## Discrete Mathematics MCQ Set 1

1. ^{d(uvw)}⁄_{dx} is where u ,v, w are the functions of x

a) u’vw + uv’w + uvw’

b) uvw + uv’w’ + u’v’w’

c) u’v’w + uv’w’ + u’vw’

d) uv’w’ + u’v’w’ + uvw

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2. ^{d(u ⁄ v)}⁄_{dx} is where u, v are the functions of x

a) ^{v’u’ – uv} ⁄ _{v2}

b) ^{vu’ – uv’} ⁄ _{v2}

c) ^{vu – u’v’} ⁄ _{v2}

d) 0

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3. If , find ^{dy}⁄_{dx} .

a) Sec^{2} (x) e^{x} [1 + Tan(x)] + e^{x} Tan(x)Sec(x)

b) Sec^{2} (x) e^{x} [Sec(x) + Tan(x)] + e^{x} Tan(x)Sec(x)

c) Sec^{2} (x) e^{2x} [Sec(x) + Tan(x)] + e^{x} Tan(x)Sec(x)

d) Sec(x) e^{x} [Sec(x) + Tan(x)] + e^{x} Tan(x)Sec(x)

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^{x}

^{dy}⁄_{dx} = Sec^{2} (x)Sec(x) e^{x} + Sec^{2} (x)Tan(x) e^{x} + e^{x} Tan(x)Sec(x)

^{dy}⁄_{dx} = Sec^{2} (x) e^{x} [Sec(x) + Tan(x)] + e^{x} Tan(x)Sec(x).

4. Value of ^{d}⁄_{dx} [(1 + xe^{x})/(1-Cos(x))].

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5. Find the derivative of Sin(x)Tan(x) w.r.t e^{x} Tan(x)

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6.

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7. Evaluate d/dx x^{x} ln(x)

a) x^{(x-1)} + x^{2x} ln(x) + x^{x} [ln(x) ]^{2}

b) x^{(x-1)} + x^{x} ln(x) + x^{x} [ln(x) ]^{2}

c) x^{(x-1)} + x^{x} ln(x) + x^{x} ln(x)

d) x^{x} + x^{x} ln(x) + x^{x} [ln(x) ]^{2}

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8. Evaluate the differentiation of

a) tan^{-1}x

b) 1

c) 0

d) -1

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9. If y = Tan(x)^{Tan(x)} then ^{dy}⁄_{dx} = ?

a) Tan(x) [1 + lnTan(x)] Tan(x)^{Tan(x)}

b) Tan^{2} (x) [1 + lnTan(x)] Tan(x)^{Tan(x)}

c) Sec^{2} (x) [1 + lnTan(x)] Tan(x)^{Tan(x)}

d) Sec(x) [1 + lnTan(x)] Tan(x)^{Tan(x)}

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^{Tan(x)}Taking ln on both side ln y = Tan(x)lnTan(x) Differentiating w.r.t x

10. Evaluate ^{d}⁄_{dx} Cot(x)Cosec(x)

a) – Cosec^{2} (x)- Cosec^{2} (x)Cot(x)

b) – Cosec^{3} (x)- ^{2} (x)Cot(x)

c) – Cosec(x) – Cosec^{2} (x)Cot(x)

d) – Cosec^{3} (x)- Cosec(x)Cot^{2} (x)

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^{d}⁄

_{dx}Cot(x)Cosec(x) = – Cosec

^{3}(x)-

^{2}(x)Cot(x).

