## Discrete Mathematics MCQ Set 1

1. Expansion of function f(x) is

a) f(0) + ^{x}⁄_{1!} f^{‘} (0) + ^{x2}⁄_{2!} f^{”} (0)…….+^{xn}⁄_{n!} f^{n} (0)

b) 1 + ^{x}⁄_{1!} f^{‘} (0) + ^{x2}⁄_{2!} f^{”} (0)…….+^{xn}⁄_{n!} f^{n} (0)

c) f(0) – ^{x}⁄_{1!} f^{‘} (0) + ^{x2}⁄_{2!} f^{”} (0)…….+(-1)^n ^{xn}⁄_{n!} f^{n} (0)

d) f(1) + ^{x}⁄_{1!} f^{‘} (1) + ^{x2}⁄_{2!} f^{”} (1)…….+^{xn}⁄_{n!} f^{n} (1)

### View Answer

^{x}⁄

_{1!}f

^{‘}(0) +

^{x2}⁄

_{2!}f

^{”}(0)…….+

^{xn}⁄

_{n!}f

^{n}(0)

2. The necessary condition for the maclaurin expansion to be true for function f(x) is

a) f(x) should be continuous

b) f(x) should be differentiable

c) f(x) should exists at every point

d) f(x) should be continuous and differentiable

### View Answer

^{x}⁄

_{1!}f

^{‘}(0) +

^{x2}⁄

_{2!}f

^{”}(0)…….+

^{xn}⁄

_{n!}f

^{n}(0)

Where, f(x) should be continuous and differentiable upto nth derivative.

3. The expansion of f(a+h) is

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^{h}⁄

_{1!}f’ (a) +

^{h2}⁄

_{2!}f

^{”}(a)…….

4. The expansion of e^{Sin(x)} is

a) 1 + x + ^{x2}⁄_{2} + ^{x4}⁄_{8} +….

b) 1 + x + ^{x2}⁄_{2} – ^{x4}⁄_{8} +….

c) 1 + x – ^{x2}⁄_{2} + ^{x4}⁄_{8} +….

d) 1 + x + ^{x3}⁄_{6} – ^{x5}⁄_{10} +….

### View Answer

^{Sin(x)}, f(0) = 1 Hence, f

^{‘}(x)=f(x)Cos(x), f

^{‘}(0) = 1

f^{”} (x)=f^{‘} (x)Cos(x) – f(x)Sin(x),f^{”} (0)=1

f^{”’} (x)=f^{”} (x)Cos(x) – 2f^{‘} (x)Sin(x) – f(x) cos(x),f^{”’} (x) = 0

f^{””} (x)=f^{”’} (x)Cos(x) – 3f^{”} (x)Sin(x) – 3f^{‘} (x) cos(x) + f(x) sin(x), f^{””} (x) = -3

Hence,

f(x) = e^{Sin(x)} = 1 + x + ^{x2}⁄_{2} – ^{x4}⁄_{8} +…. (By mclaurin’ sexpansion)

5. Expansion of y = Sin^{-1}(x) is

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^{-1}(x), hence at x = 0, y = 0 Now, differentiating it, we get

6. Find the expansion of f(x) = ln(1+e^{x})

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^{x}), f(0) = ln(2) Differentiating it we get

7. Find the expansion of e^{xSin(x)}

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

8. Given f(x)= ln(cos(x) ),calculate the value of ln(cos(^{π}⁄_{2})).

a) -1.741

b) 1.741

c) 1.563

d) -1.563

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9. Find the expansion of cos(xsin(t)).

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10. Find the expansion of Sin(lSin^{-1} (x)).

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^{-1}(x)) Now, differentiating,

11. Expand (1 + x)^{1⁄x}, gives

a) e [1 + ^{x}⁄_{2} + ^{11x2}⁄_{24} -…..].

b) e [1 – ^{x}⁄_{2} + ^{11x2}⁄_{24} -…..].

c) e [^{x}⁄_{2} – ^{11x2}⁄_{24} -…..].

d) e [^{x}⁄_{2} + ^{11x2}⁄_{24} -…..].

