Calculus - Engineering Mathematics

Q#1 Calculus GATE EE 2025 (Set 1) NAT +2 marks -0 marks

Let . The rate of change of the real valued function,

at the origin in the direction of the point is __________ (round off to the nearest integer).

Use Keypad/Enter your answer

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

Consider a vector , where represent unit vectors along the coordinate axes respectively. The directional derivative of the function at the point in the direction of is

0

7

21

Q#3 Calculus GATE EE 2024 (Set 1) MCQ +2 marks -0.66 marks

Let be a real-valued function whose second derivative is positive for . Which of the following statements is/are always true?

has at least one local minimum.

cannot have two distinct local minima.

has at least one local maximum.

The minimum value of cannot be negative.

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

Consider the function for , where denotes the maximum of and . Which of the following statements is/are true?

is not differentiable.

is differentiable and its derivative is continuous.

is differentiable but its derivative is not continuous.

and its derivative are differentiable.

Q#5 Calculus GATE EE 2023 (Set 1) MSQ +2 marks -0 marks

Consider the following equation in a 2-D real-space.

Which of the following statement(s) is/are true.

When , the area enclosed by the curve is .

When tends to , the area enclosed by the curve tends to 4.

When tends to 0, the area enclosed by the curve is 1.

When , the area enclosed by the curve is 2.

Q#6 Calculus GATE EE 2023 (Set 1) NAT +2 marks -0 marks

The closed curve shown in the figure is described by , where ;

. The magnitude of the line integral of the vector field around the closed curve is __________ (Round off to 2 decimal places).

Use Keypad/Enter your answer

Q#7 Calculus GATE EE 2023 (Set 1) NAT +2 marks -0 marks

A quadratic function of two variables is given as

The magnitude of the maximum rate of change of the function at the point is ________ (Round off to the nearest integer).

Use Keypad/Enter your answer

Q#8 Calculus GATE EE 2022 (Set 1) MCQ +2 marks -0.66 marks

Let . Then decreases in the interval.

Q#9 Calculus GATE EE 2022 (Set 1) MCQ +2 marks -0.66 marks

Let . The direction in which the function increases most rapidly at point is

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

Let be a region in the first quadrant of the plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region ?

Q#11 Calculus GATE EE 2021 (Set 1) MCQ +1 mark -0.33 marks

Let be a real-valued function such that for some and for all . Then has

two distinct local minima in

exactly one local minimum in

one local maximum in

no local minimum in

Q#12 Calculus GATE EE 2021 (Set 1) NAT +1 mark -0 marks

Suppose the circles and intersect each other orthogonally at the point (u,v). Then __________.

Use Keypad/Enter your answer

Q#13 Calculus GATE EE 2021 (Set 1) MCQ +2 marks -0.66 marks

In the open interval , the polynomial has

three real roots

two real roots

one real root

no real roots

Q#14 Calculus GATE EE 2020 (Set 1) MSQ +1 mark -0 marks

is a polynomial on real over real coefficients wherein. Which of the following statements is true?

No choice of coefficients can make all roots identical.

a, b, c, d can be chosen to ensure that all roots are complex.

d can be chosen to ensure that is a root for any given set .

c alone cannot ensure that all roots are real.

Q#15 Calculus GATE EE 2020 (Set 1) MCQ +2 marks -0.66 marks

For real numbers, and with , the maximum and minimum value of for are respectively _________.

7 and

7 and 1

-2 and

1 and

Q#16 Calculus GATE EE 2020 (Set 1) NAT +2 marks -0 marks

Let and be unit vectors along and directions, respectively. A vector function is given by .

The line integral of the above function, along the curve , which follows the parabola as shown below is __________. (rounded off to 2 decimal places).

Use Keypad/Enter your answer

Q#17 Calculus GATE EE 2019 (Set 1) NAT +1 mark -0 marks

If , the value of line integral . dr evaluated over contour C formed by the segments is ______________.

Use Keypad/Enter your answer

Q#18 Calculus GATE EE 2019 (Set 1) NAT +2 marks -0 marks

If then div(uA) at (1, 1, 1) is ____________.

Use Keypad/Enter your answer

Q#19 Calculus GATE EE 2018 (Set 1) NAT +1 mark -0 marks

Let f be a real-valued function of a real variable defined as, where denotes the largest integer less than or equal to x. The value ofis _______ (up 0.25 to 2 decimal places).

Use Keypad/Enter your answer

Q#20 Calculus GATE EE 2018 (Set 1) MCQ +1 mark -0.33 marks

Let f be a real-valued function of a real variable defined as for, andfor x < 0. Which one of the following statements is true?

f (x) is discontinuous at x=0.

f (x) is continuous but not differentiable at x=0.

f (x) is differentiable but its first derivative is not continuous at x=0.

f (x) is differentiable but its first derivative is not differentiable at x=0.

