Calculus - Engineering Mathematics

Q#1 Calculus GATE EC 2025 (Set 1) MCQ +1 mark -0.33 marks

Consider the following series:

(i)

(ii)

(iii)

Choose the correct option.

Only (ii) converges

Only (ii) and (iii) converge

Only (iii) converges

All three converge

Q#2 Calculus GATE EC 2025 (Set 1) MSQ +1 mark -0 marks

Consider the function , defined as

Which of the following statements is/are correct?

(Here, is the set of real numbers.)

has no global maximizer

has no global minimizer

is a local minimizer of

is a local maximizer of

Q#3 Calculus GATE EC 2025 (Set 1) MCQ +2 marks -0.66 marks

Consider a non-negative function which is continuous and bounded over the interval . Let and denote, respectively, the maximum and the minimum values of over the interval.

Among the combinations of and given below, choose the one(s) for which the inequality is guaranteed to hold.

Q#4 Calculus GATE EC 2024 (Set 1) MCQ +2 marks -0.66 marks

Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the N latitude circle, that contains the north pole in its interior is__________.

Q#5 Calculus GATE EC 2024 (Set 1) MSQ +2 marks -0 marks

Let be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?

For every closed path T, we have

There exists a differentiable scalar function F(x,y,z) such that

Q#6 Calculus GATE EC 2023 (Set 1) MCQ +1 mark -0.33 marks

The rate of increase, of a scalar field in the direction at a point is

2

4

Q#7 Calculus GATE EC 2023 (Set 1) MCQ +2 marks -0.66 marks

The value of the line integral along the straight line joining the points and is

20

24

29

-5

Q#8 Calculus GATE EC 2023 (Set 1) NAT +2 marks -0 marks

The value of the integral over the region , given in the figure, is __________(rounded off to the nearest integer).

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Q#9 Calculus GATE EC 2022 (Set 1) MCQ +1 mark -0.33 marks

Consider the two-dimensional vector field , where and denote the unit vectors along the -axis and the -axis, respectively. A contour in the -y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral is __________.

0

1

–1

Q#10 Calculus GATE EC 2022 (Set 1) MCQ +2 marks -0.66 marks

The function attains its minimum over the interval at __________.

(Here is the natural logarithm of .)

2

1

Q#11 Calculus GATE EC 2022 (Set 1) MSQ +2 marks -0 marks

Consider the following series:

For which of the following combinations of values does this series converge?

Q#12 Calculus GATE EC 2022 (Set 1) NAT +2 marks -0 marks

The value of the integral where is the shaded triangular region shown in the diagram, is ________ (rounded off to the nearest integer).

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Q#13 Calculus GATE EC 2021 (Set 1) MCQ +1 mark -0.33 marks

The vector function is defined over a circular arc shown in the figure.

The line integral of is

Q#14 Calculus GATE EC 2021 (Set 1) NAT +1 mark -0 marks

Consider the vector field

in a rectangular coordinate system with unit vectors , and . If the field is irrotational (conservative), then the constant (in integer) is ________.

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Q#15 Calculus GATE EC 2021 (Set 1) NAT +2 marks -0 marks

For a vector field in a cylindrical coordinate system with unit vectors and , the net flux of leaving the closed surface of the cylinder (rounded off to two decimal places) is ________.

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Q#16 Calculus GATE EC 2020 (Set 1) MCQ +1 mark -0.33 marks

For a vector field , which one of the following is False?

is another vector field.

is solenoidal if .

is irrotational if .

Q#17 Calculus GATE EC 2020 (Set 1) MCQ +1 mark -0.33 marks

The partial derivative of the function

with respect to at the point is

-1

1

0

Q#18 Calculus GATE EC 2020 (Set 1) NAT +2 marks -0 marks

For the solid S shown below, the value of is _________.

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Q#19 Calculus GATE EC 2019 (Set 1) NAT +1 mark -0 marks

The value of the integral, is equal to ______

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Q#20 Calculus GATE EC 2019 (Set 1) MCQ +2 marks -0.66 marks

Consider the line integral

The integral being taken in a counter clockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of 23 rectangle and a semi-circle of radius 1. The line integral evaluates to

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Q#21 Calculus GATE EC 2019 (Set 1) MCQ +2 marks -0.66 marks

Consider a differentiable function on the set of real numbers such that and . Given these conditions, which one of the following inequalities is necessarily true for all

Q#22 Calculus GATE EC 2018 (Set 1) NAT +1 mark -0 marks

Taylor series expansion ofdt around x = 0 has the form

The coefficient (correct to two decimal places) is equal to

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Q#23 Calculus GATE EC 2018 (Set 1) MCQ +1 mark -0.33 marks