### View Answer

^{1⁄x}

12. Find the solution of differential equation, ^{dy}⁄_{dx} = xy + x^{2}, if y = 1 at x = 0.

### View Answer

^{dy}⁄

_{dx}= xy + x

^{2}

hence, ^{dy}⁄_{dy} (x=0) = 0

and, ^{d2y}⁄_{dx2} = xy_{1} + y + 2x

hence, y_{2} = xy_{1} + y + 2x

hence, ^{d2y}⁄_{dx2} (x=0)=1

Differentiating it n times we get,

## Discrete Mathematics MCQ Set 2

1. A and B are two events such that P(A) = 0.4 and P(A ∩ B) = 0.2 Then P(A ∩ B) is equal to

a) 0.4

b) 0.2

c) 0.6

d) 0.8

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2. A problem in mathematics is given to three students A, B and C . If the probability of A solving the problem is ^{1}⁄_{2} and B not solving it is ^{1}⁄_{4} . The whole probability of the problem being solved is ^{63}⁄_{64} then what is the probability of solving it ?

a) ^{1}⁄_{8}

b) ^{1}⁄_{64}

c) ^{7}⁄_{8}

d) ^{1}⁄_{2}

### View Answer

^{1}⁄

_{2}, P(~B) =

^{1}⁄

_{4}and P(A ∪ B ∪ C) = 63/64

We know P(A ∪ B ∪ C) = 1 – P(A ∪ B ∪ C)

= 1 – P(A ∩ B ∩ C)

= 1 – P(A) P(B) P(C)

Let P(C) = p
ie ^{63}⁄_{64} = 1 – (^{1}⁄_{2})(^{1}⁄_{4})(p)

= 1 – ^{p}⁄_{8} ⇒ P =1/8 = P(C)
⇒P(C) = 1 – P = 1 – ^{1}⁄_{8} = ^{7}⁄_{8}.

3. Let A and B be two events such that P(A) = ^{1}⁄_{5} While P(A or B) = ^{1}⁄_{2}. Let P(B) = P. For what values of P are A and B independent?

a) ^{1}⁄_{10} and ^{3}⁄_{10}

b) ^{3}⁄_{10} and ^{4}⁄_{5}

c) ^{3}⁄_{8} only

d) ^{3}⁄_{10}

### View Answer

^{1}⁄

_{5}+ P (

^{1}⁄

_{5})P ⇒

^{1}⁄

_{2}=

^{1}⁄

_{5}+

^{4}⁄

_{5}P ⇒ P=

^{3}⁄

_{8}.

4. If A and B are two mutually exclusive events with P(~A) = ^{5}⁄_{6} and P(b) = ^{1}⁄_{3} then P(A /~B) is equal to

a) ^{1}⁄_{4}

b) ^{1}⁄_{2}

c) 0, since mutually exclusive

d) ^{5}⁄_{18}

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5. If A and B are two events such that P(a) = 0.2, P(b) = 0.6 and P(A /B) = 0.2 then the value of P(A /~B) is

a) 0.2

b) 0.5

c) 0.8

d) ^{1}⁄_{3}

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6. If A and B are two mutually exclusive events with P(a) > 0 and P(b) > 0 then it implies they are also independent

a) True

b) False

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7. Let A and B be two events such that the occurrence of A implies occurrence of B, But not vice-versa, then the correct relation between P(a) and P(b) is

a) P(A) < P(B)

b) P(B) ≥ P(A)

c) P(A) = P(B)

d) P(A) ≥ P(B)

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8. In a sample space S, if P(a) = 0, then A is independent of any other event

a) True

b) False

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9. If then,

a) P(a) > P(b)

b) P(a) > P(b)

c) P(a) = P(b)

d) P(~A) less than P(~B)

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10. If A is a subset of B then,

a) P(a) is greater than P(b)

b) P(~A) is greater than or equal to P(~B)

c) P(b) is equal to P(a)

d) P(b) is equal to P(~B)

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11. If A is a perfect subset of B and P(a < Pb), then P(B – A) is equal to

a) P(a) / P(b)

b) P(a)P(b)

c) P(a) + P(b)

d) P(b) – P(a)

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12. What is the probability of an impossible event?

a) 0

b) 1

c) Not defined

d) Insufficient data

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13. If A = A_{1} ∪ A_{2}……..∪ A_{n}, where A_{1}…A_{n} are mutually exclusive events then

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_{1}∪ A

_{2}……..∪ A

_{n}, where A

_{1}…A

_{n}Since A

_{1}…A

_{n}are mutually exclusive P(a) = P(A

_{1}) + P(A

_{2}) + … + P(A

_{n}) Therefore .