Q#21 Calculus GATE EE 2018 (Set 1) MCQ +1 mark -0.33 marks

The value of the directional derivative of the functionat the point in the direction of the vectoris

0

0.95

0.93

0.9

Q#22 Calculus GATE EE 2018 (Set 1) NAT +2 marks -0 marks

Let. The maximum value of f(x) over the interval [0, 2] is _____ (up to 1 decimal place).

Use Keypad/Enter your answer

Q#23 Calculus GATE EE 2018 (Set 1) NAT +2 marks -0 marks

As shown in the figure, C is the arc from the point (3, 0) to the point (0, 3) on the circle. The value of the integral is ________ (up to 2 decimal places).

Use Keypad/Enter your answer

Q#24 Calculus GATE EE 2017 (Set 1) NAT +1 mark -0 marks

Let, where R is the region shown in the figure and. The value of I equals __________. (Give the answer up to two decimal places).

Use Keypad/Enter your answer

Q#25 Calculus GATE EE 2017 (Set 1) MCQ +2 marks -0.66 marks

A function f(x) is defined as . Which one of the following statements is TRUE?

f(x) is NOT differentiable at x = 1 for any values of a and b.

f(x) is differentiable at x = 1 for the unique values of a and b.

f(x) is differentiable at x = 1 for all values of a and b such that a + b = e.

f(x) is differentiable at x = 1 for all values of a and b.

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

Consider a function f(x,y,z) given by . The partial derivative of this function with respect to x at the point, x = 2, y = 1 and z = 3 is __________ .

Use Keypad/Enter your answer

Q#27 Calculus GATE EE 2017 (Set 2) MCQ +2 marks -0.66 marks

Let

Consider the composition of f and g. i.e., . The number of discontinuities in present in the interval is:

0

1

2

4

Q#28 Calculus GATE EE 2016 (Set 1) NAT +1 mark -0 marks

The maximum value attained by the function in the interval [1, 2] is ______________.

Use Keypad/Enter your answer

Q#29 Calculus GATE EE 2016 (Set 2) MCQ +1 mark -0.33 marks

The value of the line integral along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is

0

2

4

6

Q#30 Calculus GATE EE 2016 (Set 2) NAT +2 marks -0 marks

The line integral of the vector field along a path from (0, 0, 0) to (1, 1, 1) parametrized by is ______________.

Use Keypad/Enter your answer

Q#31 Calculus GATE EE 2015 (Set 1) MCQ +1 mark -0.33 marks

If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statement is TRUE?

f(a) . f(b) = 0

f(a) . f(b) < 0

f(a) . f(b) > 0

f(a)/f(b)

Q#32 Calculus GATE EE 2015 (Set 1) MCQ +1 mark -0.33 marks

Consider a function , where r is the distance from the origin and is the unit vector is the radial direction, the divergence of this function over a sphere of radius R, which includes the origin, is

0

Q#33 Calculus GATE EE 2015 (Set 2) MCQ +1 mark -0.33 marks

Given , where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?

If f(z) is differentiable at , then g(z) and h(z) are also differentiable at .

If g(z) and h(z) are differentiable at , then f(z) is also differentiable at .

If f(z) is continuous at , then it is differentiable at .

If f(z) is differentiable at , then so are its real and imaginary parts.

Q#34 Calculus GATE EE 2015 (Set 2) NAT +2 marks -0 marks

The volume enclosed by the surface over the triangle bounded by the lines x = y; x = 0; y = 1 in the xy plane is _________.

Use Keypad/Enter your answer

Q#35 Calculus GATE EE 2014 (Set 1) MCQ +1 mark -0.33 marks

Let . The maximum value of the function in the interval is

e

Q#36 Calculus GATE EE 2014 (Set 1) MCQ +2 marks -0.66 marks

The line integral of function F=yzi, in the counter clockwise direction, along the circle at z = 1 is

Q#37 Calculus GATE EE 2014 (Set 2) MCQ +1 mark -0.33 marks

Minimum of the real valued function occurs at x equal to

0

1

Q#38 Calculus GATE EE 2014 (Set 2) MCQ +2 marks -0.66 marks

To evaluate the doubled integral, we make the substitution and . The integral will reduce to

Q#39 Calculus GATE EE 2014 (Set 2) MCQ +2 marks -0.66 marks

The minimum value of the function in the interval [-3, 3] is

20

28

16

32

Q#40 Calculus GATE EE 2014 (Set 1) NAT +1 mark -0 marks

A particle, starting from origin at t = 0s, is travelling along x-axis with velocity

At t = 3s, the difference between the distance covered by the particle and the magnitude of displacement from the origin is ___________.