Let , where a and b are constants, If at x=1 and y=2, then the relation between a and b is

Q#24 Calculus GATE EC 2018 (Set 1) MCQ +1 mark -0.33 marks

Consider with all real coefficients. It is known that its derivative has no real roots. The number of real roots ofis

0

1

2

3

Q#25 Calculus GATE EC 2018 (Set 1) NAT +2 marks -0 marks

Let and . Assume that x and y are independent variables. At , the value (correct to two decimal places) of is ________

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Q#26 Calculus GATE EC 2017 (Set 1) NAT +2 marks -0 marks

Let where x, y, z are real, and let C be the straight line segment form point A: (0, 2, 1) to point B: (4, 1, –1). The value of I is __________.

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Q#27 Calculus GATE EC 2017 (Set 1) NAT +2 marks -0 marks

A three dimensional region R of finite volume is described by

Where x, y, z are real. The volume of R (up to two decimal places) is ______________.

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Q#28 Calculus GATE EC 2017 (Set 1) MCQ +2 marks -0.66 marks

Let for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3.

Q#29 Calculus GATE EC 2017 (Set 2) NAT +1 mark -0 marks

The smaller angle (in degrees) between the planes x+y+z=1 and 2x – y+2z=0 is _________.

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Q#30 Calculus GATE EC 2017 (Set 2) MCQ +2 marks -0.66 marks

The values of the integrals and are

same and equal to 0.5

same and equal to – 0.5

0.5 and – 0.5 respectively

–0.5 and 0.5, respectively

Q#31 Calculus GATE EC 2017 (Set 2) NAT +2 marks -0 marks

The minimum value of the function in the interval occurs at x = _________.

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Q#32 Calculus GATE EC 2016 (Set 1) MCQ +1 mark -0.33 marks

Given the following statements about a function f:
, select the right option:

P: If f(x) is continuous at then it is also differentiable at

Q: If f(x) is continuous at then it may not be differentiable at

R: If f(x) is differentiable at then it is also continuous at

P is true, Q is false, R is false

P is false, Q is true, R is true

P is false, Q is true, R is false

P is true, Q is false, R is true

Q#33 Calculus GATE EC 2016 (Set 1) MCQ +1 mark -0.33 marks

Which one of the following is a property of the solutions to the Laplace equation: f = 0?

The solutions have neither maxima nor minima anywhere except at the boundaries.

The solutions are not separable in the coordinates.

The solutions are not continuous.

The solutions are not dependent on the boundary conditions.

Q#34 Calculus GATE EC 2016 (Set 1) MCQ +1 mark -0.33 marks

Consider the plot of f(x) versus x as shown below.

Suppose

Which one of the following is a graph of F(x)?

Q#35 Calculus GATE EC 2016 (Set 1) NAT +2 marks -0 marks

The integral dxdy, where D denotes the disc: , evaluates to_______________

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Q#36 Calculus GATE EC 2016 (Set 1) NAT +2 marks -0 marks

The region specified by

in cylindrical coordinates has volume of__________

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Q#37 Calculus GATE EC 2016 (Set 2) NAT +1 mark -0 marks

Consider the time –varying vector in Cartesian coordinates, where is a constant. When the vector magnitude is at its minimum value, the angle that I makes with the x axis (in degrees) such that is_______.

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Q#38 Calculus GATE EC 2016 (Set 2) MCQ +1 mark -0.33 marks

As x varies from −1 to +3, which one of the following describes the behaviour of the ?

f(x) increases monotonically.

f(x) increases, then decreases and increases again.

f(x) decreases, then increases and decreases again.

f(x) increases and then decreases.

Q#39 Calculus GATE EC 2016 (Set 2) MCQ +1 mark -0.33 marks

How many distinct values of x satisfy the equation, where x is in radians?

1

2

3

4 or more

Q#40 Calculus GATE EC 2016 (Set 2) NAT +2 marks -0 marks

Suppose C is the closed curve defined as the circle with C oriented anti-clockwise. The value of over the curve C equals ________.