11. Evaluate differentiation of x^{2} Sin(x) w.r.t Tan(x)Cosec(x)

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12. If z = e^{x} Sin(Cos(x))Cos(Sin(x)) Then find ^{dz}⁄_{dx}

a) [e^{x}Sin(Cos(x))Cos(Sin(x))-e^{x}Cos(x)Cos(Cos(x))Cos(Sin(x))-e^{x}Sin(x)Sin(Cos(x))Sin(Sin(x))].

b) [e^{x}Sin(Cos(x))Cos(Sin(x))-e^{x}Sin(x)Cos(Cos(x))Cos(Sin(x))-e^{x}Cos(x)Sin(Cos(x))Sin(Sin(x))].

c) [e^{x}Cos(Cos(x))Sin(Sin(x))-e^{x}Sin(x)Cos(Cos(x))Cos(Sin(x))-e^{x}Cos(x)Sin(Cos(x))Sin(Sin(x))].

d) [e^{x}Sin(Cos(x))Cos(Sin(x))-e^{x}Cos(x)Cos(Cos(x))Cos(Sin(x))-e^{x}Sin(x)Sin(Cos(x))Sin(Sin(x))].

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^{dz}⁄

_{dx}=

^{d}⁄

_{dx}e

^{x}Sin(Cos(x))Cos(Sin(x)) = [(e

^{x}Sin(Cos(x))Cos(Sin(x)) – e

^{x}Sin(x)Cos(Cos(x))Cos(Sin(x)) – e

^{x}Cos(x)Sin(Cos(x))Sin(Sin(x)) )].

13. If F(x) = f(x)g(x)h(x) and F’(x) = 10F(x) and f’(x) = 10f(x) , g’(x) = 10g(x) and h’(x) = 10kh(x), then find value of k.

a) 0

b) 1

c) -1

d) 2

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Putting value of F’(x), f’(x), g’(x), h’(x)

We get 10 = 10 + 10 + 10k K = -1.

## Discrete Mathematics MCQ Set 2

1. lim_{x → 1} (x-1)Tan(^{πx}⁄_{2}) is

a) 0

b) –^{1}⁄_{π}

c) –^{2}⁄_{π}

d) ^{2}⁄_{π}

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2. Value of limit always be in the range of function?

a) True

b) False

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_{x → 1} – f(x) is 1 which is not in its range.

3. Necessary Conditions of Sandwich rule is

a) All function must have common domain.

b) All function must have common range.

c) All function must have common domain and range both..

d) Function must not have common domain and range.

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4. The value of lim_{x → 0} [x]Cos(x), [x] denotes the greatest integer function

a) lies between 0 and 1

b) lies between -1 and 0

c) lies between 0 and 2

d) lies between -2 and 0

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_{x → 0} [x]Cos(x) We know that, x-1 < [x] < x

Multiplying by Cos(x), we get (x-1)Cos(x) < [x]Cos(x) < xCos(x)

Taking limits, we get
lim_{x → 0} [(x-1)Cos(x)] < lim_{x → 0} [x]Cos(x) < lim_{x → 0}[xCos(x)]
=> -1 < lim_{x → 0} [x]Cos(x) < 0.

5. Value of lim_{x → 0}[(1+xe^{x} )/(1 – Cos(x))]

a) e

b) 1

c) 2

d) Can not be solved

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_{x → 0}[(1+xe

^{x})/(1 – Cos(x))] =

^{1}⁄

_{0}(Indeterminate) => By L’Hospital rule => lim

_{x → 0}[(1+xe

^{x}) / (Sin(x))] =

^{1}⁄

_{0}(Again indeterminate) => By L’ Hospital rule => lim

_{x → 0}[((2+x)e

^{x})/ (Cos(x))] = 2.

6. The value of , [x] denotes the greatest integer function

a) 0

b) 1

c) ∞

d) – ∞

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7. Evaluate lim_{x → 0}(1+Tan(x))^{Cot(x)}

a) 1

b) e

c) ln(2)

d) e^{2}

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_{x → 0}(1+Tan(x))

^{Cot(x)}= lim

_{tan(x) → 0}(1+Tan(x))

^{1⁄Tan(x)}= lim

_{t → 0}(1 + t)

^{1⁄t}= e.

