## Discrete Mathematics MCQ Set 1

1. Integration of function is same as the

a) Joining many small entities to create a large entity.

b) Indefinetly small difference of a function

c) Multiplication of two function with very small change in value

d) Point where function neither have maximujm value nor minimum value

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2. Integration of (Sin(x) + Cos(x))e^{x} is

a) e^{x} Cos(x)

b) e^{x} Sin(x)

c) e^{x} Tan(x)

d) e^{x} (Sin(x)+Cos(x))

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^{x}Sin(x)

∫ e^{x} Sin(x)dx = e^{x} Sin(x) – ∫ e^{x} Cos(x)dx

∫ e^{x} Sin(x)dx + ∫ e^{x} Cos(x)dx = ∫ e^{x} [Cos(x)+Sin(x)]dx = e^{x} Sin(x).

3. Integration of (Sin(x) – Cos(x))e^{x} is

a) -e^{x} Cos(x)

b) e^{x} Cos(x)

c) -e^{x} Sin(x)

d) e^{x} Sin(x)

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^{x}Sin(x) ∫ e

^{x}Sin(x)dx = -e

^{x}Cos(x) + ∫ e

^{x}Cos(x)dx ∫ e

^{x}Sin(x)d-∫ e

^{x}Cos(x)dx = ∫ e

^{x}[Sin(x)-Cos(x)]dx = -e

^{x}Cos(x).

4. Value of ∫ Cos^{2} (x) Sin^{2} (x)dx

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5. If differentiation of any function is zero at any point and constant at other points then it means

a) Function is parallel to x-axis at that point

b) Function is parallel to y-axis at that point

c) Function is constant

d) Function is discontinuous at that point

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^{dy}⁄

_{dx}at that point. Hence, when

^{dy}⁄

_{dx}= 0 means slope of a function is zero i.e, parallel to x axis. Function is not a constant function since it has finite value at other points.

6. If differentiation of any function is infinite at any point and constant at other points then it means

a) Function is parallel to x-axis at that point

b) Function is parallel to y-axis at that point

c) Function is constant

d) Function is discontinuous at that point

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^{dy}⁄

_{dx}at that point.Hence,when

^{dy}⁄

_{dx}= ∞ means slope of a function is 90 degree i.e,parallel to y axis.

7. Integration of function y = f(x) from limit x1 < x < x2 , y1 < y < y2, gives

a) Area of f(x) within x1 < x < x2

b) Volume of f(x) within x1 < x < x2

c) Slope of f(x) within x1 < x < x2

d) Maximum value of f(x) within x1 < x < x2

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8. Find the value of ∫ ^{ln(x)}⁄_{x dx}

a) 3a^{2}

b) a^{2}

c) a

d) 1

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9. Find the value of ∫^{t}⁄_{(t+3)(t+2)} dt, is

a) 2 ln(t+3)-3 ln(t+2)

b) 2 ln(t+3)+3 ln(t+2)

c) 3 ln(t+3)-2 ln(t+2)

d) 3 ln(t+3)+2ln(t+2)

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^{t}= x => dx = e

^{t}dt,

10. Find the value of ∫ cot^{3}() cosec^{4} ()

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12. Find the value of

a) ^{1}⁄_{8} sin^{(-1)}(x + ^{1}⁄_{2})

b)^{1}⁄_{4} tan^{(-1)}(x + ^{1}⁄_{2})

c) ^{1}⁄_{8} sec^{(-1)}(x + ^{1}⁄_{2})

d) ^{1}⁄_{4} cos^{(-1)}(x + ^{1}⁄_{2})

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## Discrete Mathematics MCQ Set 2

1. Find the value of ∫tan^{-1}(x)dx

a) sec^{-1} (x) – ^{1}⁄_{2} ln(1 + x^{2})

b) xtan^{-1} (x) – ^{1}⁄_{2} ln(1 + x^{2})

c) xsec^{-1} (x) – ^{1}⁄_{2} ln(1 + x^{2})

d) tan^{-1} (x) – ^{1}⁄_{2} ln(1 + x^{2})

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^{-1}(x)dx

Putting, x = tan(y),

We get, dy = sec^{2}(y)dy,

∫ysec^{2}(y)dy

By integration by parts,

ytan(y) – log(sec(y)) = xtan^{-1} (x) – ^{1}⁄_{2} ln(1 + x^{2}.