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Q#41 Calculus GATE EC 2016 (Set 1) NAT +1 mark -0 marks

The integral is equal to __________

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Q#42 Calculus GATE EC 2016 (Set 1) NAT +2 marks -0 marks

A triangle in the xy-plane is bounded by the straight line 2x=3y, y=0 and x=3. The volume above the triangle and under the plane x + y + z = 6 is ___________

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Q#43 Calculus GATE EC 2015 (Set 1) MCQ +1 mark -0.33 marks

A function is defined in the closed interval [−1,1]. The value of x, in the open interval (-1,1) for which the mean value theorem is satisfied, is

-1/2

-1/3

1/3

Q#44 Calculus GATE EC 2015 (Set 1) MCQ +2 marks -0.66 marks

A vector is given by . Which one of the following statements is TRUE?

is solenoidal, but not irrotational

is irrotational, but not solenoidal

is neither solenoidal nor irrotational

is both solenoidal and irrotational

Q#45 Calculus GATE EC 2015 (Set 1) MCQ +2 marks -0.66 marks

Which one of the following graphs describes the function?

Q#46 Calculus GATE EC 2015 (Set 1) NAT +2 marks -0 marks

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse is _______.

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Q#47 Calculus GATE EC 2015 (Set 2) NAT +1 mark -0 marks

Let . If for all and c = 5, then d should be equal to _______.

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Q#48 Calculus GATE EC 2015 (Set 2) NAT +2 marks -0 marks

The value of the integral is ___________.

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Q#49 Calculus GATE EC 2015 (Set 3) MCQ +1 mark -0.33 marks

The contour on the x-y plane, where the partial derivative ofwith respect to y is equal to the partial derivative of 6y + 4x with respect to x, is

y=2

x=2

x+y=4

x– y = 0

Q#50 Calculus GATE EC 2015 (Set 3) NAT +2 marks -0 marks

A vector field exists inside a cylindrical region enclosed by the surfaces, z=0 and z=5. Let S be the surface bounding this cylindrical region. The surface integral of this field on is _______.

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Q#51 Calculus GATE EC 2014 (Set 1) MCQ +2 marks -0.66 marks

The Taylor series expansion of is

..........

...........

............

Q#52 Calculus GATE EC 2014 (Set 1) NAT +2 marks -0 marks

The volume under the surface and above the triangle in the x-y plane defined by and is ___________.

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Q#53 Calculus GATE EC 2014 (Set 2) MCQ +1 mark -0.33 marks

For , the maximum value of the function occurs at

Q#54 Calculus GATE EC 2014 (Set 2) MCQ +1 mark -0.33 marks

The value of is

1.0

Q#55 Calculus GATE EC 2014 (Set 2) NAT +2 marks -0 marks

If and , then ___________.

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Q#56 Calculus GATE EC 2014 (Set 3) NAT +1 mark -0 marks

The maximum value of the function occurs at x=______.

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Q#57 Calculus GATE EC 2014 (Set 3) MCQ +1 mark -0.33 marks

If , then

Q#58 Calculus GATE EC 2014 (Set 3) NAT +2 marks -0 marks

The maximum value of in the interval is ______.

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Q#59 Calculus GATE EC 2014 (Set 4) MCQ +1 mark -0.33 marks

The series converges to

2 ln 2

2

e

Q#60 Calculus GATE EC 2014 (Set 4) NAT +1 mark -0 marks

The magnitude of the gradient for the function at the point (1,1,1) is _________.

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Q#61 Calculus GATE EC 2014 (Set 4) NAT +1 mark -0 marks

The directional derivative of at (1, 1) in the direction of the unit vector at an angle of with y-axis, is given by ______ .

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Q#62 Calculus GATE EC 2014 (Set 4) MCQ +2 marks -0.66 marks

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is

Q#63 Calculus GATE EC 2013 (Set 1) MCQ +1 mark -0.33 marks

Consider a vector field . The closed loop line integral can be expressed as

over the closed surface bounded by the loop

over the closed volume bounded by the loop

over the open volume bounded by the loop

over the open surface bounded by the loop

Q#64 Calculus GATE EC 2013 (Set 1) MCQ +1 mark -0.33 marks

The maximum value of until which the approximation holds to within 10% error is

Q#65 Calculus GATE EC 2012 (Set 1) MCQ +2 marks -0.66 marks

The direction of vector A is radially outward from the origin, with |A| = Where and k is a constant. The value of n for which ∇.A = 0 is

–2

2

1

0

Q#66 Calculus GATE EC 2012 (Set 1) MCQ +2 marks -0.66 marks

The maximum value of in the interval [1, 6] is

21

25

41

46

Q#67 Calculus GATE EC 2011 (Set 1) MCQ +1 mark -0.33 marks

Consider a closed surface S surrounding a volume V. If is the position vector of a point inside S, with the unit normal on S, the value of the integral is

3V

5 V

10 V

15 V

Q#68 Calculus GATE EC 2010 (Set 1) MCQ +2 marks -0.66 marks

If , then y has a

maximum at

minimum at

maximum at

minimum at

Q#69 Calculus GATE EC 2010 (Set 1) MCQ +2 marks -0.66 marks

If then over the path shown in the figure is

0

1

Q#70 Calculus GATE EC 2009 (Set 1) MCQ +2 marks -0.66 marks

If a vector filed is related to another vector filed through, which of the following is true? Note: C and refer to any closed contour and any surface whose boundary is C.