8. Evaluate lim_{x → 1}[(-x^{x} + 1) / (xlog(x)) ]

a) e^{e}

b) e

c) -1

d) e^{2}

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9. Find domain of n for which lim_{x → 0}e^{nx}Cot(nx) , has non zero value.

a) n ∈ (0,∞) ∩ (1,5)

b) n ∈ (-∞,∞) ∩ (1,5)

c) n ∈ (-∞,∞)

d) n ∈ (-∞,∞) ~ 5

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10. Value of (dSin(x)Cos(x)) / dx is

a) Cos(2x)

b) Sin(2x)

c) Cos^{2}(2x)

d) Sin^{2}(2x)

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^{2}(x) – Sin

^{2}(x) = Cos(2x).

11. Evaluate

a) 1

b) e

c) 0

d) e^{2}

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12. If , then find the value of a and b.

a) 2.5, -1.5

b) -2.5, -1.5

c) -2.5, 1.5

d) 2.5, 1.5

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^{3}should be 1 ⇒ 1 + a – b=0 ⇒

^{b}⁄

_{6}–

^{a}⁄

_{2}= 1 ⇒ Solving the above two equations we get, a = -2.5, b = -1.5.

13. , then find the value of a, b and c.

a) 1.37, -4.13, 4.13

b) 1.37, 4.13, -4.13

c) -1.37, 4.13, 4.13

d) 1.37, 4.13, 4.13

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^{5}should be 1, then ⇒ B + c = 0 ⇒

^{b}⁄

_{6}+

^{c}⁄

_{2}= a ⇒

^{b}⁄

_{120}+

^{c}⁄

_{24}= 1 ⇒ By solving these 3 equations, a = 1.37, b = 4.13, c = -4.13.

## Discrete Mathematics MCQ Set 3

1. Which of the following mentioned standard Probability density functions is applicable to discrete Random Variables ?

a) Gaussian Distribution

b) Poisson Distribution

c) Rayleigh Distribution

d) Exponential Distribution

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2. What is the area under a conditional Cumulative density function ?

a) 0

b) Infinity

c) 1

d) Changes with CDF

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3. When do the conditional density functions get converted into the marginally density functions ?

a) Only if random variables exhibit statistical dependency

b) Only if random variables exhibit statistical independency

c) Only if random variables exhibit deviation from its mean value

d) If random variables do not exhibit deviation from its mean value

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4. Mutually Exclusive events

a) Contain all sample points

b) Contain all common sample points

c) Does not contain any sample point

d) Does not contain any common sample point

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5. What would be the probability of an event ‘G’ if H denotes its complement, according to the axioms of probability?

a) P (G) = 1 / P (H)

b) P (G) = 1 – P (H)

c) P (G) = 1 + P (H)

d) P (G) = P (H)

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6. A table with all possible value of a random variable and its corresponding probabilities is called

a) Probability Mass Function

b) Probability Density Function

c) Cumulative distribution function

d) Probability Distribution

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7. A variable that can assume any value between two given points is called

a) Continuous random variable

b) Discrete random variable

c) Irregular random variable

d) Uncertain random variable

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8. If a variable can certain integer values between two given points is called

a) Continuous random variable

b) Discrete random variable

c) Irregular random variable

d) Uncertain random variable

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9. The expected value of a discrete random variable ‘x’ is given by

a) P(x)

b) ∑ P(x)

c) ∑ x P(x)

d) 1

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10. If ‘X’ is a continuous random variable, then the expected value is given by

a) P(X)

b) ∑ x P(x)

c) ∫ X P(X)

d) No value such as expected value

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11. Out of the following values, which one is not possible in probability ?

a) P(x) = 1

b) ∑ x P(x) = 3

c) P(x) = 0.5

d) P(x) = – 0.5

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12. If E(x) = 2 and E(z) = 4, then E(z – x) =

a) 2

b) 6

c) 0

d) Insufficient data

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## Discrete Mathematics MCQ Set 4

1. The expansion of f(x), about x = a is

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^{h}⁄

_{1!}f’ (a) +

^{h2}⁄

_{2!}f

^{”}(a)…….