2. Integration of (Sin(x) + Cos(x))e^{x} is

a) e^{x} Cos(x)

b) e^{x} Sin(x)

c) e^{x} Tan(x)

d) e^{x} (Sin(x) + Cos(x))

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Let f(x) = e^{x} Sin(x)
∫e^{x} Sin(x)dx = e^{x} Sin(x) – ∫e^{x} Cos(x)dx

∫e^{x} Sin(x)dx + ∫e^{x} Cos(x)dx = ∫e^{x} [Cos(x) + Sin(x)]dx = e^{x} Sin(x).

3. Find the value of ∫x^{3} Sin(x)dx

a) x^{3} Cos(x) + 3x^{2} Sin(x) + 6xCos(x) – 6Sin(x)

b) – x^{3} Cos(x) + 3x^{2} Sin(x) – 6Sin(x)

c) – x^{3} Cos(x) – 3x^{2} Sin(x) + 6xCos(x) – 6Sin(x)

d) – x^{3} Cos(x) + 3x^{2} Sin(x) + 6xCos(x) – 6Sin(x)

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^{3}Sin(x) ∫x

^{3}Sin(x)dx = – x

^{3}Cos(x) + 3∫x

^{2}Cos(x)dx

∫x^{2} Cos(x)dx =x^{2} Sin(x) – 2∫xSin(x)dx

∫xSin(x)dx = – xCos(x) + ∫Cos(x)dx = – xCos(x) + Sin(x)

=> ∫x^{3} Sin(x)dx = – x^{3} Cos(x) + 3[x^{2} Sin(x) – 2[ – xCos(x) + Sin(x)]]

=> ∫x^{3} Sin(x)dx = – x^{3} Cos(x) + 3x^{2} Sin(x) + 6xCos(x) – 6Sin(x).

4. Value of ∫uv dx,where u and v are function of x

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5. Find the value of ∫x^{7} Cos(x) dx

a) x^{7} Sin(x) + 7x^{6} Cos(x) + 42x^{5} Sin(x) + 210x^{4} Cos(x) + 840x^{3} Sin(x) + 2520x^{2} Cos(x) + 5040xSin(x) + 5040Cos(x)

b) x^{7} Sin(x) – 7x^{6} Cos(x) + 42x^{5} Sin(x) – 210x^{4} Cos(x) + 840x^{3} Sin(x) – 2520x^{2} Cos(x) + 5040xSin(x) – 5040Cos(x)

c) x^{7} Sin(x) + 7x^{6} Cos(x) + 42x^{5} Sin(x) + 210x^{4} Cos(x) + 840x^{3} Sin(x) + 2520x^{2} Cos(x) + 5040xSin(x) + 5040Cos(x)

d) x^{7} Sin(x) + 7x^{6} Cos(x) + 42x^{5} Sin(x) + 210x^{4} Cos(x) + 840x^{3} Sin(x) + 2520x^{2} Cos(x) + 5040xSin(x) + 10080Cos(x)

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^{7}and v = Cos(x), ∫x

^{7}Cos(x) dx = x

^{7}Sin(x) + 7x

^{6}Cos(x) + 42x

^{5}Sin(x) + 210x

^{4}Cos(x) + 840x

^{3}Sin(x) + 2520x

^{2}Cos(x) + 5040xSin(x) + 5040Cos(x)

6. Find the value of ∫x^{3} e^{x} e^{2x} e^{3x}……..e^{nx} dx

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7. Find the area of a function f(x) = x^{2} + xCos(x) from x = 0 to a, where , a>0,

a) ^{a2}⁄_{2} + aSin(a) + Cos(a) – 1

b) ^{a3}⁄_{3} + aSin(a) + Cos(a)

c) ^{a3}⁄_{3} + aSin(a) + Cos(a) – 1

d) ^{a3}⁄_{3} + Cos(a) + Sin(a) – 1

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^{2}+ xCos(x)

Hence, F(x) = ∫x^{2} + xCos(x) dx = ^{x3}⁄_{3} + xSin(x) + Cos(x)

Hence, area inside f(x) is,

F(a) – F(0) = ^{a3}⁄_{3} + aSin(a) + Cos(a) – 1.

