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This mock test of Test: Differential Equations & Multiple Integrals- 1 for GATE helps you for every GATE entrance exam.
This contains 20 Multiple Choice Questions for GATE Test: Differential Equations & Multiple Integrals- 1 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

A triangle ABC consists of vertex points A (0,0) B(1,0) and C(0,1). The value of the integral over the triangle is

Solution:

The equation of the line AB is

QUESTION: 2

The area enclosed between the parabala y = x^{2} and the straight line y = x is

Solution:

QUESTION: 3

Changing the order of the integration in the double integral

Solution:

When

QUESTION: 4

The right circular cone of largest volume that can be enclosed by a sphere of 1 m radius has a height of

Solution:

QUESTION: 5

the parabolic arc is revolved around the x-axis. The volume of

Solution:

Differential volume

QUESTION: 6

What is the area common to the circles r = a and r = 2a cos θ?

Solution:

Area common to circles r = a

And r = 2a cos θ is 1.228 a^{2}

QUESTION: 7

The expression for the volume of a cone is equal to

Solution:

Choices (a) and (b) be correct because

for the volume of a cone is equal to

QUESTION: 8

f ( x,y ) is a continuous defined over ( x,y ) ∈ [0,1]× [0,1] . Given two constrains, x > y 2 and y > x 2 , the volume under f ( x,y ) is

Solution:

QUESTION: 9

The following differential equation has

Solution:

Order of highest derivative = 2

Hence, most appropriate answer is (b)

QUESTION: 10

A solution of the following differential equation is given by

Solution:

Let y = mx be the trial soln of the given differental equation

∴The corrosponding auxiliary equation is

QUESTION: 11

For the differential equation the boundary conditions are

(i) y = 0 for x = 0, and

(ii) y = 0 for x = a

The form of non-zero solutions of y (where m varies over all integers) are

Solution:

Here y = c_{1} coskx + c_{2} sinkx ........... (1) be the solution of the given differential equation.

Now use boundary conditions

For x = 0,y = 0 gives c_{1} = 0. Equation − (1) becomes

be the solution, n 0,1,2,3.......

QUESTION: 12

Which of the following is a solution to the differential equation

Solution:

Hints : m + 3 = 0 ⇒m = −3

QUESTION: 13

For the differential equation the general solution is

Solution:

QUESTION: 14

The solution of the differential equation

Solution:

It is a linear diff. equation

QUESTION: 15

The solution of with the condition y(1)

Solution:

Which is 1st order linear differential equation

QUESTION: 16

A technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems is called

Solution:

QUESTION: 17

The general solution of (x^{2} D^{2} – xD), y= 0 is :

Solution:

QUESTION: 18

For the particular integrals is

Solution:

QUESTION: 19

It is given that y" + 2y' + y = 0, y(0) = 0, y(1)=0. What is y (0.5)?

Solution:

Auxiliary equation is m^{2} + 2m +1 = 0⇒m = −1,−1

Using boundary condition y(0) = 0 and y(1) = 0

we get y = 0

QUESTION: 20

The partial differential equation

Solution:

The order is the highest numbered derivative in the equation (which is 2 here), while the degree is the highest power to which a derivative is raised (which is 1 here).

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