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Test: Differential Equations & Multiple Integrals- 2


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30 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Differential Equations & Multiple Integrals- 2

Test: Differential Equations & Multiple Integrals- 2 for Civil Engineering (CE) 2022 is part of GATE Mechanical (ME) 2023 Mock Test Series preparation. The Test: Differential Equations & Multiple Integrals- 2 questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Differential Equations & Multiple Integrals- 2 MCQs are made for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Differential Equations & Multiple Integrals- 2 below.
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Test: Differential Equations & Multiple Integrals- 2 - Question 1

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 1

Test: Differential Equations & Multiple Integrals- 2 - Question 2

The volume of an object expressed in spherical co-ordinates is given by 

 

The value of the integral is  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 2

Test: Differential Equations & Multiple Integrals- 2 - Question 3

By a change of variable x (u, y) = uv, y (u, v) = v/u is double integral, the integrand f(x, y) changes to f(uv, v/u) φ (u,v). Then, φ (u, v) is 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 3

Test: Differential Equations & Multiple Integrals- 2 - Question 4

Consider the shaded triangular region P shown in the figure. What is 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 4

The equation of the line AB is  

Test: Differential Equations & Multiple Integrals- 2 - Question 5

A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y)2 on path AB traversed in a counterclockwise sense is  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 5

Test: Differential Equations & Multiple Integrals- 2 - Question 6

The value of 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 6

Test: Differential Equations & Multiple Integrals- 2 - Question 7

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is     where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 7

 We know length of the curve f(x) between x = a and x = b given by 

Test: Differential Equations & Multiple Integrals- 2 - Question 8

A surface S(x,y)=2x+5y-3 is integrated once over a  path consisting of the points that satisfy ( x +1)2+ (y − 1)2 = √2 . The integral evaluates to

Test: Differential Equations & Multiple Integrals- 2 - Question 9

The value of integral  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 9

Test: Differential Equations & Multiple Integrals- 2 - Question 10

The order of the differential equation 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 10

The order of a differential equation is the order of the highest derivative involving in
equation, so answer is 2.

Test: Differential Equations & Multiple Integrals- 2 - Question 11

The solution of the differential equation    under the boundary conditions  

 (i) y =y1  At x = 0 and  

(ii) y =y2  At x = ∞, 

Where k,   y1 and y2 are constants, is 

 

Test: Differential Equations & Multiple Integrals- 2 - Question 12

A function n(x) satisfies the differential equation    where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 12

For finite solution c1 = 0

Test: Differential Equations & Multiple Integrals- 2 - Question 13

The solution of the differential equation 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 13

Given differential equation is

Integra ling we get

Test: Differential Equations & Multiple Integrals- 2 - Question 14

If     then what is y(e)?

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 14

It is linear differential equation.

Test: Differential Equations & Multiple Integrals- 2 - Question 15

 The solution of dy/dx = y2 with initial value y (0) = 1 bounded in the interval  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 15

Test: Differential Equations & Multiple Integrals- 2 - Question 16

The solution to the differential equation f’’(x)+4f’(x)+4f(x)=0 is      

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 16

Let y(x) = emx (m ≠ 0)be the trial soln .Auxiliary equation. m2 + 4m+ 4 = 0 ⇒(m+ 2)2 = 0

In particular, when A =1,B =1,then f(x) = (1 + x)e−2x

= e−2x + xe−2x

Test: Differential Equations & Multiple Integrals- 2 - Question 17

The above equation is a  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 17

Since, the differential equation cannot be expressed in x/y or y/x form, therefore, it is an example of non-homogeneous differential equation.

Test: Differential Equations & Multiple Integrals- 2 - Question 18

 Which of the following is a solution of the differential equation   

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 18

Here p = 4andq = 3.The given equation becomes

Test: Differential Equations & Multiple Integrals- 2 - Question 19

The Blasius equation, 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 19

f is non linear.

Test: Differential Equations & Multiple Integrals- 2 - Question 20

Given that .. .x+ 3x= 0, and x(0)=1, x(0) = 0 what is x(1)?  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 20

Auxiliary equn of x11 + 3x = 0 is

m2 + 3 = 0

Test: Differential Equations & Multiple Integrals- 2 - Question 21

The degree of the differential equation  

Test: Differential Equations & Multiple Integrals- 2 - Question 22

The solution for the differential equation  with the condition that y = 1 at x = 0 is  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 22

Test: Differential Equations & Multiple Integrals- 2 - Question 23

A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 23

By the given condition

Test: Differential Equations & Multiple Integrals- 2 - Question 24

Solution of the differential equation  represents a family of  

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 24

Test: Differential Equations & Multiple Integrals- 2 - Question 25

The general solution of 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 25

Let y = emx (m ≠ 0) be the trial solution.

Test: Differential Equations & Multiple Integrals- 2 - Question 26

A body originally at 60ºC cools down to 40ºC in 15 minutes when kept in air at a temperature of 25ºC. What will be the temperature of the body at the end of 30 minutes?

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 26

Test: Differential Equations & Multiple Integrals- 2 - Question 27

The partial differential equation that can be formed from z = ax + by + ab has the form 

Test: Differential Equations & Multiple Integrals- 2 - Question 28

With K as constant, the possible solution for the first order differential equation    is 

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 28

Test: Differential Equations & Multiple Integrals- 2 - Question 29

The boundary-value problem yn + λy = 0, y(0) = y(λ) = 0 will have non-zero solutions if and only if the values of λ are  

Test: Differential Equations & Multiple Integrals- 2 - Question 30

The solution of the differential equation    with the condition that y = 1 at x = 1, is

Detailed Solution for Test: Differential Equations & Multiple Integrals- 2 - Question 30

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