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# Partial Derivatives, Gradient - MCQ Test 2

## 25 Questions MCQ Test Topicwise Question Bank for GATE Civil Engineering | Partial Derivatives, Gradient - MCQ Test 2

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This mock test of Partial Derivatives, Gradient - MCQ Test 2 for Civil Engineering (CE) helps you for every Civil Engineering (CE) entrance exam. This contains 25 Multiple Choice Questions for Civil Engineering (CE) Partial Derivatives, Gradient - MCQ Test 2 (mcq) to study with solutions a complete question bank. The solved questions answers in this Partial Derivatives, Gradient - MCQ Test 2 quiz give you a good mix of easy questions and tough questions. Civil Engineering (CE) students definitely take this Partial Derivatives, Gradient - MCQ Test 2 exercise for a better result in the exam. You can find other Partial Derivatives, Gradient - MCQ Test 2 extra questions, long questions & short questions for Civil Engineering (CE) on EduRev as well by searching above.
QUESTION: 1

### f(x, y) = x2 + xyz + z Find fx at (1,1,1)

Solution:

fx = 2x + yz

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

fx = 2 + 1 = 3.

QUESTION: 2

Solution:

QUESTION: 3

### The minimum point of the function f(x) = (x2/3) – x is at

Solution:

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 = x2 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) = 2x3 – 3x2 – 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) =(x2-4)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 3x4 – 16x3 – 24x2 + 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) = 4x3 + 10y4 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 ) = 4x3+ 10y4 = 4x3 + 10(2x )4 = 4x3+ 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) = x3+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) = x2 + 2y2 + 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 a2 + b2 = 29, find the value of ab.

Solution:

2ab = (a2 + b2) - (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: