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## Discrete Mathematics MCQ Set 1

1. Let the sequence be 2, 8, 32, 128,……… then this sequence is
a) An airthmetic sequence
b) A geometic progression
c) A harmonic sequence
d) None of the mentioned

Answer: b [Reason:] The ratio of any term with previous term is same.

2. In the given Geometric progression find the number of terms
32, 256, 2048, 16384,………,250.
a) 11
b) 13
c) 15
d) None of the mentioned

Answer: d [Reason:] nth term = first term(ration – 1)., 250 = 25(23(n-1)), n=15. This implies 16th term.

3. In the given Geometric progression the term at position 11 would be
32, 256, 2048, 16384,………,250.
a) 235
b) 245
c) 35
d) None of the mentioned.

Answer: a [Reason:] nth term = first term(ration – 1)., gn = 25(23(n-1)), n=11. This implies 235.

4. For the given Geometric progression find the position of first fractional term?
250, 247, 244,………
a) 17
b) 20
c) 18
4) None of the mentioned.

Answer: c [Reason:] Let nth term=1 ,the next term would be first fractional term. Gn = 1 = 250(23(-n+1)), n=17.66.. therfore at n = 18 the first fractional term would occur.

5. For the given geometric progression find the first fractional term?
250, 247, 244,………
a) 2-1
b) 2-2
c) 2-3
4) None of the mentioned.

Answer: a [Reason:] Let nth term=1 ,the next term would be first fractional term. Gn = 1 = 250( 2 3(-n+1)), n=17.66.. therefore at n=18 the first fractional term would occur.Gn = 250( 2 3(-n+1)), G18 = 2-1.

6. State whether the given statement is true or false
1, 1, 1, 1, 1…….. is a GP series .
a) True
b) False

Answer: a [Reason:] The ratio of any term with previous term is same.

7. In the given Geometric progression, ‘225‘ would be a term in it.
32, 256, 2048, 16384,………,250.
a) True
b) False

Answer: b [Reason:] nth term = first term(ration – 1)., gn = 225 = 25 (2 3(n-1)), n=23/3, n=7.666 not an integer. Thus 225 is not a term in this series.

8. Which of the following sequeces in GP will have common ratio 3,where n is an Integer?
a) gn = 2n2 + 3n
b) gn = 2n2 + 3
c) gn = 3n2 + 3n
d) gn = 6(3n-1)

Answer: d [Reason:] gn = 6( 3n-1) it is a geometric expression with coefficient of constant as 3n-1.So it is GP with common ratio 3.

9. If a, b, c are in GP then relation between a, b, c can be
a) 2b = 2a + 3c
b) 2a = b+c
c) b =(ac)1/2
d) 2c = a + c

Answer: c [Reason:] The term b should be the geometric mean of of term a and c.

10. Let the multiplication of the 3 consecutive terms in GP be 8 then midlle of those 3 terms would be:
a) 2
b) 3
c) 4
d) 179

Answer: a [Reason:] Let a, b, c be three terms ,then a/r * a * ar = 8, b = (ac)1/2 (G M property), b3 = 8, b = 2.

## Discrete Mathematics MCQ Set 2

1. If a1, a2……… are in AP then a1-1, a2-1……… are in:
a) An airthmetic sequence
b) A geometic progression
c) Airthmetico-geometric progression
d) None of the mentioned

Answer: d [Reason:] If a1, a2……… are in AP, then a1-1, a2-1……… are in Harmonic Progression.

2. The ninth term of 13, 17, 111, 115, 119,……… is given by:
a) 135
b) 136
c) 139
d) None of the mentioned

Answer: a [Reason:] Since here a1-1, a2-1……… are in AP thus a9 = 3 + (9-1)4 = 35, 135 is h9 term of the series.

3. If for some number a and d,if first term is 1a, second term is 1/(a+d) ,thrid term is 1/(a+2d) and so on,then 5th term of the sequence is :
a) a+4d
b) a-4d
c) 1/(a+4d)
d) None of the mentioned

Answer: c [Reason:] The given sequence will form HP, thus 5th term will be (a+(5-1)d) – 1.

4. If a, b, c are in hp then a-1, b-1, c-1 are in:
a) GP
b) HP
c) AP
d) None of the mentioned

Answer: c [Reason:] If a1, a2……… are in AP then a1-1, a2-1……… are in Harmonic Progression.

5. If a, b, c are in hp, then b is related with a and c as :
a) 2(1b) = (1a + 1c)
b) 2(1c) = (1b + 1c)
c) 2(1a) = (1a + 1b)
d) None of the mentioned

Answer: a [Reason:] 1a, 1b, 1c willl be in airthmentic series and 1b will be the AM of a, c.

