Calculus MCQs Question and Answer in Engineering Maths

71

As x increased from \[ - \infty \] to \[\infty \] , the function \[{\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{e}}^{\rm{x}}}}}{{1 + {{\rm{e}}^{\rm{x}}}}}\]

A) monotonically increases B) monotonically decreases C) increases to a maximum value and then decreases D) decreases to a minimum value and then increases

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Answer: Option A

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72

The value of \[\mathop {\lim }\limits_{{\text{x}} \to 1} \frac{{{{\text{x}}^7} - 2{{\text{x}}^5} + 1}}{{{{\text{x}}^3} - 3{{\text{x}}^2} + 2}}\]

A) is 0 B) is -1 C) is 1 D) does not exist

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Answer: Option C

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73

\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{{\text{x}} - \sin {\text{x}}}}{{1 - \cos {\text{x}}}}\]

A) 0 B) 1 C) 3 D) not defined

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Answer: Option A

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74

Let f = y^x. What is \[\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}\] at x = 2, y = 1?

A) 0 B) ln 2 C) 1 D) \[\frac{1}{{\ln 2}}\]

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Answer: Option C

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75

Given y = x² + 2x + 10, the value of \[{\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 1}}\] is equal to

A) 0 B) 4 C) 12 D) 13

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Answer: Option B

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76

A sphere of unit radius is centered at the origin. The unit normal at a point (x, y, z) on the surface of the sphere is the vector

A) \[\left( {{\text{x, y, z}}} \right)\] B) \[\left( {\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right)\] C) \[\left( {\frac{{\text{x}}}{{\sqrt 3 }},\,\frac{{\text{y}}}{{\sqrt 3 }},\,\frac{{\text{z}}}{{\sqrt 3 }}} \right)\] D) \[\left( {\frac{{\text{x}}}{{\sqrt 2 }},\,\frac{{\text{y}}}{{\sqrt 2 }},\,\frac{{\text{z}}}{{\sqrt 2 }}} \right)\]

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Answer: Option A

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77

The total derivative of function xy is

A) xdy + ydx B) xdx + ydy C) dx + dy D) dxdy

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Answer: Option A

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78

\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \left( {\frac{{{{\text{e}}^{2{\text{x}}}} - 1}}{{\sin \left( {4{\text{x}}} \right)}}} \right)\] is equal to

A) 0 B) 0.5 C) 1 D) 2

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Answer: Option B

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79

The value of the function \[{\text{f}}\left( {\text{x}} \right) = \mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\text{x}}^3} + {{\text{x}}^2}}}{{2{{\text{x}}^3} - 7{{\text{x}}^2}}}\] is

A) 0 B) \[ - \frac{1}{7}\] C) \[\frac{1}{7}\] D) \[\infty \]

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Answer: Option B

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80

Consider the function f(x) = |x|³, where x is real. Then the function f(x) at x = 0 is

A) continuous but not differentiable B) once differentiable but not twice C) twice differentiable but not thrice D) thrice differentiable

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Answer: Option C

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