2. Find the expansion of e^{x} in terms of x + m, m > 0.

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^{(h-m)}By taylor theorem, putting a = -m , we get,

3. Expand ln(x) in the power of (x-m).

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^{(h+m)}By taylor theorem, putting a = m , we get,

4. Find the value of √10

a) 3.1633

b) 3.1623

c) 3.1632

d) 3.1645

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5. Expand f(x) = ^{1}⁄_{x} about x = 1.

a) 1 – (x-1) + (x-1)^{2} – (x-1)^{3} +⋯….

b) 1 + (x-1) + (x-1)^{2} + (x-1)^{3} +⋯….

c) 1 + (x-1) – (x-1)^{2} + (x-1)^{3} +⋯….

d) 1 – (x+1) + (x+1)^{2} – (x+1)^{3} +⋯….

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^{1}⁄

_{x}Let, x – 1 = h

Hence, x = 1 + h

Hence, f(x) = f(1 + h) = f(1) + ^{h}⁄_{1!} f’ (1) + ^{h2}⁄_{2!} f^{”} (1) +^{h3}⁄_{3!} f^{”’} (1)+⋯…

Now, f(1) = 1, f'(1) = -1, f”(1) = 2 ,f”'(1) = -6,…….

Hence, f(1 + h) = 1 – h + h^{2} – h^{3}+⋯…

hence, 1 – (x-1) + (x-1)^{2} – (x-1)^{3} +⋯….

6. Find the expansion of f(x) = ^{ex} ⁄_{1+ex}, given ∫f(x)dx = ln(2), for x = 0

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7. Find the value of e^{π⁄4√2}

a) 1.74

b) 1.84

c) 1.94

d) 1.64

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^{xSin(x)}, f(0) = 1

8. Find the value of ln(sin(31^{o})) if ln(2) = 0.69315

a) -0.653

b) -0.663

c) -0.764

d) -0.662

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9. The expansion of f(x,y), is

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10. The expansion of f(x, y)=e^{x Sin(y)}, is

a) x + xy + ……..

b) y + y^{2} x + ……..

c) x + x^{2} y + ……..

d) y + xy + ……..

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^{x Sin(y)}, f(0,0) = 0 Therefore, f

_{x}(x,y) = e

^{x}Sin(y), hence f

_{x}(0,0) = 0

f_{y} (x,y) = e^{x} Cos(y), hence f_{y} (0,0) = 1

f_{xx} (x,y) = e^{x} Sin(y), hence f_{xx} (0,0) = 0

f_{yy} (x,y) = -e^{x} Sin(y), hence f_{yy} (0,0) = 0

f_{xy} (x,y) = e^{x} Cos(y), hence f_{xy} (0,0) = 1
By taylor expansion,

f(x,y) = 0 + 0 + y + ^{1}⁄_{2}! [0 + 2xy + 0] +⋯.
f(x,y) = y + xy + ……..

11. The expansion of f(x, y) = e^{x} ln(1 + y), is

a) f(x,y)= y + xy – ^{y2}⁄_{2} +…….

b) f(x,y)= y – xy + ^{y2}⁄_{2} -…….

c) f(x,y)= y + x – ^{y2}⁄_{2} +……..

d) f(x,y)= x + y – ^{x2}⁄_{2} +……..

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^{x}ln(1 + y) , f(0,0) = 0 Therefore,

## Discrete Mathematics MCQ Set 5

1. For y = -x^{2} + 2x there exist a c in the interval [ – 19765, 19767] Such that f'(c) = 0

a) True

b) False

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^{2}+ 1 Observe here that on substituting – 19765 and 19767 in the equation we get (- 19766)

^{2}+ 1 and (- 19766)

^{2}respectively. As we are dealing with their squared values they have to be equal We have f(- 19765) = f(19767) Polynomial functions are continuous and differentiable over the whole domain and hence by Rolles Theorem we must have a c such that f'(c) = 0 in the interval [-19765, 19767] Hence, the claim is true.