8. Find the area ^{ln(x)}⁄_{x} from x = x = ae^{b} to a

a) ^{b2}⁄_{2}

b) ^{b}⁄_{2}

c) b

d) 1

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9. Find the area inside a function f(t) = t/(t+3)(t+2) from t = -1 to 0

a) 4 ln(3) – 5ln(2)

b) 3 ln(3)

c)3 ln(3) – 4ln(2)

d) 3 ln(3) – 5 ln(2)

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10. Find the area inside integral from x = 0 to π

a) π

b) 0

c) 1

d) 2

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11. Find the area inside function from x = 1 to a

a) ^{a2}⁄_{2} + 5a – 4ln(a)

b) ^{a2}⁄_{2} + 5a – 4ln(a) – ^{11}⁄_{2}

c) ^{a2}⁄_{2} + 4ln(a) – ^{11}⁄_{2}

d) ^{a2}⁄_{2} + 5a – ^{11}⁄_{2}

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^{x2}⁄

_{2}+ 5x – 4ln(x)

Hence, area under, x = 1 to a, is

F(a) – F(1)=^{a2}⁄_{2} + 5a – 4ln(a) – 1/2 – 5=^{a2}⁄_{2} + 5a – 4ln(a) – ^{11}⁄_{2}

12. Find the value of ∫(x^{4} – 5x^{2} – 6x)^{4} 4x^{3} – 10x – 6 dx

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13. Temperature of a rod is increased by moving x distance from origin and is given by equation T(x) = x^{2} + 2x , where x is the distance and T(x) is change of temperature w.r.t distance.If,at x = 0,temperature is 40 C,find temperature at,x=10 .

a) 473 C

b) 472 C

c) 474 C

d) 475 C

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^{2}+ 2x dx =

^{x3}⁄

_{3}+ x

^{2}+ C At x=0 given T = 40 C C = T(x = 0) = 40 C At x= 10, T(x = 10) =

^{1000}⁄

_{3}+ 100 + 43 = 473 C.

14. Find the value of

a) ^{1}⁄_{8} sin^{-1}(x + ^{1}⁄_{2})

b) ^{1}⁄_{8} tan^{-1}(x + ^{1}⁄_{2})

c) ^{1}⁄_{8} sec^{-1}(x + ^{1}⁄_{2})

d) ^{1}⁄_{4} cos^{-1}(x + ^{1}⁄_{2})

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## Discrete Mathematics MCQ Set 3

1. Find

a) 0

b) ^{π}⁄_{8}

c) ^{π}⁄_{4}

d) ^{15π}⁄_{96}

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2. Find

a) 1

b) 0

c) ^{13π}⁄_{1098}

d) ^{21π}⁄_{2048}

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3. Find

a) 0

b) 1

c)-1

d) None of these

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4. Find the value of

a) ^{1}⁄_{10!}

b) ^{5!6!}⁄_{11!}

c) ^{10!}⁄_{5!6!}

d) 0

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5. Find

a)-1

b) 1

c) 0

d) ^{1}⁄_{5} – ^{1}⁄_{3} + ^{1}⁄_{1} – ^{π}⁄_{4}

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6. Find

a) -1

b) 1

c) 0

d) 4((^{π}⁄_{2})^{3} – 3π + 1)

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7. Find

a) 1

b) 199

c) -5!

d) 5!

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8. Find

a) 0

b) 5

c) 87

d) ^{-16}⁄_{105}

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## Discrete Mathematics MCQ Set 4

1. In a simple one-constraint Lagrange multiplier setup, the constraint has to be always one dimension lesser than the objective function.

a) True

b) False

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2. Maximize the function x + y – z = 1 with respect to the constraint xy=36.

a) 0

b) -8

c) 8

d) No Maxima exists

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3. Which one of these is the right formula for the Lagrange multiplier with more than one constraint.

a) ∇f = (μ)^{2} * ∇_{g1} + ∇_{g2}

b) Cannot be applied to more than one constraint function.

c) ∇f = μ * ∇_{g1} + λ * ∇_{g2}

d) ∇f = μ * ∇_{g1} +∇_{g2}

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^{n}

_{i=1}μ

_{i}∇

_{gi}Where μ

_{i}, μ

_{2}……..μ

_{n}are appropriate constraints(scalar multiples).