## Discrete Mathematics MCQ Set 3

1. If P(^{B}⁄_{A}) = p(b), then P( A and B) =

a) p(b)

b) p(a)

c) p(b)p(a)

d) p(a) + p(b)

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2. Two unbiased coins are tossed. What is the probability of getting at most one head?

a)^{1}⁄_{2}

b)^{1}⁄_{3}

c)^{1}⁄_{6}

d)^{3}⁄_{4}

### View Answer

^{3}⁄

_{4}.

3. If A and B are two events such that p(a) > 0 and p(b) is not a sure event, then

P(~A /~B) =

a) 1 – P(A /B)

b) P(~A)/P(~B)

c) Not Defined

d) (1 – P(A or B) ) /P(~B)

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4. If A and B are two events, then the probability of exactly one of them occurs is given by

a) P(A ∩ B) + P( A ∩ B)

b) P(A) + P(B) – 2P(A) P(B)

c) P(A) + P(B) – 2P(A) P(B)

d) P(A) + P(B) – P(A ∩ B)

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5. The probability that at least one of the events M and N occur is 0.6. If M and N have probability of occurring together as 0.2, then P(~M) + P(~N) is

a) 0.4

b) 1.2

c) 0.8

d) Indeterminate

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6. A jar contains ‘y’ blue colored balls and ‘x’ red colored balls. Two balls are pulled from the jar without replacing. What is the probability that the first ball Is blue and second one is red

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7. A survey determines that in a locality, 33% go to work by Bike, 42% go by Car, and 12% use both. The probability that a random person selected uses neither of them is

a) 0.29

b) 0.37

c) 0.61

d) 0.75

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8. A coin is biased so that its chances of landing Head is ^{2}⁄_{3} . If the coin is flipped 3 times, the probability that the first 2 flips are heads and the 3rd flip is a tail is

a) ^{4}⁄_{27}

b) ^{8}⁄_{27}

c) ^{4}⁄_{9}

d) ^{2}⁄_{9}

### View Answer

^{2}⁄

_{3}x

^{2}⁄

_{3}x

^{1}⁄

_{3}=

^{4}⁄

_{27}.

9. Husband and wife apply for two vacant spots in a company. If the probability of wife getting selected and husband getting selected are 3/7 and 2/3 respectively, what is the probability that neither of them will be selected?

a) ^{2}⁄_{7}

b) ^{5}⁄_{7}

c) ^{4}⁄_{21}

d) ^{17}⁄_{21}

### View Answer

^{2}⁄

_{3}) x (1 –

^{3}⁄

_{7}) =

^{4}⁄

_{21}.

10. For two events A and B, if P (B) = 0.5 and P (A ∪ B) = 0.5, then

P (A|B) =

a) 0.5

b) 0

c) 0.25

d) 1

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11. A fair coin is tossed thrice, what is the probability of getting all 3 same outcomes?

a) ^{3}⁄_{4}

b) ^{1}⁄_{4}

c) ^{1}⁄_{2}

d) ^{1}⁄_{6}

### View Answer

^{2}⁄

_{8}=

^{1}⁄

_{4}.

12. A bag contains 5 red and 3 yellow balls. Two balls are picked at random. What is the probability that both are of the same colour?

### View Answer

^{8}C

_{2}Probability of picking both balls as red =

^{5}C

_{2}/

^{8}C

_{2}Probability of picking both balls as yellow =

^{3}C

_{2}/

^{8}C

_{2}∴ required probability .

