## Discrete Mathematics MCQ Set 1

1. is

a) 0

b) 1

c) 2

d) 3

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2. Value of lim_{x → 0}(1+Sin(x))^{Cosec(x)}

a) e

b) 0

c) 1

d) ∞

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_{x → 0}(1+Sin(x))

^{Cosec(x)}

Put sin(x) = t we get

lim_{t → 0}(1+t)^{(1⁄t)} = e.

3. Value of lim_{x → 0}(1 + cot(x))^{sin(x)}

a) e

b) e^{2}

c) ^{1}⁄_{e}

d) Can not be solved

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4.

a) True

b) False

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5.

a) True

b) False

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6. Evaluate lim_{x → 1}[(x^{x} – 1) / (xlog(x))]

a) e^{e}

b) e

c) 1

d) e^{2}

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_{x → 1}[(x

^{x}– 1) / (xlog(x))] = (

^{0}⁄

_{0})

By L hospital rule,

lim_{x → 1} [x^{x} (1+xlog(x))/ (1+xlog(x))] = lim_{x → 1} [x^{x}] = 1.

7. Find n for which , has non zero value.

a) >=1

b) >=2

c) <=2

d) ~2

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8. Find the value of lim_{x → 0}(Sin(2x))^{Tan2 (2x)} ?

a) e^{0.5}

b) e^{-0.5}

c) e^{-1}

d) e

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9. Evaluate

a) ^{1}⁄_{4}

b) ^{1}⁄_{3}

c) ^{1}⁄_{2}

d)1

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## Discrete Mathematics MCQ Set 2

1. Distance travelled by any object is

a) Double integral of its accelecration

b) Double integral of its velocity

c) Double integral of its Force

d) Double integral of its Momentum

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2. Find the distance travelled by a car moving with acceleration given by a(t)=t^{2} + t, if it moves from t = 0 sec to t = 10 sec, if velocity of a car at t = 0sec is 40 km/hr.

a) 743.3km

b) 883.3km

c) 788.3km

d) 783.3km

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3. Find the distance travelled by a car moving with acceleration given by a(t)=Sin(t), if it moves from t = 0 sec to t = π/2 sec, if velocity of a car at t=0sec is 10 km/hr.

a) 10.19 km

b) 19.13 km

c) 15.13 km

d) 13.13 km

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4. Find the distance travelled by a car moving with acceleration given by a(t)=t^{2} – t, if it moves from t = 0 sec to t = 1 sec, if velocity of a car at t = 0sec is 10 km/hr.

a) ^{119}⁄_{22} km

b) ^{119}⁄_{15} km

c) ^{129}⁄_{12} km

d) ^{119}⁄_{12} km

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5. Find the value of ∫∫ xy dxdy over the area bpunded by parabola y=x^{2} and x = -y^{2},is

a) ^{1}⁄_{67}

b) ^{1}⁄_{24}

c) –^{1}⁄_{6}

d) –^{1}⁄_{12}

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6. Find the value of ∫∫ xydxdy over the area b punded by parabola x = 2a and x^{2} = 4ay, is

a) ^{a4}⁄_{4}

b) ^{a4}⁄_{3}

c) ^{a5}⁄_{3}

d) ^{a2}⁄_{3}

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7. Find the integration of ∫∫_{0}^{x} (x^{2} + y^{2}) dxdy, where x varies from 0 to 1

a) ^{4}⁄_{3}

b) ^{5}⁄_{3}

c) ^{2}⁄_{3}

d) 1

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8. Evaluate the value of , where y varies from 0 to 1.

a) ^{11}⁄_{12}

b) ^{14}⁄_{6}

c) ^{11}⁄_{6}

d) ^{11}⁄_{7}

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9. Evaluate ∫∫[x^{2} + y^{2} – a^{2} ]dxdy where, x and y varies from –a to a.

a) –^{2}⁄_{3} a^{4}

b) –^{4}⁄_{3} a^{4}

c) –^{4}⁄_{3} a^{5}

d) –^{2}⁄_{3} a^{5}

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^{2}+ b

^{2}= a

^{2}

10. Find the area inside a ellipse of minor-radius ‘b’ and major-radius ‘a’.

a) –^{4}⁄_{3} a^{2}

b) –^{4}⁄_{3} ab^{2}

c) ^{4}⁄_{3} ab

d) –^{4}⁄_{3}

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11. Find the value of where, y varies from 0 to 1.