4. Maximum value of a 3-d plane is to be found over a circular region. Which of the following happens if we increase the radius of the circular region.

a) Maximum value is invariant

b) Maximum value decreases

c) Maximum value increases and minimum value goes lesser

d) minimum value goes higher

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5. Find the points on the plane x + y + z = 9 which are closest to origin.

a) (3,3,3)

b) (2,1,3)

c) (2,2,2)

d) (3,4,1)

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^{2}+ y

^{2}+ z

^{2}compute gradient ∇f = 2x i + 2y j + 2z k Now compute gradient of the function x + y + z = 9 which is = i + j + k Using Lagrange condition we have ∇f = λ . ∇

_{g}2x i + 2y j + 2z k = λ * (i + j + k) ⇒ x = y =z Put this back into constraint function we get 3x = 9 ⇒ (x,y,z) = (3,3,3).

6. Consider the points closest to the origin on the planes x + y + z =a.

a) The closest point travels farther as **a** is increased

b) The closest point travels nearer as **a** is increased

c) The closest point is independent of **a** as **a** is not there in the expression of the gradient.

d) Varies as **a ^{2}**, away from the origin.

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**a**value. Hence, we may conclude that the closest point of lower

**a**value plane would be closer to the origin. The Lagrange multiplier set up can be used to verified this.

7. The span of a Astroid is increased along both the x and y axes equally. Then the maximum value of: z = x + y along the Astroid

a) Increases

b) Decreases

c) Invariant

d) The scaling of Astroid is irrelevant

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^{2/3}+ y

^{2/3}= a

^{2/3}and then equating them by Lagrange condition. we can conclude that the maximum value increases.

8. The extreme value of the function With respect to the constraint Σ^{m}_{i=1} (x_{i})^{2} = 1 where **m** always stays lesser than **n** and as **m,n** tends to infinity is:

a) 1

b) 2/3√3

c) 2

d) 1 ⁄ 2

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## Discrete Mathematics MCQ Set 5

1. Laplace of function f(t) is given by

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2. Laplace transform any function changes it domain to s-domain.

a) True

b) False

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3. Laplace transform if sin(at)u(t) is

a) s ⁄ a^{2}+s^{2}

b) a ⁄ a^{2}+s^{2}

c) s^{2} ⁄ a^{2}+s^{2}

d) a^{2} ⁄ a^{2}+s^{2}

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4. Laplace transform if cos(at)u(t) is

a) s ⁄ a^{2}+s^{2}

b) a ⁄ a^{2}+s^{2}

c) s^{2} ⁄ a^{2}+s^{2}

d) a^{2} ⁄ a^{2}+s^{2}

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5. Find the laplace transform of e^{t} Sin(t).

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6. Laplace transform of t^{2} sin(2t)

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7. Find the laplace transform of t^{5⁄2}

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8. Value of ∫_{-∞}^{∞}e^{t} Sin(t)Cos(t)dt = ?

a) 0.5

b) 0.75

c) 0.2

d) 0.71

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_{-∞}

^{∞}e

^{-st}Sin(2t)dt = 2/(s

^{2}+ 4) Putting s=-1 ∫

_{-∞}

^{∞}e

^{t}Sin(2t)dt = 0.4 hence, ∫

_{-∞}

^{∞}e

^{-st}Sin(t)Cos(t)dt = 0.2.

9. Value of ∫_{-∞}^{∞}e^{t} Sin(t) dt = ?

a) 0.50

b) 0.25

c) 0.17

d) 0.12

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_{-∞}

^{∞}e

^{-st}Sin(t)dt = 1/(s

^{2}+ 1) Putting s = -1 ∫

_{-∞}

^{∞}e

^{t}Sin(t)dt = 0.5.

10. Value of ∫_{-∞}^{∞}e^{t} log(1+t)dt = ?

a) Sum of infinite integers

b) Sum of infinite factorials

c) Sum of squares of Integers

d) Sum of square of factorials

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11. Find the laplace transform of y(t)=e^{t}.t.Sin(t)Cos(t)

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12. Find the value of ∫_{0}^{∞} tSin(t)Cos(t)

a) s ⁄ s^{2}+2^{2}

b) a ⁄ a^{2}+s^{4}

c) 1

d) 0

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13. Find the laplace transform of y(t)=e^{|t-1|} u(t).