## Discrete Mathematics MCQ Set 4

1. nth derivative of Sinh(x) is

a) 0.5(e^{x} – e^{-x})

b) 0.5(e^{-x} – e^{x})

c) 0.5(e^{x} – (-1)^{n} e^{-x})

d) 0.5((-1)^{-n} e^{-x} -e^{x})

### View Answer

^{x}– e

^{-x}]. y

_{1}= 0.5 [e

^{x}– (-1)e

^{-x}]. y

_{2}= 0.5 [e

^{x}– (-1)

^{2}e

^{-x}]. Similarly, y

_{n}= 0.5 [e

^{x}– (-1)

^{n}e

^{-x}].

2. If y=log(^{x}⁄_{(x2 – 1)}) , then nth derivative of y is ?

a) (-1)^{(n-1)} (n-1)!(x^{(-n)} + (x-1)^{(-n)} + (x+1)^{(-n)} )

b) (-1)^{n} (n)! (x^{(-n-1)} + (x-1)^{(-n-1)} + (x+1)^{(-n-1)} )

c) (-1)^{(n+1)} (n+1)!(x^{(-n)} + (x-1)^{(-n)} + (x+1)^{(-n)} )

d) (-1)^{n}(n)! (x^{(-n-1)} + (x-1)^{(-n+1)} + (x+1)^{(-n+1)} )

### View Answer

^{2}– 1) y

_{1}= x

^{(-1)}-2x/(x

^{2}-1) y

_{1}= x

^{(-1)}-(x-1)

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

^{(-1)}y

_{n}= (-1)

^{(n-1)}(n-1)!(x

^{(-n)}-(x-1)

^{(-n)}+ (x+1)

^{(-n)}).

3. If x = a(Cos(t) + t^{2}) and y = a(Sin(t) + t^{2} + t^{3}) then dy/dx equals to

a) (Cos(t) + 3t^{2} + 2t) / (-Sin(t) + 2t)

b) (Sin(t) + 3t^{2} + 2t) / (-Cos(t) + 2t)

c) (Sin(t) + 3t^{2} + 2t) / (Cos(t) + 2t)

d) (Cos(t) + 3t^{2} + 2t) / (Sin(t) + 2t)

### View Answer

^{2}) Then, dy/dx = (Cos(t) + 3t

^{2}+2t)/(-Sin(t) + 2t).

4. If y=tan^{(-1)}(x) , then which one is correct ?

a) y_{3} + y_{1}^{2} + 4xy_{2} y_{1}=0

b) y_{3} + y_{1}^{2} + xy_{2} y_{1}=0

c) y_{3} + 2y_{1}^{2} + xy_{2} y_{1}=0

d) y_{2} + 2y_{1}^{2} + 4xy_{2} y_{1}=0

### View Answer

^{(-1)}(x)

5. What is the value of ^{dn (xm)}⁄_{dxn} for m<n, m=n, m>n ?

a) 0, n!, m_{Pn} x^{(m-n)}

b) m_{Pn} x^{(m-n)}, n!, 0

c) 0, n!, m_{Cn} x^{(m-n)}

d) m_{Cn} x^{(m-n)}, n!, 0

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6. Which of the following is true

a) Value of ^{dm (Sin(nx))}⁄_{dxm} is always positive for m=0, 1, 4, 5, 8, 9… for 0 < nx < ^{π}⁄_{2} and n<0.

b) Value of ^{dm (Sin(nx))}⁄_{dxm } is always positive for m=2, 3, 6, 7, 10, 11… for 0 < nx < ^{π}⁄_{2} and n>0.

c) Value of ^{dm (Sin(nx))}⁄_{dxm} is always positive for m=0, 1, 4, 5, 8, 9… for 0 < nx < ^{π}⁄_{2} and n>0.

d) Value of ^{dm (Sin(nx))}⁄_{dxm} is always positive for m=2, 3, 6, 7, 10, 11… for 0 < nx < ^{π}⁄_{2} and n<0.

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7. If nth derivative of e^{ax} sin(bx+c) cos(bx+c) is, e^{ax} r^{n} sin(bx+c+^{nα}⁄_{2}) cos(bx+c+^{nα}⁄_{2}) then,

### View Answer

^{ax}sin(bx+c) cos(bx+c) y = e

^{ax}sin2(bx+c)/2 y

_{n}= e

^{ax}r

^{n}sin(2(bx+c+nα/2))/2 y

_{n}= e

^{ax}r

^{n}sin(bx+c+nα/2) cos(bx+c+nα/2) where r = √(a

^{2}+4b

^{2}) , α = tan

^{(-1)}2b/a.