a) ^{16}⁄_{946}

b) ^{8}⁄_{945}

c) ^{16}⁄_{45}

d) ^{16}⁄_{945}

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12. Find the value of integral

a) ^{3}⁄_{15}

b) ^{2}⁄_{15}

c) ^{2}⁄_{30}

d) ^{1}⁄_{15}

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## Discrete Mathematics MCQ Set 3

1. Three companies A, B and C supply 25%, 35% and 40% of the notebooks to a school. Past experience shows that 5%, 4% and 2% of the notebooks produced by these companies are defective. If a notebook was found to be defective, what is the probability that the notebook was supplied by A?

a) ^{44}⁄_{69}

b) ^{25}⁄_{69}

c) ^{13}⁄_{24}

d) ^{11}⁄_{24}

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= ^{25}⁄_{69}.

2. A box of cartridges contains 30 cartridges, of which 6 are defective. If 3 of the cartridges are removed from the box in succession without replacement, what is the probability that all the 3 cartridges are defective?

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^{6}⁄

_{30}) * (

^{5}⁄

_{29}) * (

^{4}⁄

_{28})

= ^{(6 * 5 * 4)}⁄_{(30 * 29 * 28)}.

3. Two boxes containing candies are placed on a table. The boxes are labelled B_{1} and B_{2}. Box B^{1} contains 7 cinnamon candies and 4 ginger candies. Box B_{2} contains 3 cinnamon candies and 10 pepper candies. The boxes are arranged so that the probability of selecting box B_{1} is ^{1}⁄_{3} and the probability of selecting box B_{2} is ^{2}⁄_{3}. Suresh is blindfolded and asked to select a candy. He will win a colour TV if he selects a cinnamon candy. What is the probability that Suresh will win the TV (that is, she will select a cinnamon candy)?

a) ^{7}⁄_{33}

b) ^{6}⁄_{33}

c) ^{13}⁄_{33}

d) ^{20}⁄_{33}

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Then, P(B1) =^{1⁄3} and P(B2) = ^{2⁄3} P(A) = P(A ∩ B1) + P(A ∩ B2)
= P(A|B1) * P(B1) + P(A|B2)*P(B2)
= (^{7}⁄_{11}) * (^{1}⁄_{3}) + (^{3}⁄_{11}) * (^{2}⁄_{3})

= ^{13}⁄_{33}.

4. Two boxes containing candies are placed on a table. The boxes are labelled B_{1} and B_{2}. Box B_{1} contains 7 cinnamon candies and 4 ginger candies. Box B_{2} contains 3 cinnamon candies and 10 pepper candies. The boxes are arranged so that the probability of selecting box B_{1} is ^{1⁄3} and the probability of selecting box B_{2} is ^{2⁄3}. Suresh is blindfolded and asked to select a candy. He will win a colour TV if he selects a cinnamon candy. If he wins a colour TV, what is the probability that the marble was from the first box?

a) ^{7}⁄_{13}

b) ^{13}⁄_{7}

c) ^{7}⁄_{33}

d) ^{6}⁄_{33}

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_{1}be the event of selecting box B

_{1}. Let B

_{2}be the event of selecting box B

_{2}.

Then, P(B_{1}) = ^{1⁄3} and P(B_{2}) = ^{2⁄3} Given that Suresh won the TV, the probability that the cinnamon candy was selected from B1 is
P(B_{1}|A) = (P(A|B_{1}) * P( B_{1}) ) /( P(A│B_{1} ) * P( B_{1} ) + P(A│B_{1} ) * P(B_{2}) ) = ^{7}⁄_{13}.

5. Suppose box A contains 4 red and 5 blue coins and box B contains 6 red and 3 blue coins. A coin is chosen at random from the box A and placed in box B. Finally, a coin is chosen at random from among those now in box B. What is the probability a blue coin was transferred from box A to box B given that the coin chosen from box B is red?

a) ^{15}⁄_{29}

b) ^{14}⁄_{29}

c) ^{1}⁄_{2}

d) ^{7}⁄_{10}

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^{7}⁄

_{9}and the probability of choosing a blue coin from box A is P(B) =

^{5}⁄

_{9}. If a red coin was moved from box A to box B, then box B has 7 red coins and 3 blue coins. Thus the probability of choosing a red coin from box B is

^{7}⁄

_{10}. Similarly, if a blue coin was moved from box A to box B, then the probability of choosing a red coin from box B is

^{6}⁄

_{10}. Hence, the probability that a blue coin was transferred from box A to box B given that the coin chosen from box B is red is given by =

^{15}⁄

_{29}.

