Calculus MCQs Question and Answer in Engineering Maths

1

\[\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + \frac{1}{{\text{x}}}} \right)^{2{\text{x}}}}\] is equal to

A) e^-2 B) e C) 1 D) e²

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Answer & Solution

Answer: Option D

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2

\[\mathop {\lim }\limits_{{\text{x}} \to \infty } \frac{{{\text{x}} - \sin {\text{x}}}}{{{\text{x}} + \cos {\text{x}}}}\] equals

A) 1 B) -1 C) \[\infty \] D) \[ - \infty \]

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Answer: Option A

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3

For the spherical surface x² + y² + z² = 1, the unit outward normal vector at the point \[\left( {\frac{1}{{\sqrt 2 }},\,\frac{1}{{\sqrt 2 }},\,0} \right)\] is given by

A) \[\frac{1}{{\sqrt 2 }}{\rm{\hat i}} + \frac{1}{{\sqrt 2 }}{\rm{\hat j}}\] B) \[\frac{1}{{\sqrt 2 }}{\rm{\hat i}} - \frac{1}{{\sqrt 2 }}{\rm{\hat j}}\] C) \[{{\rm{\hat k}}}\] D) \[\frac{1}{{\sqrt 3 }}{\rm{\hat i}} + \frac{1}{{\sqrt 3 }}{\rm{\hat j}} + \frac{1}{{\sqrt 3 }}{\rm{\hat k}}\]

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Answer & Solution

Answer: Option A

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4

Divergence of the vector field \[{{\rm{x}}^2}{\rm{z\hat i}} + {\rm{xy\hat j}} - {\rm{y}}{{\rm{z}}^2}{\rm{\hat k}}\] at (1, -1, 1) is

A) 0 B) 3 C) 5 D) 6

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Answer & Solution

Answer: Option C

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5

\[\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{1 - \cos {\text{x}}}}{{{{\text{x}}^2}}}} \right)\] is

A) \[\frac{1}{4}\] B) \[\frac{1}{2}\] C) 1 D) 2

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Answer & Solution

Answer: Option B

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6

As x varies from -1 to +3, which one of the following describes the behaviour of the function f(x) = x³ - 3x² + 1?

A) f(x) increases monotonically B) f(x) increases, then decreases and increases again C) f(x) decreases, then increases and decreases again D) f(x) increases and then decreases

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Answer & Solution

Answer: Option B

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7

The following plot shows a function y which varies linearly with x. The value of the integral \[{\text{I}} = \int\limits_1^2 {{\text{y dx}}} \] is
Calculus mcq question image

A) 1.0 B) 2.5 C) 4.0 D) 5.0

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Answer: Option B

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8

Given:
\[{\text{x}}\left( {\text{t}} \right) = 3\sin \left( {1000\pi {\text{t}}} \right)\] and \[{\text{y}}\left( {\text{t}} \right) = 5\cos \left( {1000\pi {\text{t}} + \frac{\pi }{4}} \right)\]
The X-Y plot will be

A) a circle B) a multi-loop closed curve C) a hyperbola D) an ellipse

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Answer & Solution

Answer: Option D

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9

The infinite series \[1 + {\text{x}} + \frac{{{{\text{x}}^2}}}{{2!}} + \frac{{{{\text{x}}^3}}}{{3!}} + \frac{{{{\text{x}}^4}}}{{4!}} + \,...\] corresponds to

A) sec x B) e^x C) cos x D) 1 + sin²x

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Answer: Option B

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10

If \[\phi = 2{{\text{x}}^3}{{\text{y}}^2}{{\text{z}}^4}\] then \[{\nabla ^2}\phi \] is

A) 12xy²z⁴ + 4x²z² + 20x³y²z³ B) 2x²y²z + 4x³z⁴ + 24x³y²z² C) 12xy²z⁴ + 4x³z⁴ + 24x³y²z² D) 4xy²z + 4x²z² + 24x³y²z²

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Answer & Solution

Answer: Option C

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