8. If y=^{x4}⁄_{x2-1} , Then,

a) 0.5*(-1)^{n} (n-1)! [(x-1)^{-n-1} + (x+1)^{-n-1} ].

b) 0.5*(-1)^{n} (n-1)! [x^{– n-1} + (x-1)^{-n-1} + (x+1)^{-n-1}].

c) 0.5*(-1)^{n} (n-1)! [(x-1)^{-n} + (x+1)^{-n}) ].

d) 0.5*(-1)^{n} (n-1)! [x^{-n} + (x-1)^{-n} + (x+1)^{-n}].

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9. If y=sin^{(-1)}(x) then select the true statement

a) y_{2} = xy_{1}^{3}

b) y_{3} = xy_{2}^{3}

c) y_{2} = xy_{1}^{2}

d) y_{3} = xy_{1}^{2}

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10. nth derivative of y = sin^{2}x cos^{3}x is

a) ^{1}⁄_{8} cos(x + ^{nπ}⁄_{2}) –^{1}⁄_{16} 5^{n} cos(x + ^{nπ}⁄_{2}) – ^{1}⁄_{16} 3^{n} cos(3x + ^{nπ}⁄_{2})

b) ^{1}⁄_{8} sin(x+^{nπ}⁄_{2}) –^{1}⁄_{16} 5^{n} cos(x + ^{nπ}⁄_{2}) – ^{1}⁄_{16} 3^{n} cos(3x + ^{nπ}⁄_{2})

c) ^{1}⁄_{8} cos(x+^{nπ}⁄_{2}) –^{1}⁄_{16} 5^{n} sin(x + ^{nπ}⁄_{2}) – ^{1}⁄_{16} 3^{n} sin(3x + ^{nπ}⁄_{2})

d) ^{1}⁄_{8} sin(x + ^{nπ}⁄_{2}) –^{1}⁄_{16} 5^{n} sin(x + ^{nπ}⁄_{2}) – ^{1}⁄_{16} 3^{n} sin(3x + ^{nπ}⁄_{2})

### View Answer

^{2}x cos

^{2}x cos(x) y =

^{1}⁄

_{4}sin

^{2}2x cos

^{2}x cos(x) y =

^{1}⁄

_{8}(2sin

^{2}2x) cos(x) y =

^{1}⁄

_{8}(1 – cos4x) cos(x) y =

^{1}⁄

_{8}(1 – cos4x) cos(x) y =

^{1}⁄

_{8}cos(x) –

^{1}⁄

_{8}cos4x cos(x) y =

^{1}⁄

_{8}cos(x) –

^{1}⁄

_{16}(cos5x + cos(3x)) Now, nth derivative is y

_{n}=

^{1}⁄

_{8}cos(x +

^{nπ}⁄

_{2}) –

^{1}⁄

_{16}5

^{n}cos(x +

^{nπ}⁄

_{2}) –

^{1}⁄

_{16}3

^{n}cos(3x +

^{nπ}⁄

_{2}).

11. If , Then value of ‘c’ equals to

a) 1

b) 2

c) 3

d) 4

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12. If y = e^{x}Sin^{-1}(x) and , Then find the value of ‘c’ ?

a) -2

b) 2

c) -0.5

d) 0.5

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13. Find nth derivative of y = Sin(x) Cos^{3}(x)

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14. If nth derivative of then find the value of a and b

a) -1, -2

b) 2, 1

c) 1, 2

d) -2, -1

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## Discrete Mathematics MCQ Set 5

1. Mean Value Theorem tells about the

a) Existence of point c in a curve where slope of a tangent to curve is equal to the slope of line joining two points in which curve is continuous and differentiable

b) Existence of point c in a curve where slope of a tangent to curve is equal to zero

c) Existence of point c in a curve where curve meets y axis

d) Existence of point c in a curve where curve meets x axis

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2. If f(a) is euquals to f(b) in Mean Value Theorem, then it becomes

a) Lebniz Theorem

b) Rolle’s Theorem

c) Taylor Series of a function

d) Leibnit’x Theorem

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3. Mean Value theorem is applicable to the

a) Functions differentiable in closed interval [a, b] and continuous in open interval (a, b).