Use Keypad/Enter your answer

Q#41 Calculus GATE EE 2014 (Set 3) MCQ +1 mark -0.33 marks

Let , where f and v are scalar and vector fields respectively. If , then is

2xy + 2yz + 2zx

x + y + z

0

Q#42 Calculus GATE EE 2013 (Set 1) MCQ +1 mark -0.33 marks

The curl of the gradient of the scalar field defined by is

0

Q#43 Calculus GATE EE 2013 (Set 1) MCQ +1 mark -0.33 marks

Given a vector field , the line integral ∫F.dl evaluated along a segment on the x-axis from x=1 to x=2 is

-2.33

0

2.33

7

Q#44 Calculus GATE EE 2013 (Set 1) MCQ +2 marks -0.66 marks

A function is defined over an open interval x= (1, 2). At least at one point in this interval, is exactly

20

25

30

35

Q#45 Calculus GATE EE 2012 (Set 1) MCQ +2 marks -0.66 marks

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

21

25

41

46

Q#46 Calculus GATE EE 2012 (Set 1) MCQ +2 marks -0.66 marks

The direction of vector A is radially outward from the origin, with where and k is a constant. The value of n for which is

-2

2

1

0

Q#47 Calculus GATE EE 2011 (Set 1) MCQ +1 mark -0.33 marks

Roots of the algebraic equation are

Q#48 Calculus GATE EE 2011 (Set 1) MCQ +2 marks -0.66 marks

The function has

a maxima at x = 1 and a minima at x = 5

a maxima at x = 1 and a minima at x = -5

only a maxima at x = 1

only a minima at x = 1

Q#49 Calculus GATE EE 2011 (Set 1) MCQ +2 marks -0.66 marks

The two vectors [1, 1, 1] and , where , are

Orthonormal

Orthogonal

Parallel

Collinear

Q#50 Calculus GATE EE 2010 (Set 1) MCQ +1 mark -0.33 marks

The value of the quantity P, where. Is equal to

0

1

e

1/e

Q#51 Calculus GATE EE 2010 (Set 1) MCQ +1 mark -0.33 marks

Divergence of the three-dimensional radial vector field is

3

1/r

Q#52 Calculus GATE EE 2010 (Set 1) MCQ +2 marks -0.66 marks

At t = 0, the function has

a minimum

a discontinuity

a point of inflection

a maximum

Q#53 Calculus GATE EE 2009 (Set 1) MCQ +2 marks -0.66 marks

f (x, y) is a continuous function defined over. Given the two constraintsand , the volume under f(x, y) is

Q#54 Calculus GATE EE 2009 (Set 1) MCQ +2 marks -0.66 marks

A cubic polynomial with real coefficients

Can possibly have no Extrema and no zero crossings

May have up to three Extrema and up to 2 zero crossings

Cannot have more than two Extrema and more than three zero crossings

Will always have an equal number of Extrema and zero crossings

Q#55 Calculus GATE EE 2009 (Set 1) MCQ +2 marks -0.66 marks

It's line integral over the straight line from evaluates to

−8

4

8

0

Q#56 Calculus GATE EE 2008 (Set 1) MCQ +2 marks -0.66 marks

Consider function where x is a real number. Then the function has

Only one minimum

Only two minima

Three minima

Three maxima

Q#57 Calculus GATE EE 2007 (Set 1) MCQ +1 mark -0.33 marks

Divergence of the vector field

is

None of these

Q#58 Calculus GATE EE 2007 (Set 1) MCQ +2 marks -0.66 marks

The integral equals

0

Q#59 Calculus GATE EE 2006 (Set 1) MCQ +2 marks -0.66 marks

The expression for the volume of a cone is equal to

Q#60 Calculus GATE EE 2006 (Set 1) MCQ +2 marks -0.66 marks

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

0

Q#61 Calculus GATE EE 2005 (Set 1) MCQ +1 mark -0.33 marks

If , then S has the value

1

Q#62 Calculus GATE EE 2005 (Set 1) MCQ +2 marks -0.66 marks

For the function, the maximum occurs when x is equal to:

2

1

0

–1

Q#63 Calculus GATE EE 2005 (Set 1) MCQ +2 marks -0.66 marks

For the scalar field , the magnitude of the gradient at the point (1,3) is:

ExamDost Learning Portal
GATE EE

Engineering Mathematics
Calculus

63

0

0

0

ExamDost Learning Portal GATE EE

Engineering Mathematics Calculus

63

0

0

0

Report Question

Monday Activity

Notification Title

Welcome Aboard, User!

Badge Title

Hang on, we are picking questions...

Login Account

Package

Purchasing Terms

No Refunds

No Account Sharing

Ban Policy

Fair Use

Create Your Account

Why Create an Account?

Personal Learning Experience

Exclusive Access

Video Solution Please adjust or seek the video; sometimes, the answers appear 20 seconds earlier or later.

ExamDost Learning Portal
GATE EE

Engineering Mathematics
Calculus

Video Solution
Please adjust or seek the video; sometimes, the answers appear 20 seconds earlier or later.