Engineering Mathematics
Calculus
Practice questions from Calculus.
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IncorrectGiven that , the integral over the unit sphere centered at the origin evaluates to
(Round off to one decimal place)
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Sign in to UnlockThe integral evaluates to (Round off to two decimal places)
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Sign in to UnlockLet . 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).
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Sign in to UnlockConsider 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
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Sign in to UnlockLet 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.
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Sign in to UnlockConsider 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.
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Sign in to UnlockConsider 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.
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Sign in to UnlockThe 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).
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Sign in to UnlockA 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).
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Sign in to UnlockLet . Then decreases in the interval.
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Sign in to UnlockLet . The direction in which the function increases most rapidly at point is
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Sign in to UnlockLet 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 ?
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Sign in to UnlockLet 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
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Sign in to UnlockSuppose the circles and intersect each other orthogonally at the point (u,v). Then __________.
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Sign in to UnlockIn the open interval , the polynomial has
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Sign in to Unlockis 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.
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Sign in to UnlockFor real numbers, and with , the maximum and minimum value of for are respectively _________.
7 and
7 and 1
-2 and
1 and
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Sign in to UnlockLet 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).
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Sign in to UnlockIf , the value of line integral . dr evaluated over contour C formed by the segments is ______________.
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Sign in to UnlockIf then div(uA) at (1, 1, 1) is ____________.
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Sign in to UnlockLet 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).
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Sign in to UnlockLet f be a real-valued function of a real variable defined as for, andfor x < 0. Which one of the following statements is true?
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Sign in to UnlockThe value of the directional derivative of the functionat the point in the direction of the vectoris
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Sign in to UnlockLet. The maximum value of f(x) over the interval [0, 2] is _____ (up to 1 decimal place).
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Sign in to UnlockAs 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).
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Sign in to UnlockLet, where R is the region shown in the figure and. The value of I equals __________. (Give the answer up to two decimal places).
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Sign in to UnlockA function f(x) is defined as . Which one of the following statements is TRUE?
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Sign in to UnlockConsider 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 __________ .
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Sign in to UnlockLet
Consider the composition of f and g. i.e., 
. The number of discontinuities in 
present in the interval is:
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Sign in to UnlockThe maximum value attained by the function in the interval [1, 2] is ______________.
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Sign in to UnlockThe value of the line integral along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is
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Sign in to UnlockThe line integral of the vector field along a path from (0, 0, 0) to (1, 1, 1) parametrized by is ______________.
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Sign in to UnlockIf 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)
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Sign in to UnlockConsider 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
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Sign in to UnlockGiven , 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.
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Sign in to UnlockThe volume enclosed by the surface over the triangle bounded by the lines x = y; x = 0; y = 1 in the xy plane is _________.
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Sign in to UnlockLet . The maximum value of the function in the interval is
e
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Sign in to UnlockThe line integral of function F=yzi, in the counter clockwise direction, along the circle at z = 1 is
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Sign in to UnlockMinimum of the real valued function occurs at x equal to
0
1
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Sign in to UnlockTo evaluate the doubled integral, we make the substitution and . The integral will reduce to
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Sign in to UnlockThe minimum value of the function in the interval [-3, 3] is
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Sign in to UnlockA 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 ___________.
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Sign in to UnlockLet , where f and v are scalar and vector fields respectively. If , then is
2xy + 2yz + 2zx
x + y + z
0
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Sign in to UnlockThe curl of the gradient of the scalar field defined by is
0
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Sign in to UnlockGiven a vector field , the line integral ∫F.dl evaluated along a segment on the x-axis from x=1 to x=2 is
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Sign in to UnlockA function is defined over an open interval x= (1, 2). At least at one point in this interval, is exactly
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Sign in to UnlockThe maximum value of in the interval [1, 6] is
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Sign in to UnlockRoots of the algebraic equation are
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Sign in to UnlockThe function has
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Sign in to UnlockThe two vectors [1, 1, 1] and , where , are
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Sign in to UnlockThe value of the quantity P, where. Is equal to
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Sign in to UnlockDivergence of the three-dimensional radial vector field is
3
1/r
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Sign in to UnlockAt t = 0, the function has
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Sign in to Unlockf (x, y) is a continuous function defined over. Given the two constraintsand , the volume under f(x, y) is
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Sign in to UnlockA cubic polynomial with real coefficients
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Sign in to UnlockIt's line integral over the straight line from evaluates to
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Sign in to UnlockConsider function where x is a real number. Then the function has
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Sign in to UnlockDivergence of the vector field
is
None of these
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Sign in to UnlockThe integral equals
0
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Sign in to UnlockThe expression for the volume of a cone is equal to
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Sign in to UnlockA surface S(x, y) = 2x + 5y − 3 is integrated once over a path consisting of the points that satisfy. The integral evaluates to
0
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Sign in to UnlockIf , then S has the value
1
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Sign in to UnlockFor the function, the maximum occurs when x is equal to:
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Sign in to UnlockFor the scalar field , the magnitude of the gradient at the point (1,3) is:
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