Calculus MCQs Question and Answer in Engineering Maths

61

The value of the integral of the function g(x, y) = 4x³ + 10y⁴ along the straight line segment from the point (0, 0) to the point (1, 2) in the x - y plane is

A) 33 B) 35 C) 40 D) 56

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Answer: Option A

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62

The value of \[\mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{1 - \cos \left( {{{\rm{x}}^2}} \right)}}{{2{x^4}}}\] is

A) 0 B) \[\frac{1}{2}\] C) \[\frac{1}{4}\] D) undefined

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Answer: Option C

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63

Consider points P and Q in the x-y plane, with P = (1, 0) and Q = (0, 1). The line integral \[2\int\limits_{\rm{P}}^{\rm{Q}} {\left( {{\rm{xdx}} + {\rm{ydy}}} \right)} \] along the semicircle with the line segment PQ as its diameter

A) is -1 B) is 0 C) is 1 D) depends on the direction (clockwise or anticlockwise) of the semicircle

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Answer: Option B

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64

A surface S(x, y) = 2x + 5y - 3 is integrated once over a path consisting of the points that satisfy (x + 1)² + (y - 1)² = √2. The integral evaluates to

A) 17√2 B) \[\frac{{17}}{{\sqrt 2 }}\] C) \[\frac{{\sqrt 2 }}{{17}}\] D) 0

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Answer: Option D

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65

Let w = f(x, y), where x and y are functions of t. Then, according to the chain rule, \[\frac{{{\rm{dw}}}}{{{\rm{dt}}}}\] is equal

A) \[\frac{{{\rm{dw}}}}{{{\rm{dx}}}}\frac{{{\rm{dx}}}}{{{\rm{dt}}}} + \frac{{{\rm{dw}}}}{{{\rm{dy}}}}\frac{{{\rm{dt}}}}{{{\rm{dt}}}}\] B) \[\frac{{\partial {\rm{w}}}}{{\partial {\rm{x}}}}\frac{{\partial {\rm{x}}}}{{\partial {\rm{t}}}} + \frac{{\partial {\rm{w}}}}{{\partial {\rm{y}}}}\frac{{\partial {\rm{y}}}}{{\partial {\rm{t}}}}\] C) \[\frac{{\partial {\rm{w}}}}{{\partial {\rm{x}}}}\frac{{{\rm{dx}}}}{{{\rm{dt}}}} + \frac{{\partial {\rm{w}}}}{{\partial {\rm{y}}}}\frac{{{\rm{dy}}}}{{{\rm{dt}}}}\] D) \[\frac{{{\rm{dw}}}}{{{\rm{dx}}}}\frac{{\partial {\rm{x}}}}{{\partial {\rm{t}}}} + \frac{{{\rm{dw}}}}{{{\rm{dy}}}}\frac{{\partial {\rm{y}}}}{{\partial {\rm{t}}}}\]

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Answer: Option C

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66

For the parallelogram OPQR shown in the sketch, \[\overline {{\rm{OP}}} = {\rm{a\hat t}} + {\rm{b\hat j}}\] and \[\overline {{\rm{OR}}} = {\rm{c\hat t}} + {\rm{d\hat j}}{\rm{.}}\] The area of the parallelogram is
Calculus mcq question image

A) ad - bc B) ac + bd C) ad + bc D) ab - cd

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Answer: Option A

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67

The distance between the origin and the point nearest to it on the surface z² = 1 + xy is

A) 1 B) \[\frac{{\sqrt 3 }}{2}\] C) \[\sqrt 3 \] D) 3

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Answer: Option A

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68

The value of \[\int\limits_0^3 {\int\limits_0^{\rm{x}} {\left( {6 - {\rm{x}} - {\rm{y}}} \right)} {\rm{dx\,dy}}} \] is

A) 13.5 B) 27.0 C) 40.5 D) 54.0

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Answer: Option A

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69

If for non-zero x, \[{\rm{af}}\left( {\rm{x}} \right) + {\rm{bf}}\left( {\frac{1}{{\rm{x}}}} \right) = \frac{1}{{\rm{x}}} - 25\] where a ≠ b then \[\int\limits_1^2 {{\rm{f}}\left( {\rm{x}} \right){\rm{dx}}} \] is

A) \[\frac{1}{{{{\rm{a}}^2} - {{\rm{b}}^2}}}\left[ {a\left( {\ln 2 - 25} \right) + \frac{{47{\rm{b}}}}{2}} \right]\] B) \[\frac{1}{{{{\rm{a}}^2} - {{\rm{b}}^2}}}\left[ {a\left( {2\ln 2 - 25} \right) - \frac{{47{\rm{b}}}}{2}} \right]\] C) \[\frac{1}{{{{\rm{a}}^2} - {{\rm{b}}^2}}}\left[ {a\left( {2\ln 2 - 25} \right) + \frac{{47{\rm{b}}}}{2}} \right]\] D) \[\frac{1}{{{{\rm{a}}^2} - {{\rm{b}}^2}}}\left[ {a\left( {\ln 2 - 25} \right) - \frac{{47{\rm{b}}}}{2}} \right]\]

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Answer: Option A

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70

The improper integral \[\int\limits_0^\infty {{{\rm{e}}^{ - 2{\rm{t}}}}} {\rm{dt}}\] converges to

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

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Answer: Option C

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