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This mock test of Partial Derivatives, Gradient - MCQ Test - 2 for GATE helps you for every GATE entrance exam.
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QUESTION: 1

f(x, y) = x^{2} + xyz + z Find f_{x} at (1,1,1)

Solution:

f_{x} = 2x + yz

Put (x,y,z) = (1,1,1)

f_{x} = 2 + 1 = 3.

QUESTION: 2

Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by:

Solution:

QUESTION: 3

The minimum point of the function f(x) = (x^{2/3}) – x is at

Solution:

**Correct Answer :- a**

**Explanation : **f(x) = (x^2/3) - x

f'(x) = 2/3(x^{-1/2}) - 1

f"(x) = -1/3(x^{-3/2})

For critical points. f′(x)=0

=> 2/3(x^{-1/2}) - 1 = 0

f has minimum value of x = 1

QUESTION: 4

If x=a(θ+ sin θ) and y=a(1-cosθ), then dy/dx will be equal

Solution:

QUESTION: 5

The minimum value of function y = x^{2} in the interval [1, 5] is

Solution:

y =x 2 is strictly increasing function on [1,5]

∴ y= x 2 has minimum value at x = 1 is 1.

QUESTION: 6

The function f(x) = 2x^{3} – 3x^{2} – 36x + 2 has its maxima at

Solution:

QUESTION: 7

What should be the value of λ such that the function defined below is continuous at x = π/22?

Solution:

By the given condition

QUESTION: 8

Consider function f(x) =(x^{2}-4)^{2} where x is a real number. Then the function has

Solution:

QUESTION: 9

If f where ai (i = 0 to n) are constants, then

Solution:

- Euler’s theorem for homogeneous function

QUESTION: 10

Solution:

QUESTION: 11

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct exterma for the curve 3x^{4} – 16x^{3} – 24x^{2} + 37 is

Solution:

QUESTION: 12

∇ × ∇ × P, where P is a vector, is equal to

Solution:

QUESTION: 13

The value of the integral of the function g(x, y) = 4x^{3} + 10y^{4} along the straight line segment from the point (0, 0) to the point (1, 2) in the x-y plane is

Solution:

The equation of the line passing through (0,0) and (1,2) is y = 2x

Given y x, y ) = 4x^{3}+ 10y^{4} = 4x^{3} + 10(2x )^{4} = 4x^{3}+ 160xy

QUESTION: 14

If is a differentiable vector function and f is a sufficient differentiable scalar function, then curl

Solution:

QUESTION: 15

The temperature field in a body varies according to the equation T(x,y) = x^{3}+4xy. The direction of fastest variation in temperature at the point (1,0) is given by

Solution:

QUESTION: 16

The divergence of vector

Solution:

QUESTION: 17

The divergence of the vector

Solution:

QUESTION: 18

Among the following, the pair of vectors orthogonal to each other is

Solution:

Then we say that they are orthogonal. Choice (c) is correct.

QUESTION: 19

The directional derivative of the scalar function f(x, y, z) = x^{2} + 2y^{2} + z at the point P = (1,1, 2) in the direction of the vector

Solution:

Required directional derivatives at P(1,1,-1)

=2

QUESTION: 20

The Gauss divergence theorem relates certain

Solution:

QUESTION: 21

If P, Q and R are three points having coordinates (3, –2, –1), (1, 3, 4), (2, 1, –2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by

Solution:

The equation of the plane OQR is (O being origin).

QUESTION: 22

Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product.

Then the determinant det

Solution:

QUESTION: 23

If a - b = 3 and a^{2} + b^{2} = 29, find the value of ab.

Solution:

2*ab* = (*a*^{2} + *b*^{2}) - (*a* - *b*)^{2}

= 29 - 9 = 20

*ab* = 10.

QUESTION: 24

If a vector R(t) ^{→} has a constant magnitude, then

Solution:

On analysing the given (a) option, we find that will give constant magnitude, so first

differentiation of the integration will be zero.

QUESTION: 25

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

Solution:

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