6. State whether the given statement is true or false
For number A, C if H is harmonic mean ,G is geometric mean then H>=G.
a) True
b) False

Answer: b [Reason:] Geometric mean is always greater than or equal to harmonic mean.

7. State whether the given statement is true or false
For number B, C if H is harmonic mean, A is the airthmetic mean then H>=A.
a) True
b) False

Answer: b [Reason:] Airthmetic mean is always greater than or equal to harmonic mean.

8. Which of the following gives the right inequality for AM, GM, HM?
a) AM>=HM>=GM
b) GM>=AM>=HM
c) AM>=GM>=HM
d) GM>=HM>=AM

Answer: c [Reason:] Airthmetic mean is always greater than or equal to geometric mean,geometric mean is always greater than or equal to harmonic mean.

9. For two number a,b HM between them is given by:
a) (2b+2a )/3b
b) 2ab/(a+b)
c) (a+b)/2ab
d) 2b/(a+b)

Answer: b [Reason:] Let c be the hm, 2c = 1a + 1b (AM property), c = 2b/(a+b).

10. If A, G, H are the AM, GM, HM between a and b respectively then:
a) A, G, H are in hp
b) A, G, H are in gp
c) A, G, H are in ap
d) None of the mentioned

Answer: b [Reason:] A = (a+b)/2, G = (ab)1/2, H = 2b/(a+b), clearly AxH = G2 thus A, G, H are in gp.

## Discrete Mathematics MCQ Set 3

1. Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”
a) Modus tollens
b) Modus ponens
c) Disjunctive syllogism
d) Simplification

Answer: b [Reason:] (M ∧ (M → N)) → N is Modus ponens.

2. Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”
a) Modus tollens
b) Conjunction
c) Hypothetical syllogism
d) Simplification

Answer: a [Reason:] (¬N ∧ (M → N)) → ¬M is Modus tollens.

3. Which rule of inference is used,”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”
a) Simplification
b) Conjunction
d) Disjunctive syllogism

4. What rule of inference is used here?
“It is cloudy and drizzling now. Therefore, it is cloudy now.”?
b) Simplification
c) Resolution
d) Conjunction

Answer: b [Reason:] (p ∧ q) → p argument is Simplification.

5. What rule of inference is used in this argument?
“If I go for a balanced diet, then I will be fit. If I will be fit, then I will remain healthy. Therefore, if I go for a balanced diet, then I will remain healthy.”
a) Modus tollens
b) Modus ponens
c) Disjunctive syllogism
d) Hypothetical syllogism

Answer: d [Reason:] ((p → q) ∧ (q → r)) → (p → r) argument is ‘Hypothetical syllogism’.

6. What rules of inference are used in this argument?
“All students in this science class has taken a course in physics” and “Marry is a student in this class” imply the conclusion “Marry has taken a course in physics.”
a) Universal instantiation
b) Universal generalization
c) Existential instantiation
d) Existential generalization

Answer: a [Reason:] ∀xP (x), ∴ P (c) Universal instantiation.

7. What rules of inference are used in this argument?
“It is either colder than Himalaya today or the pollution is harmful. It is hotter than Himalaya today. Therefore, the pollution is harmful.”
a) Conjunction
b) Modus ponens
c) Disjunctive syllogism
d) Hypothetical syllogism

Answer: c [Reason:] ((p ∨ q) ∧ ¬p) → q argument is Disjunctive syllogism.

8. The premises (p ∧ q) ∨ r and r → s imply which of the conclusion?
a) p ∨ r
b) p ∨ s
c) q ∨ s
d) q ∨ r

Answer: b [Reason:] The premises (p ∧ q) ∨ r as two clauses, p ∨ r and q ∨ r. We can also replace r → s by the equivalent clause ¬r ∨ s. using the two clauses p ∨ r and ¬r ∨ s, we can use resolution to conclude p ∨ s.

9. What rules of inference are used in this argument?
“Jay is an awesome student .Jay is also a good dancer. Therefore, Jay is an awesome student and a good dancer.”
a) Conjunction
b) Modus ponens
c) Disjunctive syllogism
d) Simplification

Answer: a [Reason:] ((p) ∧ (q)) → (p ∧ q) argument is conjunction.

10. “Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that
a) Parul is out for trip.
b) Raju is playing chess
c) Parul is out for a trip and Raju is playing chess.
d) Parul is out for a trip or Raju is playing chess.

Answer: d [Reason:] Let p be “It is snowing,” q be “Parul is out for a trip,” and r the proposition “Raju is playing chess.” The hypotheses as ¬p ∨ q and p ∨ r, respectively. Using resolution, the proposition q ∨ r is, “Parul is out for a trip or Raju is playing chess.”

