly 9 heads) = C(10,9).(3/4)9.(1/4)

b, c, d

27) Let R be a relation from the set A to the set B and R-1 be inverse relation of R. Select correct statements a. RU(~R) = A, where ~R is the complementary relation of R b. R-1 is symmetric if R is symmetric c. The complementary relation ~R of R is reflexive if R is irreflexive d. RUR-1 = A e. If R-1 is reflexive then the complementary relation ~R of R is reflexive

b

28) Let A = {1,2,3} ; (i) How many different relations on A are there? ; (ii) How many different relations on A containing (1,2)? a. (i) 23 (ii) 22 b. (i) 29 (ii) 28 c. (i) 29 (ii) 29 - 1 d. (i) 26 (ii) 25

c

29) A recursive definition with initial condition of the set S = {1, 111, 11111, 1111111, ...} is a. 1 in S; if x in S then x1 in S b. 1 in S; if x in S then x111 in S c. 1 in S; if x in S then 100x + 11 in S d. 1 in S; if x in S then x11 in S

e

3) What is the probability that a card selected from a desk is an queen or diamond? a. 17/52 b. 17/C(52,2) c. none of the others d. 16/C(52,2) e. 16/52

b, c

32) Suppose you have 30 different books a. none of the others b. there are 30! ways to put the 30 books in a row on a shelf c. There are C(30,4) ways to get a buch of four books to give to a friend d. There are C(30,1) ways to put the 30 books in a row on a shelf e. There are C(26,4) ways to get a buch of four books to give to a friend

b, c, d

35) Describe each sequence recursively. Include initial conditions and assume that the sequences begin with a(1) a. If the sequence are 1, 101, 10101, 1010101,... the a(n) = 10a(n-1) + 1, a(1) = 1 b. If the sequence are 0.1, 0.11, 0.111, 0.1111, ... then a(n) = a(n-1) + 1/10n, a(1) = 0.1 c. If the sequence are 0, 1, 0, 1, 0, 1,... then a(n) = a(n-2), a(1) = 0, a(2) = 1 d. If the sequence are 1, 111, 11111, 1111111,... then a(n) = 100a(n-1) + 11

b

36) Which of the following is the recurrence relation with inition condition for the number of permutations of a set with n elements? a. an = an-1n b. an = an-1n, a0 = 1 c. an = n! d. none of the others

c

37) A solution of A(n) = 4A(n-1) - 3A(n-2) is... a. A(n) = 12 b. A(n) = 9 c. A(n) = 3^n + 7 d. none of the others e. A(n) = 6

c

38) A, B, C, D are sets ; |A| = 12, |B| = 18, |A^B| = 6 ; |C| = 20, |D| = 15, |C^D| = 15 a. |CuD| = 15 b. |CuD| = 18 c. none of the others d. |AuB| = 22 e. |AuB| = 20

b

39) Recurrence relation of 1, 101, 10101, 1010101, ... is... a. An = 100An-1 - 99 b. An = 100An-1 + 1 c. none of the others

c

4) ... Is the probability that a five card poker hand contains at least one ace a. none of the others b. C(48,5)/C(52,5) c. 1-C(48,5)/C(52,5) d. 1-C(48,4)/C(52,5) e. C(48,4)/C(52,5)

e

40) How many bit strings of length 14 start with 1111 and end with 0000 a. 512 b. 128 c. none of the others d. 256 e. 64

a, c

41) How many bit strings with length 10 containing 7 0s and 3 1s and end with 00? a. C(8,5) b. C(9,3) c. C(8,3) d. C(10,7) e. C(9,6)

c

42) Find f(2) and f(3) if f(n) = 2f(n-1) + 6, f(0) = 3 a. f(2) = 5, f(3) = 21 b. f(2) = 66, f(3) = 30 c. f(2) = 30, f(3) = 66 d. f(2) = 21, f(3) = 5

b

43) Suppose that an equivalence relation R has three partitions A1 = {1,2}, A2 = {0}, and A3 = {3}. List all the ordered pairs in R a. R = {0,1,2,3} b. R = {(0,0), (1,1), (2,2), (1,2), (2,1), (3,3)} c. R = {(0,0), (1,2), (2,1), (3,3)} d. none of the others

b, c

44) Select incorrect statements a. The "less than" relation on the set of positive integers is reflexive b. Let A be {1,2,3}. The relation R = {(2,3), (1,2), (2,1), (3,2)} is irreflexive c. The "same language" relation on the set of all people is transitive d. A relation R on the set A is called reflexive if (a,a) is in R for some a in A

a

5) Suppose you and a friend each choose at random one integer from 1 to 8. a. All of the others b. p(both numbers match) = 1/8 c. p(you pick 5 and your friend picks 8) = 1/64 d. p(sum of the two numbers picked is < 4) = 3/64

d

6) Select a correct statements. Three coins are tossed a. ¼ is the probability that no head show (should be 1/8) b. all of the others c. 3/9 is the probability that exactly two heads show (should be 1/4) d. Sample space is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

a, c

7) Let R be a relation from A to B Let S be a relation from B to C Choose the incorrect statements a. SoR is a relation from C to A b. RoS is a relation from C to A c. RoS is a relation from A to C d. SoR is a relation from A to C

b, c

9) Let ~p be not p. Select correct statements a. x+xy = x for all values of Boolean variables x,y b. Let F(x,y,z) be Boolean function such that F(x,y,z) = 1 if and only if xyz = 0. The sum of products expansion of F has 7 products c. xy = x + y if and only if x = y = 1, where x,y are Boolean variables d. {~} is a functionally complete set of operators