6. An urn B_{1} contains 2 white and 3 black chips and another urn B_{2} contains 3 white and 4 black chips. One urn is selected at random and a chip is drawn from it. If the chip drawn is found black, find the probability that the urn chosen was B_{1}.

a) ^{4}⁄_{7}

b) ^{3}⁄_{7}

c) ^{20}⁄_{41}

d) ^{21}⁄_{41}

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_{1}, E

_{2}denote the vents of selecting urns B

_{1}and B

_{2}respectively. Then P(E

_{1}) = P(E

_{2}) =

^{1}⁄

_{2}Let B denote the event that the chip chosen from the selected urn is black . Then we have to find P(E

_{1}/B). By hypothesis P(B /E

_{1}) =

^{3}⁄

_{5}and P(B /E

_{2}) =

^{4}⁄

_{7}By Bayes theorem P(E

_{1}/B) = (P(E1)*P(B│E1))/((P(E1) * P(B│E1)+P(E2) * P(B│E2)) ) = ((1/2) * (3/5))/((1/2) * (3/5)+(1/2)*(4/7) ) = 21/41.

7. At a certain university, 4% of men are over 6 feet tall and 1% of women are over 6 feet tall. The total student population is divided in the ratio 3:2 in favour of women. If a student is selected at random from among all those over six feet tall, what is the probability that the student is a woman?

a) ^{2}⁄_{5}

b) ^{3}⁄_{5}

c) ^{3}⁄_{11}

d) ^{1}⁄_{100}

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^{2}⁄

_{5}P(F) =

^{3}⁄

_{5}P(T|M) =

^{4}⁄

_{100}P(T|F) =

^{1}⁄

_{100}

P(F│T) = (P(T│F) * P(F))/(P(T│F) * P(F) + P(T│M) * P(M))

= ((1/100) * (3/5))/((1/100) * (3/5) + (4/100) * (2/5) )

= ^{3}⁄_{11}.

8. Previous probabilities in Bayes Theorem that are changed with help of new available information are classified as

a) independent probabilities

b) posterior probabilities

c) interior probabilities

d) dependent probabilities

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## Discrete Mathematics MCQ Set 4

1. Transfer function may be defined as

a) Ratio of out to input

b) Ratio of laplace transform of output to input

c) Ratio of laplace transform of output to input with zero initial conditions

d) None of the above

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2. Poles of any transfer function is define as the roots of equation of denominator of transfer function.

a) True

b) False

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3. Zeros of any transfer function is define as the roots of equation of numerator of transfer function.

a) True

b) False

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4. Find the poles of transfer function which is defined by input x(t)=5Sin(t)-u(t) and output y(t)=Cos(t)-u(t).

a) 4.79, 0.208

b) 5.73, 0.31

c) 5.89, 0.208

d) 5.49, 0.308

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Hence, transfer function

Roots of equation s^{2} – 5s + 1 = 0 is s = 4.79, 0.208.

5. Find the equation of transfer function which is defined by y(t)-∫_{0}^{t} y(t)dt + ^{d}⁄_{dt} x(t) – 5Sin(t) = 0

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6. Find the poles of transfer function given by system ^{d2}⁄_{dt2} y(t) – ^{d}⁄_{dt} y(t) + y(t) – ∫_{0}^{t} x(t)dt = x(t)

a) 0, 0.7 ± 0.466

b) 0, 2.5 ± 0.866

c) 0, 0 .5 ± 0.866

d) 0, 1.5 ± 0.876

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7. Find the transfer function of a system given by equation ^{d2}⁄_{dt2} y(t-a) + x(t) + 5 ^{d}⁄_{dt} y(t) = x(t-a).

a)(e^{-as}-s)/(1+e^{-as} s^{2})

b)(e^{-as}-5s)/(e^{-as} s^{2})

c) (e^{-as}-s)/(2+e^{-as} s^{2})

d) (e^{-as}-5s)/(1+e^{-as} s^{2})

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^{d2}⁄

_{dt2}y(t-a) + x(t) + 5

^{d}⁄

_{dt}y(t) = x(t-a).

Taking laplace transform, s^{2} Y(s) e^{-sa} + X(s) + 5sY(s) = e^{-as} X(s)

Hence, H(s) = ^{Y(s)}⁄_{X(s)} =(e^{-as}-5s)/(1+e^{-as} s^{2}).