Q#71 Calculus GATE EC 2008 (Set 1) MCQ +1 mark -0.33 marks

For real value of x, the minimum value of the function is

2

1

0.5

0

Q#72 Calculus GATE EC 2008 (Set 1) MCQ +2 marks -0.66 marks

The value of the integral of the function along the straight line segment from the point (0, 0) to the point (1, 2) in the x-y plane is

33

35

40

56

Q#73 Calculus GATE EC 2008 (Set 1) MCQ +2 marks -0.66 marks

Consider points P and Q in the x-y plane, with and . The line integral along the semicircle with the line segment PQ as its diameter

is –1

is 0

is 1

depends on the direction (clockwise or anti-clockwise) of the semicircle

Q#74 Calculus GATE EC 2007 (Set 1) MCQ +1 mark -0.33 marks

The following plot shows a function y which varies linearly with x. The value of the integral is

1.0

2.5

4.0

5.0

Q#75 Calculus GATE EC 2007 (Set 1) MCQ +1 mark -0.33 marks

For |x|≪ 1, coth(x) can be approximated as

x

Q#76 Calculus GATE EC 2007 (Set 1) MCQ +1 mark -0.33 marks

is

0.5

1

2

not defined

Q#77 Calculus GATE EC 2007 (Set 1) MCQ +1 mark -0.33 marks

Which one of the following functions is strictly bounded?

Q#78 Calculus GATE EC 2007 (Set 1) MCQ +1 mark -0.33 marks

For the function the linear approximation around x = 2 is

1-x

Q#79 Calculus GATE EC 2007 (Set 1) MCQ +2 marks -0.66 marks

Consider the function The maximum value of f(x) in the closed interval [-4, 4] is

18

10

–2.25

indeterminate

Q#80 Calculus GATE EC 2007 (Set 1) MCQ +2 marks -0.66 marks

Three functions and which are zero outside the interval [0,T], are shown in the figure. Which of the following statements is correct?

Q26

and are orthogonal

and are orthonormal

Q#81 Calculus GATE EC 2006 (Set 1) MCQ +1 mark -0.33 marks

, where P is a vector, is equal to

Q#82 Calculus GATE EC 2006 (Set 1) MCQ +1 mark -0.33 marks

, where P is a vector, is equal to

Q#83 Calculus GATE EC 2006 (Set 1) MCQ +2 marks -0.66 marks

The integral is given by

Q#84 Calculus GATE EC 2006 (Set 1) MCQ +2 marks -0.66 marks

As x is increased from to, the function

Monotonically increases

Monotonically decreases

Increases to a maximum value and then decreases

Decreases to a maximum value and then increases

Q#85 Calculus GATE EC 2005 (Set 1) MCQ +2 marks -0.66 marks

The value of the integral is

1

2

Q#86 Calculus GATE EC 2005 (Set 1) MCQ +2 marks -0.66 marks

The derivative of the symmetric function drawn in figure will look like

Q#87 Calculus GATE EC 1994 (Set 1) NAT +1 mark -0 marks

(True=1. False =0)

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Q#88 Calculus GATE EC 1993 (Set 1) MCQ +1 mark -0.33 marks

Which of the following improper integrals is (are) convergent?

Q#89 Calculus GATE EC 1993 (Set 1) MCQ +1 mark -0.33 marks

The function , has

No local maximum

One local minimum but no local maximum

One local maximum but no local minimum

One local minimum but one local maximum

Q#90 Calculus GATE EC 1993 (Set 1) NAT +1 mark -0 marks

is ___________.

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Q#91 Calculus GATE EC 1993 (Set 1) MCQ +1 mark -0.33 marks

If the linear velocity is given by the angular velocity at the point is _______.

None

Q#92 Calculus GATE EC 1993 (Set 1) NAT +1 mark -0 marks

The radius of convergence of the power series is _______

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Q#93 Calculus GATE EC 1993 (Set 1) NAT +1 mark -0 marks

The value of the double integral is __________

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Q#94 Calculus GATE EC 1993 (Set 1) NAT +1 mark -0 marks

Given, and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of the integral is _________

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