## Discrete Mathematics MCQ Set 4

1. For a matrix A, B and identity matrix I, if a matrix AB=I=BA then:
a) B is inverse of A
b) A is inverse of B
c) A-1 = B, B-1 = A
d) All of the mentioned

Answer: d [Reason:] Since AB = I, A = B-1 Similarly A is the inverse of B.

2. For matrix A,(A3) = I, A-1 is equals to:
a) A2
b) A-2
c) Can’t say
d) None of the mentioned

Answer: a [Reason:] A(A2) = I this implies A-1 = A2.

3. Let A = [0 1 0 0 ], A-1 is equal to:
a) Null matrix
b) Identity matrix
c) Does not exist
d) None of the mentioned

Answer: c [Reason:] Since A is singular matrix, inverse does not exists.

4. If A is an invertible square matrix then:
a) (AT)-1 = (A-1)T
b) (AT)T = (A-1)T
c) (AT)-1 = (A-1)-1
d) None of the mentioned

Answer: a [Reason:] For invertible matrix A, AT is also inveritble.

5. If matrix A, B and C are invertible matrix of same order then (ABC)-1=
a) CBA
b) C-1 B-1 A-1
c) CT B-1 AT
4) None of the mentioned

Answer: b [Reason:] Reversal rule holds for inverse multiplication of the matrices.

6. State True or False?
If A is non singular matrix then AB = AC implies B = C.
a) True
b) False

Answer: a [Reason:] Pre-multipliying by A-1 we get B = C.

7. State True or False:
For a matrix A of order n, the det(adj(A)) = (det(A))n, where adj() is adjoint of matrix.
a) True
b) False

Answer: b [Reason:] For a matrix A of order n, the det(adj(A)) = (det(A))n-1.

8. For a non-singular matrix A, A-1 is equal to:
c) det(A)*A
d) None of the mentioned

9. Let I3 be the Identity matrix of order 3 then (I3)-1 is equal to:
a) 0
b) 3I3
c) I3
d) None of the mentioned.

Answer: c [Reason:] Idenity matrices are self invertible that is I3 x I3 = I3 .

10. If for a square matrix A(non-singular) and B, null matrix O, AB = O then:
a) B is a null matrix
b) B is a non singular matrix
c) B is a identity matrix
d) All of the mentioned

Answer: a [Reason:] Given det(A) is not equal to zero. A-1 exists, A-1(AB) = O, B = O.

## Discrete Mathematics MCQ Set 5

1. A Least Common Multiple of a,b is defined as:
a) It is the smallest integer divisible by both a and b
b) It is the greatest integer divisible by both a and b
c) It is the sum of the number a and b
d) None of the mentioned

Answer: a [Reason:] Defination of LCM(a, b)-smallest multiple of a and b.

2. The LCM of two number 1, b(integer) are
a) b + 2
b) 1
c) b
d) None of the mentioned

Answer: c [Reason:] Since b is the smallest integer divisible by 1 and b.

3. If a, b are integers such that a > b then lcm(a, b) lies in
a) a>lcm(a, b)>b
b) a>b>lcm(a, b)
c) lcm(a, b)>=a>b
d) None of the mentioned

Answer: c [Reason:] LCM of number is either equal to biggest number or greater than all.

4. LCM of 6, 10 is:
a) 60
b) 30
c) 10
d) 6

Answer: b [Reason:] Since 30 is the smallest integer divisible by 6 and 10.

5. The product of two numbers are 12 and there Greatest common divisior is 2 then LCM is:
a) 12
b) 2
c) 6
d) None of the mentioned.

Answer: c [Reason:] The lcm of two number a and b is given by lcm(a, b) = ab/(GCD(a, b)).

6. If LCM of two number is 14 and GCD is 1 then the product of two numbers is :
a) 14
b) 15
c) 7
d) 49

Answer: a [Reason:] The lcm of two number a and b is given by lcm(a,b) = ab/(GCD(a,b)), this implies ab = lcm(a, b) * gcd(a, b).

7. If a number is 22 x 31 x 50 and b is 21 x 31 x 51 then lcm of a, b is:
a) 22 x 31 x 51
b) 22 x 32 x 52
c) 23 x 31 x 50
d) 22 x 32 x 50

Answer: a [Reason:] Lcm is the product of sets having highest exponent value among a and b.

8. State whether the given statement is True or False.
LCM (a, b, c, d) = LCM(a,(LCM(b,(LCM(c, d)))).
a) True
b) False

Answer: a [Reason:] LCM function can be reursively defined.

9. LCM(a, b) is equals to :
a) ab/(GCD(a, b))
b) (a+b)/(GCD(a, b))
c) (GCD(a, b))/ab
d) None of the mentioned