8. Any system is said to be stable if and only if

a) It poles lies at the left of imaginary axis

b) It zeros lies at the left of imaginary axis

c) It poles lies at the right of imaginary axis

d) It zeros lies at the right of imaginary axis

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9. The system given by equation 5 ^{d3}⁄_{dt3} y(t) + 10 ^{d}⁄_{dt} y(t) – 5y(t) = x(t) + ∫_{0}^{t} x(t)dt, is

a) Stable

b) Unstable

c) Has poles 0, 0.455, -0.236±1.567

d) Has zeros 0, 0.455, -0.226±1.467

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10. Find the laplace transform of input x(t) if the system given by ^{d3}⁄_{dt3} y(t) – 2 ^{d2}⁄_{dt2} y(t) –^{d}⁄_{dt} y(t) + 2y(t) = x(t), is stable.

a) s + 1

b) s – 1

c) s + 2

d) s – 2

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^{d3}⁄

_{dt3}y(t) – 2

^{d2}⁄

_{dt2}y(t) –

^{d}⁄

_{dt}y(t) + 2y(t) = x(t),

Taking laplae transform,

(s^{3} – 2s^{2} – s + 2)Y(s) = X(s)
H(s) = ^{Y(s)}⁄_{X(s)} = ^{1}⁄_{(s-1)(s+1)(s+2)}

For the system to be stable, X(s) = s – 1.

11. The system given by equation y(t – 2a) – 3y(t – a) + 2y(t) = x(t – a),is

a)Stable

b)Unstable

c) Marginally stable

d) 0

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## Discrete Mathematics MCQ Set 5

1. The curvature of a function f(x) is zero. Which of the following functions could be f(x)

a) ax + b

b) ax^{2} + bx + c

c) sin(x)

d) cos(x)

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^{”}(x) = 0 f(x) = a + b.

2. The curvature of the function f(x) = x^{2} + 2x + 1 at x = 0 is

a) ^{3}⁄_{2}

b) 2

c) ∣2 / 5^{3⁄2}∣

d) 0

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3.The curvature of a circle depends inversely upon its radius r

a) True

b) False

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4.Find the curvature of the function f(x) = 3x^{3} + 4680x^{2} + 1789x + 181 at x = -520

a) 1

b) 0

c) ∞

d) -520

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^{-b}⁄

_{3a}is zero because f

^{”}is zero at that point. Looking at the form of the given point we can see that x =

^{-4680}⁄

_{3*3}Thus, curvature is zero.

5.Let c(f(x)) denote the curvature function of given curve f(x). The value of c(c(f(x))) is observed

to be zero. Then which of the following functions could be f(x)

a) f(x) = x^{3} + x + 1

b) f(x)^{2} + y^{2} = 23400

c) f(x) = x^{19930} + x + 90903

d) No such function exist

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6. The curvature of the function f(x) = x^{3} – x + 1 at x = 1 is given by

a) ∣^{6}⁄_{5} ∣

b) ∣^{3}⁄_{5} ∣

c) ∣^{6}⁄ 5^{3⁄2} ∣

d) ∣^{3}⁄ 5^{3⁄2}∣

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7.The curvature of a function depends directly on leading coefficient when x=0 which of the following could be f(x)

a) y = 323x^{3} + 4334x + 10102

b) y = x^{5} + 232x^{4} + 232x^{2} + 12344

c) y = ax^{5} + c

d) f(x) = x^{3} – x + 1

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y = 33x^{2} + 112345x + 8945

Observe numerator which is Now this second derivative must be non zero for the above condition asked in the question Looking at all the options we see that only quadratic polynomials can satisfy this.

8. Given x = k1e^{a1t} : y = k2e^{a2t} it is observed that the curvature function obtained is zero. What is the relation between a_{1} and a_{2}

a) a_{1} ≠ a_{2}

b) a_{1} = a_{2}

c) a_{1} = (a_{2})^{2}

d) a_{2} = (a_{1})^{2}

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9.The curvature function of some function is given to be k(x) = ^{1} ⁄ _{ [2 + 2x + x2]3 ⁄ 2} then which of the following functions

could be f(x)

a) ^{x2}⁄_{2} + x + 101

b) ^{x2}⁄_{4} + 2x + 100

c) x^{2} + 13x + 101

d) x^{3} + 4x^{2} + 1019

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10. Consider the curvature of the function f(x) = e^{x} at x=0. The graph is scaled up by a factor of and the curvature is measured again at x=0.

What is the value of the curvature function at x=0 if the scaling factor tends to infinity

a) a

b) 2

c) 1

d) 0

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^{ax}Now using curvature formula we have