2. For the function f(x) = sin(x)⁄x^{2} How many points exist in the interval [0, 7π] Such that f’ (c) = 0

a) 8

b) 0

c) 7

d) 6

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^{2}. Observe that the sine takes the value of zero at integral arguments, hence at every interval of the form We have f(nπ) = f((n + 1) π) The sine and the polynomial combination is continuous and differentiable at every point except x = 0 Every such interval has a point such that f’ (c) = 0 Hence, by Rolles theorem, in every interval of the form [nπ, (n + 1)π] we must have a point such that Leaving the interval [0, π] we are left with six such intervals from 0 to 7π.

3. f(x) = sin(x)⁄x, How many points exist such that f’ (c) = 0 in the interval [0, 18π]

a) 18

b) 17

c) 8

d) 9

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_{x→0}sin(x)⁄x reaching one. So leaving the first interval [0, π], for every other interval of the form [nπ (n + 1)π ] we must have f(nπ) = f((n + 1)π) By Rolles theorem we have f’ (c) = 0 For every interval of the form [nπ (n + 1)π ] There are 17 such intervals.

4. Let f(x) = x + sin(x) Every point on the graph is rotated by 45 degree with respect to the origin along the radius equal to the radius vector at that point. How many c that belong to [0, 11π] exist Such that f'(c) = 0

a) 10

b) 11

c) 110

d) 9

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5. A Function f(x) has the property f(a) = f(b) for ∀a,b…∊….I and a + b = 20 then which of the following even degree polynomials could be f(x)

a) ^{x4}⁄_{4} – 10x^{3} + 150x^{2} – 1000x + 10131729

b) x^{2} + 5x + 6

c) x^{2} + x + 1

d) Polynomial functions are inadequate representations

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^{3}– 30x

^{2}+ 300x – 1000 Equating to zero, we see that x = 10 satisfies the equation (10)

^{3}– 30(10)

^{2}+ 300(10) – 1000 = 0.

6. For some function f(x) we have f(a) = f(b) for ∀a,b…∊….I and a + b = 2 then which of the following even degree polynomials could f(x) be

a) x^{2} + 3x +1

b) ^{5x2}⁄_{2} – 5x + 101

c) x^{2} + 2x + 1

d) Even degree polynomials of such kind cannot exist

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^{-3}⁄

_{2}≠ we get 5x – 5 = 0 x =1.

7. For all second degree polynomials with y = ax^{2} + bx + k, it is seen that the Rolles’ point is at c = 0 . Also the value of k is zero. Then what is the value of b

a) 0

b) 1

c) -1

d) 56

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^{2}+ bx Now differentiating the function yields y’ = 2ax + b = 0 Equating it to zero we get the Rolle point which is also zero 2a(0) + b = 0 b = 0 .

8. For second degree polynomial it is seen that the roots are equal. Then what is the relation between the Rolles point c and the root x

a) c = x

b) c = x^{2}

c) They are independent

d) c = sin(x)

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^{2}Where a is the repeated root. Differentiating it and equating the function to zero we get the Rolles point, f'(x) = 2(x – a) = 0 x = a = c.

9. For any second degree polynomial with two real unequal roots. The relation between Rolles point r_{1} and the two roots r_{2} is

a) They are independent

b) c = r_{1} – r_{2}

c) c = r_{1} * ^{1}⁄_{r2}

d) c = r_{1} + ^{r2}⁄_{2}

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_{1}+

^{r2}⁄

_{2}.

10. If the domain of a function can be broken into infinite number of disjoint subsets such that every subset has a Rolles point then the function cannot be in a

polynomial structure.

a) True

b) False