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Civil Engineering (CE)  >  GATE Mechanical (ME) 2023 Mock Test Series  >  Test: Partial Derivatives, Gradient- 2 Download as PDF

Test: Partial Derivatives, Gradient- 2


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25 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Partial Derivatives, Gradient- 2

Test: Partial Derivatives, Gradient- 2 for Civil Engineering (CE) 2023 is part of GATE Mechanical (ME) 2023 Mock Test Series preparation. The Test: Partial Derivatives, Gradient- 2 questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Partial Derivatives, Gradient- 2 MCQs are made for Civil Engineering (CE) 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Partial Derivatives, Gradient- 2 below.
Solutions of Test: Partial Derivatives, Gradient- 2 questions in English are available as part of our GATE Mechanical (ME) 2023 Mock Test Series for Civil Engineering (CE) & Test: Partial Derivatives, Gradient- 2 solutions in Hindi for GATE Mechanical (ME) 2023 Mock Test Series course. Download more important topics, notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free. Attempt Test: Partial Derivatives, Gradient- 2 | 25 questions in 75 minutes | Mock test for Civil Engineering (CE) preparation | Free important questions MCQ to study GATE Mechanical (ME) 2023 Mock Test Series for Civil Engineering (CE) Exam | Download free PDF with solutions
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Test: Partial Derivatives, Gradient- 2 - Question 1
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 f(x, y) = x2 + xyz + z Find fx at (1,1,1)

Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 1

fx = 2x + yz

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

fx = 2 + 1 = 3.

Test: Partial Derivatives, Gradient- 2 - Question 2
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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:

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Test: Partial Derivatives, Gradient- 2 - Question 3
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The minimum point of the function f(x) = (x2/3) – x is at 

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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

Test: Partial Derivatives, Gradient- 2 - Question 4
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If x=a(θ+ sin θ) and y=a(1-cosθ), then dy/dx will be equal 
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Test: Partial Derivatives, Gradient- 2 - Question 5
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The minimum value of function y = x2 in the interval [1, 5] is  
Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 5

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

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

Test: Partial Derivatives, Gradient- 2 - Question 6
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The function f(x) = 2x3 – 3x2 – 36x + 2 has its maxima at  
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Test: Partial Derivatives, Gradient- 2 - Question 7
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What should be the value of λ such that the function defined below is continuous at x = π/22? 
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By the given condition 

Test: Partial Derivatives, Gradient- 2 - Question 8
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Consider function f(x) =(x2-4)2 where x is a real number. Then the function has  
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Test: Partial Derivatives, Gradient- 2 - Question 9
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If f     where ai (i = 0 to n) are constants, then  

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  - Euler’s theorem for homogeneous function 

Test: Partial Derivatives, Gradient- 2 - Question 10
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Test: Partial Derivatives, Gradient- 2 - Question 11
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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 
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Test: Partial Derivatives, Gradient- 2 - Question 12
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∇ × ∇ × P, where P is a vector, is equal to  
Test: Partial Derivatives, Gradient- 2 - Question 13
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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  
Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 13

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

Test: Partial Derivatives, Gradient- 2 - Question 14
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If    is a differentiable vector function and f is a sufficient differentiable scalar function, then curl  

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Test: Partial Derivatives, Gradient- 2 - Question 15
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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 
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Test: Partial Derivatives, Gradient- 2 - Question 16
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The divergence of vector  

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Test: Partial Derivatives, Gradient- 2 - Question 17
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The divergence of the vector 

Test: Partial Derivatives, Gradient- 2 - Question 18
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 Among the following, the pair of vectors orthogonal to each other is 
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Then we say that they are orthogonal.  Choice (c) is correct. 

Test: Partial Derivatives, Gradient- 2 - Question 19
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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 

Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 19

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

=2

Test: Partial Derivatives, Gradient- 2 - Question 20
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The Gauss divergence theorem relates certain  
Test: Partial Derivatives, Gradient- 2 - Question 21
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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  
Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 21

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

Test: Partial Derivatives, Gradient- 2 - Question 22
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Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. 

Then the determinant det 

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Test: Partial Derivatives, Gradient- 2 - Question 23
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If a - b = 3 and a2 + b2 = 29, find the value of ab.
Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 23

2ab = (a2 + b2) - (a - b)2

   = 29 - 9 = 20

   ab = 10.

Test: Partial Derivatives, Gradient- 2 - Question 24
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If a vector R(t) → has a constant magnitude, then  
Detailed Solution for Test: Partial Derivatives, Gradient- 2 - Question 24

On analysing the given (a) option, we find that    will give constant magnitude, so first 
differentiation of the integration will be zero. 

Test: Partial Derivatives, Gradient- 2 - Question 25
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For the scalar field    magnitude of the gradient at the point(1,3) is   
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