b) Functions continuous in closed interval [a, b] only and having same value at point ‘a’ and ‘b’

c) Functions continuous in closed interval [a, b] and differentiable in open interval (a, b).

d) Functions differentiable in open interval (a, b) only and having same value at point ‘a’ and ‘b’

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4. Mean Value theorem is also known as

a) Rolle’s Theorem

b) Lagrange’s Theorem

c) Taylor Expansion

4) Leibnit’s Theorem

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5. Find the point c in the curve f(x) = x^{3} + x^{2} + x + 1 in the interval [0, 1] where slope of a tangent to a curve is equals to the slope of a line joining (0,1)

a) 0.64

b) 0.54

c) 0.44

d) 0.34

### View Answer

^{3}+ x

^{2}+ x + 1 f(x) is continuous in given interval [0,1]. f’(x) = 3x

^{2}+2x+1 Since, value of f’(x) is always finite in interval (0, 1) it is differentiable in interval (0, 1). f(0) = 1 f(1) = 4 By mean value theorem, f’(c) = 3c

^{2}+ 2c + 1 = (4-1)/(1-0) = 3 ⇒ c = 0.548,-1.215 Since c belongs to (0, 1) c = 0.54.

6. Find the equation of curve whose roots gives the point which lies in the curve f(x) = xSin(x) in the interval [0, ^{π}⁄_{2}] where slope of a tangent to a curve is equals to the slope of a line joining (0, ^{π}⁄_{2})

a) c = -Sec(c) – Tan(c)

b) c = -Sec(c) – Tan(c)

c) c = Sec(c) +Tan(c)

d) c = Sec(c) – Tan(c)

### View Answer

_{1}(x) = x and f

_{2}(x)=Sin(x) both are continuous in interval [0,

^{π}⁄

_{2}], the curve f(x)=f

_{1}(x)f

_{2}(x) is also continuous. f’(x) = xCos(x) + Sin(x) f’(x) always have finite value in interval [0,

^{π}⁄

_{2}] hence it is differentiable in interval (0,

^{π}⁄

_{2}). f(0) = 0 f(

^{π}⁄

_{2}) =

^{π}⁄

_{2}

By mean value theorem,
f’(c) = cCos(c) + Sin(c) = (^{π}⁄_{2} – 0)/(^{π}⁄_{2} – 0)=1
Hence, c = Sec(c) – Tan(c) is the required curve.

7. Can Mean Value Theorem be applied in the curve

a) True

b) False

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8. Find point c between [2,9] where, the slope of tangent to the function f(x)=1+∛x-1 at point c is equals to the slope of a line joining point (2,f(2)) and (9,f(9)).

(Providing given function is continuous and differentiable in given interval).

a) -2.54

b) 4.56

c) 4.0

d) 4.9

### View Answer

f(9) = 3
Applying mean value theorem,
f’(c) = 1/3∛(c-1)^{2} = [f(9)-f(2)]/(9-2) = 1/7

c = 1 ± (7/3)^{(3/2)}

c = 4.56,-2.54 Since c lies in (2,9), c = 4.56.

9. Find point c between [-1,6] where, the slope of tangent to the function f(x) = x^{2}+3x+2 at point c is equals to the slope of a line joining point (-1,f(-1)) and (6,f(6)).

(Providing given function is continuous and differentiable in given interval).

a) 2.5

b) 0.5

c) -0.5

d) -2.5

### View Answer

f(6) = 56 Applying mean value theorem,

f’(c) = 2c+3 = [f(6)-f(-1)]/[6-(-1)] = 56/7 = 8 c = 5/2 c = 2.5.

10. If f(x) = Sin(x)Cos(x) is continuous and differentiable in interval (0, x) then

### View Answer

^{(Cos(x)Sin(x))}⁄

_{x}……………………. (1)

Now, Given 0 < c < x Multiplying by 2 and taking Cos, We get 1 < Cos(2c) < Cos(2x)

1 < ^{(Cos(x)Sin(x))}⁄_{x} < Cos(2x).