Integral Calculus NAT Level - 1


10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Integral Calculus NAT Level - 1


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This mock test of Integral Calculus NAT Level - 1 for Physics helps you for every Physics entrance exam. This contains 10 Multiple Choice Questions for Physics Integral Calculus NAT Level - 1 (mcq) to study with solutions a complete question bank. The solved questions answers in this Integral Calculus NAT Level - 1 quiz give you a good mix of easy questions and tough questions. Physics students definitely take this Integral Calculus NAT Level - 1 exercise for a better result in the exam. You can find other Integral Calculus NAT Level - 1 extra questions, long questions & short questions for Physics on EduRev as well by searching above.
*Answer can only contain numeric values
QUESTION: 1

Find the area between the curves r = 3 cos θ and r = 2 - cos θ  is απ + β√3. Find the value of  α + β.


Solution:

The points of intersection are

Total Area = 2(Area of I + Area of II)

Area of Region I

Area of Region II


Area of Region 

The correct answer is: -0.75

*Answer can only contain numeric values
QUESTION: 2

Find by Double Integration, whole area of the curve a2x2 = y3(2a – y). Let area be of the form λa2. Find value of λ.


Solution:

As, it contains only even powers of x, hence, it is symmetrical about y-axis
Total Area = 2 × area OAB



The correct answer is: 3.142

*Answer can only contain numeric values
QUESTION: 3

Find the whole area included between the curve x2y2 = a2(y2 – x2) and its asymptotes. It is of form λa2. Find value of λ.


Solution:

The curve is
x2y2 = a2(y2 – x2)
(1) Symmetry about both the axes at even powers of x and y occur.
(2) Asymptotes are x = ±a


The correct answer is: 4

*Answer can only contain numeric values
QUESTION: 4

Find the area bounded by the curve r = 4cos3θ. as given below :


Solution:

Hence, the Area bounded is given as

As the curve is symmetrical.
Total Area = 3 × Area of loop OA




 

= 12.571
The correct answer is: 12.571 

*Answer can only contain numeric values
QUESTION: 5

If the area common to the circles r = a√2 and r = 2a cosθ is a2λ. Find the value of λ.


Solution:

Given curves are

Solving the two equations,

Here Required area will be

For Region I

For Region II


 

Hence, λ = 2.142

The correct answer is: 2.142

*Answer can only contain numeric values
QUESTION: 6

The area bounded by the curve √x + √y = 1 and the coordinates axes is


Solution:

This can be done by two ways:
Ist Way:


IInd Way :
Putting x = 0 and y = 0, we find that the given curve meets y and x-axes in (0, 1) and (1, 0) respectively.

The correct answer is: 0.167

*Answer can only contain numeric values
QUESTION: 7

  where D is the region that lies inside the circle r = 3cosθ and outside the cadioid r = 1 + cos θ is  Find the value of α . β.


Solution:

Points of intersection

Notice that D is the region such that for each fixed angle θ between  varies from 1 + cos θ (cardioid) to 3 cos θ (circle).


The correct answer is: 8

*Answer can only contain numeric values
QUESTION: 8

Find the value of  where R is a circular disk x2 + y2 = a2 is  Find the value of α.


Solution:

Changing to polar coordinates, we get

Putting r2=t

2r dr = dt

The correct answer is: 1

*Answer can only contain numeric values
QUESTION: 9

 over the area of cardioid r = a(1 + cosθ) above initial line is λa3.  Find the value of λ.


Solution:

The region of integration A can be covered by radial strips whose ends are at r = 0, r = a (1 + cos θ). The strips like between θ = 0 and θ = π.

Thus,


The correct answer is: 1.333

*Answer can only contain numeric values
QUESTION: 10

Evaluate  over the domain {{x, y) : x > 0, y > 0, x2 + y2 < 1}


Solution:

The region of integration is

Now, changing to polar coordinates, we get

The correct answer is: 0.0327

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